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Some Starlikeness Criterions for Analytic Functions
Journal of Inequalities and Applications volume 2010, Article number: 175369 (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.
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, Obradović [5] proved that
if and . Recently, Obradović and Owa [6] proved that
if and .
In this paper we find a condition on , , , and for which
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]).
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 Obradović [5], and Obradović 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
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Singh V: Univalent functions with bounded derivative in the unit disc. Indian Journal of Pure and Applied Mathematics 1977, 8(11):1370–1377.
Fournier R: On integrals of bounded analytic functions in the closed unit disc. Complex Variables. Theory and Application 1989, 11(2):125–133.
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/S0017089500009472
Obradović M: A class of univalent functions. Hokkaido Mathematical Journal 1998, 27(2):329–335.
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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