Generalized Zalcman conjecture for some classes of analytic functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:450

Article

Ravichandran, V.¹  Verma, Shelly¹ 

[1] Univ Delhi, Dept Math, Delhi 110007, India

关键词: Univalent functions;    Starlike functions;    Convex functions;    Coefficient bounds;    Closed convex hull;    Coefficient functional;   

DOI  :  10.1016/j.jmaa.2017.01.053

来源: Elsevier

【 摘 要 】

For functions f (z) = z + alpha(2)z(2) + alpha(3)z(3) + ... in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional phi(f, n, m; lambda) = vertical bar lambda a(n)a(m) - a(n+m-1)vertical bar. For all real parameters lambda and beta < 1, we provide the sharp upper bound of phi(f, n, m; lambda) for functions f satisfying Ref (z) > beta and hence settle the open problem of estimating phi( f, n, m; lambda) recently proposed by Agrawal and Sahoo (2016) [1]. For all real values of lambda, the estimations of phi(f, n, m; lambda) are provided for starlike and convex functions of order a (alpha < 1) which are sharp for lambda <= 0 or for certain positive values of lambda. Moreover, for certain positive A, the sharp estimation of 0(f, n, m; A) is given when f is a typically real function or a univalent function with real coefficients or is in some subclasses of close-to-convex functions. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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