# Some Starlikeness Criterions for Analytic Functions

## 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

(1.1)

that are analytic in the unit disk . For , let

(1.2)

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

(1.3)

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

(1.4)

is the order of starlikeness of . Now, we define

(1.5)

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

(1.6)

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

(1.7)

if and .

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

(1.8)

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

(2.1)

Then

(2.2)

where

(2.3)

Theorem 2.2.

Let , , , and

(2.4)

where

(2.5)

If and are analytic in , satisfy

(2.6)
(2.7)

where , then

(2.8)

Proof.

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

(2.9)

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

(2.10)

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

(2.11)

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

(2.12)

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

(2.13)

for all and . Now, if we define

(2.14)

then we have

(2.15)

where

(2.16)

Since

(2.17)

the denominator of is positive. Further, let

(2.18)

We have

(2.19)

If

(2.20)

we get

(2.21)

where

(2.22)

Note that

(2.23)

We obtain

(2.24)

If

(2.25)

we have

(2.26)

Hence we obtain

(2.27)

where

(2.28)

If

(2.29)

we have . It follows that .

Therefore we obtain for if or . It follows that

(2.30)

If , we have

(2.31)

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

(2.32)

Then there exists a unique , such that

(2.33)

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

(2.34)

or

(2.35)

By a simple calculation, we may obtain

(2.36)

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

(2.37)

Theorem 2.3.

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

(2.38)

where , then .

Proof.

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

(2.39)

then by some transformations and (2.38) we get

(2.40)

By Lemma 2.1, we obtain

(2.41)

Let

(2.42)

Then we have

(2.43)

By Theorem 2.2, we get

(2.44)

It follows that .

For , we get the following corollary.

Corollary 2.4.

Let , , and let

(2.45)

where

(2.46)

If satisfies

(2.47)

where , then .

Corollary 2.5.

Let , , and let

(2.48)

If satisfies

(2.49)

where , then .

Proof.

Note that

(2.50)

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

(2.51)

If satisfies

(2.52)

where , and

(2.53)

then .

Proof.

Let

(2.54)

Then from (2.52) and (2.53) we obtain

(2.55)

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

(2.56)

and the desired result easily follows from Corollary 2.5.

For , we have the following corollary.

Corollary 2.8.

Let , , and let

(2.57)

If satisfies

(2.58)

where , and

(2.59)

then .

## References

1. 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.

2. Singh V: Univalent functions with bounded derivative in the unit disc. Indian Journal of Pure and Applied Mathematics 1977, 8(11):1370â€“1377.

3. Fournier R: On integrals of bounded analytic functions in the closed unit disc. Complex Variables. Theory and Application 1989, 11(2):125â€“133.

4. 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/S0017089500009472

5. ObradoviÄ‡ M: A class of univalent functions. Hokkaido Mathematical Journal 1998, 27(2):329â€“335.

6. ObradoviÄ‡ M, Owa S: Some sufficient conditions for strongly starlikeness. International Journal of Mathematics and Mathematical Sciences 2000, 24(9):643â€“647. 10.1155/S0161171200004154

7. Hallenbeck 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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Correspondence to Gejun Bao or Yi Ling.

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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