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

Open Access Research

On certain univalent functions with missing coefficients

Yi-Ling Cang1 and Jin-Lin Liu2*

Author Affiliations

1 Department of Mathematics, Suqian College, Suqian, Jiangsu, 223800, P.R. China

2 Department of Mathematics, Yangzhou University, Yangzhou, 225002, P.R. China

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Advances in Difference Equations 2013, 2013:89  doi:10.1186/1687-1847-2013-89

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/89


Received:12 January 2013
Accepted:23 March 2013
Published:3 April 2013

© 2013 Cang and Liu; 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 main object of the present paper is to show certain sufficient conditions for univalency of analytic functions with missing coefficients.

MSC: 30C45, 30C55.

Keywords:
analytic; univalent; subordination

1 Introduction

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M1">View MathML</a> be the class of functions of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M2">View MathML</a>

(1.1)

which are analytic in the unit disk <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M3">View MathML</a>. We write <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M4">View MathML</a>.

A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M5">View MathML</a> is said to be starlike in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M6">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M7">View MathML</a>) if and only if it satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M8">View MathML</a>

(1.2)

A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M5">View MathML</a> is said to be close-to-convex in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M6">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M7">View MathML</a>) if and only if there is a starlike function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M13">View MathML</a>

(1.3)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12">View MathML</a> be analytic in U. Then we say that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a> is subordinate to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12">View MathML</a> in U, written <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M18">View MathML</a>, if there exists an analytic function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M19">View MathML</a> in U, such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M20">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M21">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M22">View MathML</a>). If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12">View MathML</a> is univalent in U, then the subordination <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M18">View MathML</a> is equivalent to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M25">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M26">View MathML</a>.

Recently, several authors showed some new criteria for univalency of analytic functions (see, e.g., [1-7]). In this note, we shall derive certain sufficient conditions for univalency of analytic functions with missing coefficients.

For our purpose, we shall need the following lemma.

Lemma (see [8,9])

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M12">View MathML</a>be analytic inUwith<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M25">View MathML</a>. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M30">View MathML</a>is starlike inUand<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M31">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M32">View MathML</a>

(1.4)

2 Main results

Our first theorem is given by the following.

Theorem 1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M33">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M34">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M35">View MathML</a>. If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M36">View MathML</a>

(2.1)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M37">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a>is univalent inU.

Proof

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M39">View MathML</a>

(2.2)

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M40">View MathML</a> is analytic in U. By integration from 0 to zn-times, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M41">View MathML</a>

(2.3)

Thus, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M42">View MathML</a>

(2.4)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M43">View MathML</a>

(2.5)

It is easily seen from (2.1), (2.2) and (2.5) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M44">View MathML</a>

(2.6)

and, in consequence,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M45">View MathML</a>

Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M46">View MathML</a>

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M47">View MathML</a>

and so

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M48">View MathML</a>

(2.7)

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M49">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M50">View MathML</a>.

Now it follows from (2.4) and (2.7) that

Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a> is univalent in U. The proof of the theorem is complete. □

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M53">View MathML</a> denote the class of functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M33">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M34">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M35">View MathML</a>, which satisfy the condition (2.1) given by Theorem 1.

Next we derive the following.

Theorem 2Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M57">View MathML</a>. Then, for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M22">View MathML</a>,

(2.8)

(2.9)

(2.10)

Proof In view of (2.1), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M62">View MathML</a>

(2.11)

Applying Lemma to (2.11), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M63">View MathML</a>

(2.12)

By using the lemma repeatedly, we finally have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M64">View MathML</a>

(2.13)

According to a result of Hallenbeck and Ruscheweyh [[1], Theorem 1], (2.13) gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M65">View MathML</a>

(2.14)

i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M66">View MathML</a>

(2.15)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M19">View MathML</a> is analytic in U and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M68">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M22">View MathML</a>).

Now, from (2.15), we can easily derive the inequalities (2.8), (2.9) and (2.10). □

Finally, we discuss the following theorem.

Theorem 3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M70">View MathML</a>and have the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M71">View MathML</a>

(2.16)

(i) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M72">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a>is starlike in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M74">View MathML</a>;

(ii) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M75">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a>is close-to-convex in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M77">View MathML</a>.

Proof

If we put

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M78">View MathML</a>

(2.17)

then by (2.1) and the proof of Theorem 2 with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M80">View MathML</a>

(2.18)

It follows from the lemma that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M81">View MathML</a>

(2.19)

which implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M82">View MathML</a>

(2.20)

(i) Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M72">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M84">View MathML</a>

(2.21)

Then by (2.20), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M85">View MathML</a>

(2.22)

Also, from (2.8) in Theorem 2 with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M79">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M87">View MathML</a>

(2.23)

and so

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M88">View MathML</a>

(2.24)

Therefore, it follows from (2.22) and (2.24) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M89">View MathML</a>

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M90">View MathML</a>. This proves that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a> is starlike in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M90">View MathML</a>.

(ii) Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M75">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M94">View MathML</a>

(2.25)

Then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M95">View MathML</a>

Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M96">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M97">View MathML</a>. This shows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M14">View MathML</a> is close-to-convex in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/89/mathml/M97">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have made the same contribution. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

We would like to express sincere thanks to the referees for careful reading and suggestions, which helped us to improve the paper.

References

  1. Dziok, J, Srivastava, HM: Certain subclasses of analytic functions associated with the generalized hypergeometric function. Integral Transforms Spec. Funct.. 14, 7–18 (2003). Publisher Full Text OpenURL

  2. Nunokawa, M, Obradovič, M, Owa, S: One criterion for univalency. Proc. Am. Math. Soc.. 106, 1035–1037 (1989). Publisher Full Text OpenURL

  3. Obradovič, M, Pascu, NN, Radomir, I: A class of univalent functions. Math. Jpn.. 44, 565–568 (1996)

  4. Owa, S: Some sufficient conditions for univalency. Chin. J. Math.. 20, 23–29 (1992)

  5. Samaris, S: Two criteria for univalency. Int. J. Math. Math. Sci.. 19, 409–410 (1996). Publisher Full Text OpenURL

  6. Silverman, H: Univalence for convolutions. Int. J. Math. Math. Sci.. 19, 201–204 (1996). Publisher Full Text OpenURL

  7. Yang, D-G, Liu, J-L: On a class of univalent functions. Int. J. Math. Math. Sci.. 22, 605–610 (1999). Publisher Full Text OpenURL

  8. Hallenbeck, DJ, Ruscheweyh, S: Subordination by convex functions. Proc. Am. Math. Soc.. 51, 191–195 (1975). Publisher Full Text OpenURL

  9. Suffridge, TJ: Some remarks on convex maps of the unit disk. Duke Math. J.. 37, 775–777 (1970). Publisher Full Text OpenURL