- Research Article
- Open Access
Some Starlikeness Criterions for Analytic Functions
© Gejun Bao et al. 2010
- Received: 26 October 2010
- Accepted: 16 December 2010
- Published: 22 December 2010
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.
- Positive Integer
- Analytic Function
- Unit Disk
- Simple Calculation
- Minimum Point
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 .
if and .
For our results we need the following lemma.
Lemma 2.1 (see ).
we have . It follows that .
where , then .
It follows that .
For , we get the following corollary.
where , then .
where , then .
Putting in Theorem 2.3, we obtain the above corollary.
and the desired result easily follows from Corollary 2.5.
For , we have the following corollary.
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).
- 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
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.