# Some Starlikeness Criterions for Analytic Functions

- Gejun Bao
^{1}Email author, - Lifeng Guo
^{1}and - Yi Ling
^{2}Email author

**2010**:175369

https://doi.org/10.1155/2010/175369

© Gejun Bao et al. 2010

**Received: **26 October 2010

**Accepted: **16 December 2010

**Published: **22 December 2010

## Abstract

## 1. Introduction

denote the class of starlike function of order and .

Let and be analytic in ; then we say that the function is subordinate to in , if there exists an analytic function in such that , and , denoted that or . If is univalent in , then the subordination is equivalent to and [1].

implies and extend some results of Obradović and Owa [5, 6]. Also, we obtain some new results on starlikeness criterions.

## 2. Main Results

For our results we need the following lemma.

Lemma 2.1 (see [6]).

Theorem 2.2.

Proof.

Theorem 2.3.

Proof.

For , we get the following corollary.

Corollary 2.4.

Corollary 2.5.

Proof.

Putting in Theorem 2.3, we obtain the above corollary.

Remark 2.6.

Our results extend the results given by Obradović [5], and Obradović and Owa [6].

Theorem 2.7.

Proof.

and the desired result easily follows from Corollary 2.5.

For , we have the following corollary.

Corollary 2.8.

## Declarations

### Acknowledgment

The authors would like to thank the referee for giving them thoughtful suggestions which greatly improved the presentation of the paper. Bao Gejun was supported by NSF of P.R.China (no. 11071048).

## Authors’ Affiliations

## References

- Pommerenke C:
*Univalent Functions with a Chapter on Quadratic Differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher*.*Volume 20*. Vandenhoeck & Ruprecht, Göttingen, Germany; 1975:376.Google Scholar - Singh V: Univalent functions with bounded derivative in the unit disc.
*Indian Journal of Pure and Applied Mathematics*1977, 8(11):1370–1377.MathSciNetMATHGoogle Scholar - Fournier R: On integrals of bounded analytic functions in the closed unit disc.
*Complex Variables. Theory and Application*1989, 11(2):125–133.MathSciNetView ArticleMATHGoogle Scholar - Fournier R: The range of a continuous linear functional over a class of functions defined by subordination.
*Glasgow Mathematical Journal*1990, 32(3):381–387. 10.1017/S0017089500009472MathSciNetView ArticleMATHGoogle Scholar - Obradović M: A class of univalent functions.
*Hokkaido Mathematical Journal*1998, 27(2):329–335.MathSciNetView ArticleMATHGoogle Scholar - Obradović M, Owa S: Some sufficient conditions for strongly starlikeness.
*International Journal of Mathematics and Mathematical Sciences*2000, 24(9):643–647. 10.1155/S0161171200004154MathSciNetView ArticleMATHGoogle Scholar - Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions.
*Proceedings of the American Mathematical Society*1975, 52: 191–195. 10.1090/S0002-9939-1975-0374403-3MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.