tailieunhanh - Báo cáo: Some Starlikeness Criterions for Analytic Functions

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 175369, 11 pages doi: Research Article Some Starlikeness Criterions for Analytic Functions Gejun Bao,1 Lifeng Guo,1 and Yi Ling2 1 2 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Department of Mathematical Science, Delaware State University, Dover, DE 19901, USA Correspondence should be addressed to Gejun Bao, baogj@ and Yi Ling, yiling@ Received 26 October 2010; Accepted 16 December 2010 Academic Editor: Vijay Gupta Copyright q 2010 Gejun Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction. | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 175369 11 pages doi 2010 175369 Research Article Some Starlikeness Criterions for Analytic Functions Gejun Bao 1 Lifeng Guo 1 and Yi Ling2 1 Department of Mathematics Harbin Institute of Technology Harbin 150001 China 2 Department of Mathematical Science Delaware State University Dover DE 19901 USA Correspondence should be addressed to Gejun Bao baogj@ and Yi Ling yiling@ Received 26 October 2010 Accepted 16 December 2010 Academic Editor Vijay Gupta Copyright 2010 Gejun Bao et al. 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. We determine the condition on a p f and 1 for which l 1-a z f z 1 a zf r z f z z f z 1-11 1 implies f z e S f where S f is the class of starlike functions of order f. Some results of Obradovic and Owa are extended. We also obtain some new results on starlikeness criterions. 1. Introduction Let n be a positive integer and let Hn denote the class of function to f z z 2 ak 1zk 1 k n that are analytic in the unit disk U z z 1 . For 0 f 1 let _ Í zf zl 1 S f f e H1 Re-fz f z e U f z denote the class of starlike function of order f and S 0 S . Let f z and F z be analytic in U then we say that the function f z is subordinate to F z in U if there exists an analytic function w z in U such that w z z and f z F w z denoted that f F or f z F z . If F z is univalent in U then the subordination is equivalent to f 0 F 0 and f U c FfU 1 . 2 Journal of Inequalities and Applications Let SX - f e H1 f z - 1 X z e U . Singh 2 proved that SX c S if 0 1 2 C5. More recently Fournier 3 4 proved that 2 Sx c S 0 X i 1 - X 1 - X 2 1 - X2 4 1 2 1 - 5 4 X2 . 1 - X2 4 if 0 X 3 2 if 3 X 1 is the order of starlikeness of SX. Now we define U X ý n f e Hn zf zU z Ỵ 1 f z V z 7 X z