On the generalized Zalcman functional λan2 − a2n−1 in the close-to-convex family

Article Type

Research Article

Publication Title

Proceedings of the American Mathematical Society

Abstract

Let S denote the class of all functions f(z) = z + ∑n=2∞ anzn analytic and univalent in the unit disk D. For f ∈ S, Zalcman conjectured that |an2 − a2n−1| ≤ (n − 1)2 for n ≥ 3. This conjecture has been verified for only certain values of n for f ∈ S and for all n ≥ 4 for the class C of closeto-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional |λan2 −a2n−1| for functions in C and for all n ≥ 3, where λ is a positive constant.

First Page

833

Last Page

846

DOI

10.1090/proc/13260

Publication Date

1-1-2017

Comments

Open Access, Green

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