Extremal problems on the class of convex functions of order ?/2
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- 作者：Yusuf Abu Muhanna (1)
Liulan Li (2)
Saminathan Ponnusamy (3) - 关键词：Primary 30C65 ; 30C45 ; Secondary 30C20 ; 30C55 ; 31A05 ; 31B05 ; 31C05 ; Univalent ; Convex ; Starlike ; Close ; to ; convex ; Extreme points ; Hayman index ; Arclength ; Zalcman functional ; Harmonic function
- 刊名：Archiv der Mathematik
- 出版年：2014
- 出版时间：December 2014
- 年：2014
- 卷：103
- 期：6
- 页码：461-471
- 全文大小：224 KB
- 参考文献：1. Y. Abu Muhanna and S. Ponnusamy, Extreme points method and univalent harmonic mappings, Preprint.
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文摘
Lawrence Zalcman’s conjecture states that if \({f(z)=z+\sum\nolimits_{n=2}^{\infty}a_{n}z^{n}}\) is analytic and univalent in the unit disk \({|z| , then \({|a_n^2-a_{2n-1}|\leq (n-1)^2,}\) for each \({n\geq 2}\) , with equality only for the Koebe function \({k(z)=z/(1-z)^2}\) and its rotations. This conjecture remains open although it has been verified for a few geometric subclasses of the class of univalent analytic functions. In this paper, we consider this problem for the family of normalized functions f analytic and univalent in the unit disk |z|? $${\rm Re }\left( 1+\frac{zf''(z)}{f'(z)}\right) > -\frac{1}{2}\,\,\,\,\,{\rm for}\,\,\,\,\,|z| Functions satisfying this condition are known to be convex in some direction (and hence close-to-convex and univalent) in |z|?