- Research Article
- Open Access

# 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

We determine the condition on , , , and for which implies , where is the class of starlike functions of order . Some results of Obradović and Owa are extended. We also obtain some new results on starlikeness criterions.

## Keywords

- Positive Integer
- Analytic Function
- Unit Disk
- Simple Calculation
- Minimum Point

## 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].

if and .

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.

we have . It follows that .

Theorem 2.3.

where , then .

Proof.

It follows that .

For , we get the following corollary.

Corollary 2.4.

where , then .

Corollary 2.5.

where , then .

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.

then .

Proof.

and the desired result easily follows from Corollary 2.5.

For , we have the following corollary.

Corollary 2.8.

then .

## 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

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## 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.