This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Subordination properties for a general class of integral operators involving meromorphically multivalent functions

Nak Eun Cho1* and Rekha Srivastava2

Author Affiliations

1 Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea

2 Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

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Advances in Difference Equations 2013, 2013:93  doi:10.1186/1687-1847-2013-93


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/93


Received:14 January 2013
Accepted:14 March 2013
Published:5 April 2013

© 2013 Cho and Srivastava; 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

The purpose of the present paper is to investigate some subordination- and superordination-preserving properties of certain nonlinear integral operators defined on the space of meromorphically p-valent functions in the punctured open unit disk. The sandwich-type theorems associated with these integral operators are established. Relevant connections of the various results presented here with those involving relatively simpler nonlinear integral operators are also indicated.

MSC: 30C45, 30C80.

Keywords:
differential subordination; differential superordination; meromorphic functions; integral operators; convex functions; close-to-convex functions; subordination (or Löwner) chain

1 Introduction, definitions and preliminaries

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M1">View MathML</a> denote the class of analytic functions in the open unit disk

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M2">View MathML</a>

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M3">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M4">View MathML</a>, let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M5">View MathML</a>

Let f and F be members of ℋ. The function f is said to be subordinate to F, or F is said to be superordinate to f, if there exists a function w analytic in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>, with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M7">View MathML</a>

such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M8">View MathML</a>

In such a case, we write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M9">View MathML</a>

Furthermore, if the function F is univalent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>, then we have (cf.[1,2] and [3])

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M11">View MathML</a>

Definition 1 (Miller and Mocanu [1])

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M12">View MathML</a>

and let h be univalent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14">View MathML</a> is analytic in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a> and satisfies the differential subordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M16">View MathML</a>

(1.1)

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14">View MathML</a> 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/2013/1/93/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M18">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14">View MathML</a> satisfying (1.1). A dominant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M20">View MathML</a> that satisfies the following condition:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M21">View MathML</a>

for all dominants q of (1.1) is said to be the best dominant.

Definition 2 (Miller and Mocanu [4])

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M22">View MathML</a>

and let h be analytic in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>. If p and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M24">View MathML</a> are univalent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a> and satisfy the differential superordination

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M26">View MathML</a>

(1.2)

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14">View MathML</a> is called a solution of the differential superordination. An analytic function q is called a subordinant of the solutions of the differential superordination or, more simply, a subordinant if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M28">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14">View MathML</a> satisfying (1.2). A univalent subordinant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M20">View MathML</a> that satisfies the following condition:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M31">View MathML</a>

for all subordinants q of (1.2) is said to be the best subordinant.

Definition 3 (Miller and Mocanu [4])

We denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M32">View MathML</a> the class of functions f that are analytic and injective on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M33">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M34">View MathML</a>

and are such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M35">View MathML</a>

We also denote the class <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M36">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M37">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M38">View MathML</a> denote the class of functions of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M39">View MathML</a>

which are analytic in the punctured open unit disk <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M40">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M41">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M42">View MathML</a> be the subclasses of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M43">View MathML</a> consisting of all functions which are, respectively, meromorphically starlike and meromorphically convex in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M44">View MathML</a> (see, for details, [1,5]).

For a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M45">View MathML</a>, we introduce the following general integral operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M47">View MathML</a>

(1.3)

Several members of the family of integral operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M48">View MathML</a> defined by (1) have been extensively studied by many authors (see, for example, [6-10]; see also [11] and [12]) with suitable restrictions on the parameters α, β, γ and δ, and for f belonging to some favored subclasses of meromorphic functions. In particular, Bajpai [6] showed that the integral operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M49">View MathML</a> belongs to the classes <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M41">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M42">View MathML</a>, whenever f belongs to the classes <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M41">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M42">View MathML</a>, respectively.

Making use of the principle of subordination between analytic functions, Miller et al.[13] and, more recently, Owa and Srivastava [14] obtained some interesting subordination-preserving properties for certain integral operators. Moreover, Miller and Mocanu [4] considered differential superordinations as the dual concept of differential subordinations (see also [15]). It should be remarked that in recent years several authors obtained many interesting results involving various integral operators associated with differential subordination and superordination (for example, see [5,16-18]). In the present paper, we obtain the subordination- and superordination-preserving properties of the general integral operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46">View MathML</a> defined by (1) with the sandwich-type theorem.

The following lemmas will be required in our present investigation.

