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# Some Starlikeness Criterions for Analytic Functions

*Journal of Inequalities and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 175369 (2010)

## Abstract

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

## 1. Introduction

Let be a positive integer, and let denote the class of function

that are analytic in the unit disk . For , let

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].

Let

Singh [2] proved that if . More recently, Fournier [3, 4] proved that

is the order of starlikeness of . Now, we define

Clearly, . In 1998, ObradoviÄ‡ [5] proved that

if and . Recently, ObradoviÄ‡ and Owa [6] proved that

if and .

In this paper we find a condition on , , , and for which

implies and extend some results of ObradoviÄ‡ 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]).

Let be analytic in , , and satisfy the condition

Then

where

Theorem 2.2.

Let , , , and

where

If and are analytic in , satisfy

where , then

Proof.

If , it is easy to see the result is true. Now, assume . Let

If there exists , such that , then we will show that

for . Note that for ; it is sufficient to show that

for . Let , ; then, the left-hand side of (2.11) is

Suppose that and note that ; then inequality (2.11) is equivalent to

for all and . Now, if we define

then we have

where

Since

the denominator of is positive. Further, let

We have

If

we get

where

Note that

We obtain

If

we have

Hence we obtain

where

If

we have . It follows that .

Therefore we obtain for if or . It follows that

If , we have

by (2.13) and (2.21) for and by (2.23) for . Note that is an continuous increasing function for , and

Then there exists a unique , such that

Thus, is the global minimum point of on [. It follows from (2.33) that

or

By a simple calculation, we may obtain

for . It follows from (2.30) and (2.36) that that inequality (2.13) holds. This shows that inequality (2.10) holds, which contradicts with (2.7). Hence we must have

Theorem 2.3.

Let , , , and be defined as in Theorem 2.2. If satisfies

where , then .

Proof.

If , and the result is trivial. Now, assume . If we put

then by some transformations and (2.38) we get

By Lemma 2.1, we obtain

Let

Then we have

By Theorem 2.2, we get

It follows that .

For , we get the following corollary.

Corollary 2.4.

Let , , and let

where

If satisfies

where , then .

Corollary 2.5.

Let , , and let

If satisfies

where , then .

Proof.

Note that

Putting in Theorem 2.3, we obtain the above corollary.

Remark 2.6.

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

Theorem 2.7.

Let , , , and let

If satisfies

where , and

then .

Proof.

Let

Then from (2.52) and (2.53) we obtain

Hence, by using Theorem 1 given by Hallenbeck and Ruscheweyh [7], we have that

and the desired result easily follows from Corollary 2.5.

For , we have the following corollary.

Corollary 2.8.

Let , , and let

If satisfies

where , and

then .

## References

Pommerenke C:

*Univalent Functions with a Chapter on Quadratic Differentials by Gerd Jensen, Studia Mathematica/Mathematische LehrbÃ¼cher*.*Volume 20*. Vandenhoeck & Ruprecht, GÃ¶ttingen, Germany; 1975:376.Singh V: Univalent functions with bounded derivative in the unit disc.

*Indian Journal of Pure and Applied Mathematics*1977, 8(11):1370â€“1377.Fournier R: On integrals of bounded analytic functions in the closed unit disc.

*Complex Variables. Theory and Application*1989, 11(2):125â€“133.Fournier R: The range of a continuous linear functional over a class of functions defined by subordination.

*Glasgow Mathematical Journal*1990, 32(3):381â€“387. 10.1017/S0017089500009472ObradoviÄ‡ M: A class of univalent functions.

*Hokkaido Mathematical Journal*1998, 27(2):329â€“335.ObradoviÄ‡ M, Owa S: Some sufficient conditions for strongly starlikeness.

*International Journal of Mathematics and Mathematical Sciences*2000, 24(9):643â€“647. 10.1155/S0161171200004154Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions.

*Proceedings of the American Mathematical Society*1975, 52: 191â€“195. 10.1090/S0002-9939-1975-0374403-3

## 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).

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Bao, G., Guo, L. & Ling, Y. Some Starlikeness Criterions for Analytic Functions.
*J Inequal Appl* **2010**, 175369 (2010). https://doi.org/10.1155/2010/175369

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