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Differential subordinations using the Ruscheweyh derivative and the generalized Sălăgean operator

Loriana Andrei

Author Affiliations

Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii street, Oradea, 410087, Romania

Advances in Difference Equations 2013, 2013:252  doi:10.1186/1687-1847-2013-252


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/252


Received:13 June 2013
Accepted:6 August 2013
Published:20 August 2013

© 2013 Andrei; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the present paper, we study the operator, using the Ruscheweyh derivative <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M1">View MathML</a> and the generalized Sălăgean operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M2">View MathML</a>, denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M3">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M6">View MathML</a> is the class of normalized analytic functions. We obtain several differential subordinations regarding the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M7">View MathML</a>.

MSC: 30C45, 30A20, 34A40.

Keywords:
differential subordination; convex function; best dominant; differential operator; generalized Sălăgean operator; Ruscheweyh derivative

1 Introduction

Denote by U the unit disc of the complex plane, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M8">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M9">View MathML</a> the space of holomorphic functions in U.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M6">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M11">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M12">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M13">View MathML</a>.

Denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M14">View MathML</a> the class of normalized convex functions in U.

If f and g are analytic functions in U, we say that f is subordinate to g, written <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M15">View MathML</a>, if there is a function w analytic in U, with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M16">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M17">View MathML</a>, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M19">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>. If g is univalent, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M15">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M22">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M23">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M24">View MathML</a>, and let h be an univalent function in U. If p is analytic in U and satisfies the (second-order) differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M25">View MathML</a>

(1.1)

then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M26">View MathML</a> for all p satisfying (1.1).

A dominant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M27">View MathML</a> that satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M28">View MathML</a> for all dominants q of (1.1) is said to be the best dominant of (1.1). The best dominant is unique up to a rotation of U.

Definition 1.1 (Al-Oboudi [1])

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M30">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M32">View MathML</a> is defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M34">View MathML</a>

Remark 1.1 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M37">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Remark 1.2 For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M39">View MathML</a>, in the definition above, we obtain the Sălăgean differential operator [2].

Definition 1.2 (Ruscheweyh [3])

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M42">View MathML</a> is defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M43">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M44">View MathML</a>

Remark 1.3 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M47">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Definition 1.3[4]

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>. Denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M7">View MathML</a> the operator given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M3">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M53">View MathML</a>

Remark 1.4 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M56">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

This operator was studied also in [4-6] and [7].

Remark 1.5 For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M58">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M59">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a> and for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M62">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M39">View MathML</a>, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M65">View MathML</a>, which was studied in [8-11].

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M66">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M67">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Lemma 1.1 (Hallenbeck and Ruscheweyh [[12], Th. 3.1.6, p.71])

Lethbe a convex function with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M69">View MathML</a>, and let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M70">View MathML</a>be a complex number with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M71">View MathML</a>. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M72">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M73">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M74">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M75">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Lemma 1.2 (Miller and Mocanu [12])

Letgbe a convex function inU, and let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M77">View MathML</a>, for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M79">View MathML</a>andnis a positive integer.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M80">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, is holomorphic inUand

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M82">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M83">View MathML</a>

and this result is sharp.

2 Main results

Theorem 2.1Letgbe a convex function, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M84">View MathML</a>, and lethbe the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M85">View MathML</a>, for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M90">View MathML</a>

(2.1)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M91">View MathML</a>

and this result is sharp.

Proof By using the properties of operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M7">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M93">View MathML</a>

Consider <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M94">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

We deduce that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M96">View MathML</a>.

Differentiating we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M97">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Then (2.1) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M99">View MathML</a>

By using Lemma 1.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M100">View MathML</a>

 □

Theorem 2.2Lethbe a holomorphic function, which satisfies the inequality<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M107">View MathML</a>

(2.2)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M108">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M109">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M110">View MathML</a>

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M96">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M97">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and (2.2) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M115">View MathML</a>

Using Lemma 1.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M116">View MathML</a>

and q is the best dominant. □

Corollary 2.3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M117">View MathML</a>be a convex function inU, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M118">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M119">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M122">View MathML</a>

(2.3)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M123">View MathML</a>

whereqis given by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M124">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Following the same steps as in the proof of Theorem 2.2 and considering <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M126">View MathML</a>, the differential subordination (2.3) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M127">View MathML</a>

By using Lemma 1.1, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M128">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M129">View MathML</a>, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M130">View MathML</a>

 □

Remark 2.1 For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M131">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M132">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M133">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M134">View MathML</a>, we obtain the same example as in [[13], Example 4.2.1, p.125].

Theorem 2.4Letgbe a convex function such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M135">View MathML</a>, and lethbe the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M136">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M141">View MathML</a>

(2.4)

holds, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M142">View MathML</a>

and this result is sharp.