Lemma 1 (Miller and Mocanu [19])

Suppose that the function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M55">View MathML</a>

satisfies the following condition:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M56">View MathML</a>

for all realsand for alltwith

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M57">View MathML</a>

If the function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M58">View MathML</a>

is analytic in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M60">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M61">View MathML</a>

Lemma 2 (Miller and Mocanu [20])

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M62">View MathML</a>

If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M63">View MathML</a>

then the solution of the differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M64">View MathML</a>

is analytic in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>and satisfies the following inequality:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M66">View MathML</a>

Lemma 3 (Miller and Mocanu [1])

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M67">View MathML</a>

and let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M68">View MathML</a>

be analytic in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M69">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M70">View MathML</a>

If the functionqis not subordinate to<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M14">View MathML</a>, then there exist points

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M72">View MathML</a>

for which

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M73">View MathML</a>

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M74">View MathML</a>

If R is the univalent function defined in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M76">View MathML</a>

then the open door function (see [1]) is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M77">View MathML</a>

(1.4)

Remark 1 The function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M78">View MathML</a> defined by (1.4) is univalent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M80">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M81">View MathML</a> is the complex plane with slits along the half-lines given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M82">View MathML</a>

Lemma 4 (Totoi [21])

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M83">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M84">View MathML</a>

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M85">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M86">View MathML</a>

(1.5)

and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M87">View MathML</a>is defined by (1.4) with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M88">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M89">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M90">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M91">View MathML</a>is the integral operator defined by (1).

A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M93">View MathML</a> is the subordination chain (or Löwner chain) if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M94">View MathML</a> is analytic and univalent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M96">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M97">View MathML</a> is continuously differentiable on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M98">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M99">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M100">View MathML</a>

(1.6)

Lemma 5 (Miller and Mocanu [4])

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M101">View MathML</a>

Also set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M102">View MathML</a>

If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M103">View MathML</a>

is a subordination chain and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M104">View MathML</a>

then the following subordination condition:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M105">View MathML</a>

(1.7)

implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M106">View MathML</a>

Furthermore, if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M107">View MathML</a>

has a univalent solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M108">View MathML</a>, thenqis the best subordinant.

Lemma 6 (Pommerenke [22])

The function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M109">View MathML</a>

with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M110">View MathML</a>

Suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M94">View MathML</a>is analytic in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M113">View MathML</a>and that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M97">View MathML</a>is continuously differentiable on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M98">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M99">View MathML</a>. If the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92">View MathML</a>satisfies the following inequalities:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M118">View MathML</a>

(1.8)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M119">View MathML</a>

(1.9)

for some positive constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M120">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M121">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92">View MathML</a>is a subordination chain.

2 Main results and their corollaries and consequences

We begin by proving a general subordination property involving the integral operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46">View MathML</a> defined by (1), which is contained in Theorem 1 below.

Theorem 1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M124">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125">View MathML</a>is defined by (1.5). Suppose also that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M126">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M127">View MathML</a>

(2.2)

Then the subordination relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M128">View MathML</a>

(2.3)

implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M129">View MathML</a>

(2.4)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46">View MathML</a>is the integral operator defined by (1). Moreover, the function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M131">View MathML</a>

is the best dominant.

Proof Let us define the functions F and G, respectively, by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M132">View MathML</a>

(2.5)

We first show that if the function q is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M133">View MathML</a>

(2.6)

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M134">View MathML</a>

From the definition of (1), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M135">View MathML</a>

(2.7)

We also have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M136">View MathML</a>

(2.8)

It follows from (2.7) and (2.8) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M137">View MathML</a>

(2.9)

Now, by a simple calculation with (2.9), we obtain the following relationship:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M138">View MathML</a>

(2.10)

Thus, from (2.1), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M139">View MathML</a>

and by using Lemma 2, we conclude that the differential equation (2.10) has a solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M140">View MathML</a> with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M141">View MathML</a>

Put

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M142">View MathML</a>

(2.11)

where ρ is given by (2.2). From (2.1), (2.10) and (2.11), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M143">View MathML</a>

We now proceed to show that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M144">View MathML</a>

(2.12)

Indeed, from (2.11), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M145">View MathML</a>

(2.13)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M146">View MathML</a>

(2.14)

For ρ given by (2.2), we note that the coefficient of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M147">View MathML</a> in the quadratic expression <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M148">View MathML</a> given by (2.14) is positive or equal to zero and also <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M148">View MathML</a> is a perfect square. Hence from (2.13), we see that (2.12) holds true. Thus, by using Lemma 1, we conclude that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M150">View MathML</a>

That is, the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M151">View MathML</a> defined by (2.5) is convex in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>.

We next prove that the subordination condition (2.3) implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M153">View MathML</a>

(2.15)

for the functions F and G defined by (2.5). Without loss of generality, we can assume that G is analytic and univalent on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M154">View MathML</a> and that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M155">View MathML</a>

We now consider the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M157">View MathML</a>

We note that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M158">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M159">View MathML</a>

Furthermore, since G is convex, by using the well-known growth and distortion sharp inequalities for convex functions (see [23]), we can prove that the second condition of Lemma 6 is satisfied. Therefore, by virtue of Lemma 6, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92">View MathML</a> is a subordination chain. We observe from the definition of a subordination chain that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M161">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M162">View MathML</a>

This implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M163">View MathML</a>

We now suppose that F is not subordinate to G. Then, in view of Lemma 3, there exist points <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M164">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M165">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M166">View MathML</a>

Hence we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M167">View MathML</a>

by virtue of the subordination condition (2.3). This contradicts the above observation that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M168">View MathML</a>

Therefore, the subordination condition (2.3) must imply the subordination given by (2.15). Considering <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M169">View MathML</a>, we see that the function G is the best dominant. This evidently completes the proof of Theorem 1. □

We next prove a solution to a dual problem of Theorem 1 in the sense that the subordinations are replaced by superordinations.

