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

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

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 be the class of functions of the form

(1.1)

which are analytic in the unit disk . We write .

A function is said to be starlike in () if and only if it satisfies

(1.2)

A function is said to be close-to-convex in () if and only if there is a starlike function such that

(1.3)

Let and be analytic in U. Then we say that is subordinate to in U, written , if there exists an analytic function in U, such that and (). If is univalent in U, then the subordination is equivalent to and .

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])

Letandbe analytic inUwith. Ifis starlike inUand, then

(1.4)

### 2 Main results

Our first theorem is given by the following.

Theorem 1Letwithfor. If

(2.1)

where, thenis univalent inU.

Proof

Let

(2.2)

then is analytic in U. By integration from 0 to zn-times, we obtain

(2.3)

Thus, we have

(2.4)

where

(2.5)

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

(2.6)

and, in consequence,

Since

we get

and so

(2.7)

for and .

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

Hence, is univalent in U. The proof of the theorem is complete. □

Let denote the class of functions with for , which satisfy the condition (2.1) given by Theorem 1.

Next we derive the following.

Theorem 2Let. Then, for,

(2.8)

(2.9)

(2.10)

Proof In view of (2.1), we have

(2.11)

Applying Lemma to (2.11), we get

(2.12)

By using the lemma repeatedly, we finally have

(2.13)

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

(2.14)

i.e.,

(2.15)

where is analytic in U and ().

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

Finally, we discuss the following theorem.

Theorem 3Letand have the form

(2.16)

(i) If, thenis starlike in;

(ii) If, thenis close-to-convex in.

Proof

If we put

(2.17)

then by (2.1) and the proof of Theorem 2 with , we have

(2.18)

It follows from the lemma that

(2.19)

which implies that

(2.20)

(i) Let and

(2.21)

Then by (2.20), we have

(2.22)

Also, from (2.8) in Theorem 2 with , we obtain

(2.23)

and so

(2.24)

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

for . This proves that is starlike in .

(ii) Let and

(2.25)

Then we have

Thus, for . This shows that is close-to-convex in . □

### 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

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