Proof For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M36">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M145">View MathML</a>

Consider <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M146">View MathML</a>, and we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M147">View MathML</a>

Relation (2.4) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M148">View MathML</a>

By using Lemma 1.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M149">View MathML</a>

 □

Theorem 2.5Lethbe a holomorphic function, which satisfies the inequality<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M156">View MathML</a>

(2.5)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M157">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M109">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M146">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M161">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M162">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and (2.5) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M164">View MathML</a>

Using Lemma 1.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M165">View MathML</a>

and q is the best dominant. □

Theorem 2.6Letgbe a convex function such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M84">View MathML</a>, and lethbe the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M85">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M172">View MathML</a>

(2.6)

holds, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M173">View MathML</a>

This result is sharp.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M174">View MathML</a>. We deduce that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M176">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Using the notation in (2.6), the differential subordination becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M178">View MathML</a>

By using Lemma 1.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M179">View MathML</a>

and this result is sharp. □

Theorem 2.7Lethbe an holomorphic function, which satisfies the inequality<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M186">View MathML</a>

(2.7)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M187">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M109">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M174">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M176">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and (2.7) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M194">View MathML</a>

Using Lemma 1.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M195">View MathML</a>

and q is the best dominant. □

Theorem 2.8Letgbe a convex function such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M84">View MathML</a>, and lethbe the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M197">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M202">View MathML</a>

(2.8)

holds, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M203">View MathML</a>

This result is sharp.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M204">View MathML</a>. We deduce that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M161">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M206">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Using the notation in (2.8), the differential subordination becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M208">View MathML</a>

By using Lemma 1.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M209">View MathML</a>

and this result is sharp. □

Theorem 2.9Lethbe a holomorphic function, which satisfies the inequality<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M101">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M216">View MathML</a>

(2.9)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M217">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M218">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M204">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M206">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and (2.9) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M224">View MathML</a>

Using Lemma 1.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M225">View MathML</a>

and q is the best dominant. □

Corollary 2.10Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M117">View MathML</a>be a convex function inU, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M118">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M231">View MathML</a>

(2.10)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M232">View MathML</a>

whereqis given by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M233">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Following the same steps as in the proof of Theorem 2.9 and considering <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M235">View MathML</a>, the differential subordination (2.10) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M236">View MathML</a>

By using Lemma 1.1 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M237">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M129">View MathML</a>, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M239">View MathML</a>

 □

Example 2.1 Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M240">View MathML</a> be a convex function in U with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M241">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M242">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M243">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M131">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M246">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M132">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M133">View MathML</a>, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M249">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M251">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M252">View MathML</a>

We have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M253">View MathML</a>.

Using Theorem 2.9, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M254">View MathML</a>

induce

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M255">View MathML</a>

Theorem 2.11Letgbe a convex function such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M256">View MathML</a>, and lethbe the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M197">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M262">View MathML</a>

(2.11)

holds, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M263">View MathML</a>

This result is sharp.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M264">View MathML</a>. We deduce that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M266">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Using the notation in (2.11), the differential subordination becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M268">View MathML</a>

By using Lemma 1.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M269">View MathML</a>

and this result is sharp. □

Theorem 2.12Lethbe a holomorphic function, which satisfies the inequality<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M270">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M272">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M276">View MathML</a>

(2.12)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M277">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M218">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M264">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M266">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and (2.12) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M284">View MathML</a>

Using Lemma 1.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M285">View MathML</a>

and q is the best dominant. □

Corollary 2.13Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M117">View MathML</a>be a convex function inU, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M118">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M291">View MathML</a>

(2.13)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M292">View MathML</a>

whereqis given by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M233">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Following the same steps as in the proof of Theorem 2.12 and considering <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M295">View MathML</a>, the differential subordination (2.13) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M296">View MathML</a>

By using Lemma 1.1 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M237">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M129">View MathML</a>, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M299">View MathML</a>

 □

Example 2.2 Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M240">View MathML</a> be a convex function in U with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M241">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M242">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M243">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M131">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M246">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M132">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M133">View MathML</a>, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M249">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M251">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M312">View MathML</a>

We have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M253">View MathML</a>.

Using Theorem 2.12, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M314">View MathML</a>

induce

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M315">View MathML</a>

Theorem 2.14Letgbe a convex function such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M256">View MathML</a>, and lethbe the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M317">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M320">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M323">View MathML</a>

(2.14)

holds, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M324">View MathML</a>

This result is sharp.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M325">View MathML</a>. We deduce that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M161">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>.

Using the notation in (2.14), the differential subordination becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M329">View MathML</a>

By using Lemma 1.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M330">View MathML</a>

and this result is sharp. □

Theorem 2.15Lethbe a holomorphic function, which satisfies the inequality<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M270">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M103">View MathML</a>.

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M320">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M29">View MathML</a>and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M338">View MathML</a>

(2.15)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M339">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M340">View MathML</a>. The functionqis convex, and it is the best dominant.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M325">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M175">View MathML</a>.

Differentiating, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M344">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M5">View MathML</a>, and (2.15) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M346">View MathML</a>

Using Lemma 1.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/252/mathml/M347">View MathML</a>

and q is the best dominant. □

Competing interests

The author declares that she has no competing interests.

Author’s contributions

The author drafted the manuscript, read and approved the final manuscript.

Acknowledgements

The author thanks the referee for his/her valuable suggestions to improve the present article.

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