Theorem 2Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M124">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125">View MathML</a>is defined by (1.5). Suppose also that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M172">View MathML</a>

whereρis given by (2.2) and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173">View MathML</a>is univalent in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M69">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M175">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46">View MathML</a>is the integral operator defined by (1). Then the superordination relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M177">View MathML</a>

(2.16)

implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M178">View MathML</a>

Moreover, the function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M179">View MathML</a>

is the best subordinant.

Proof Let the functions F and G be given by (2.5). We first note from (2.7) and (2.8) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M180">View MathML</a>

(2.17)

After a simple calculation, equation (2.17) yields the following relationship:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M181">View MathML</a>

where the function q is given in (2.6). Then, by using the same method as in the proof of Theorem 1, we can prove that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M182">View MathML</a>

that is, G defined by (2.5) is convex (univalent) in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>.

We next prove that the superordination condition (2.16) implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M184">View MathML</a>

(2.18)

For this purpose, we consider the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M186">View MathML</a>

Since G is convex and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M187">View MathML</a>, we can prove easily that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M92">View MathML</a> is a subordination chain as in the proof of Theorem 1. Therefore, according to Lemma 5, we conclude that the superordination condition (2.16) must imply the superordination given by (2.18). Furthermore, since the differential equation (2.17) has the univalent solution G, it is the best subordinant of the given differential superordination. Hence we complete the proof of Theorem 2. □

If we combine Theorem 1 and Theorem 2, then we obtain the following sandwich-type theorem.

Theorem 3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M189">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190">View MathML</a>), where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125">View MathML</a>is defined by (1.5). Suppose also that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M192">View MathML</a>

(2.19)

whereρis given by (2.2) and the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173">View MathML</a>is univalent in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M69">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M195">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46">View MathML</a>is the integral operator defined by (1). Then the subordination relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M197">View MathML</a>

implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M198">View MathML</a>

Moreover, the functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M199">View MathML</a>

are the best subordinant and the best dominant, respectively.

The assumption of Theorem 3, that is, the functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M200">View MathML</a>

need to be univalent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>, will be replaced by another set of conditions in the following result.

Corollary 1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M189">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190">View MathML</a>), where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M125">View MathML</a>is defined by (1.5). Suppose also that the condition (2.19) is satisfied and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M205">View MathML</a>

(2.20)

whereρis given by (2.2). Then the subordination relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M206">View MathML</a>

implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M207">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M46">View MathML</a>is the integral operator defined by (1). Moreover, the functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M209">View MathML</a>

are the best subordinant and the best dominant, respectively.

Proof In order to prove Corollary 1, we have to show that the condition (2.20) implies the univalence of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M210">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M211">View MathML</a>

(2.21)

By noting that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M212">View MathML</a> from (2.2), we obtain from the condition (2.20) that ψ is a close-to-convex function in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a> (see [24]), and hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173">View MathML</a> is univalent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>. Furthermore, by using the same techniques as in the proof of Theorem 3, we can prove the convexity (univalence) of F and so the details may be omitted. Therefore, by applying Theorem 3, we obtain Corollary 1. □

By setting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M216">View MathML</a> in Theorem 3, we have the following result.

Corollary 2Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M217">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190">View MathML</a>), where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M219">View MathML</a>is defined by (1.5) with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M220">View MathML</a>. Suppose also that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M221">View MathML</a>

and the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173">View MathML</a>is univalent in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M224">View MathML</a>

(2.22)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M225">View MathML</a>is the integral operator defined by (1) with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M220">View MathML</a>. Then the subordination relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M227">View MathML</a>

implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M228">View MathML</a>

Moreover, the functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M229">View MathML</a>

are the best subordinant and the best dominant, respectively.

If we take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M230">View MathML</a> in Theorem 3, then we are easily led to the following result.

Corollary 3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M231">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M190">View MathML</a>), where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M233">View MathML</a>is defined by (1.5) with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M234">View MathML</a>. Suppose also that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M235">View MathML</a>

(2.23)

and the function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M173">View MathML</a>is univalent in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M6">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M238">View MathML</a>

(2.24)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M239">View MathML</a>is the integral operator defined by (1) with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M234">View MathML</a>. Then the subordination relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M241">View MathML</a>

implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M242">View MathML</a>

Moreover, the functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/93/mathml/M243">View MathML</a>

are the best subordinant and the best dominant, respectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors jointly worked on the results and they read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was supported by the Basic Science Research Program through the National Research Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).

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