Generalized Zalcman conjecture for convex functions of order α

Article Type

Research Article

Publication Title

Acta Mathematica Hungarica

Abstract

Let S denote the class of all functions of the form f(z) = z+ a2z2+ a3z3+ ⋯ which are analytic and univalent in the open unit disk D and, for λ> 0 , let Φλ(n,f)=λan2-a2n-1 denote the generalized Zalcman coefficient functional. Zalcman conjectured that if f∈ S, then | Φ 1(n, f) | ≤ (n- 1) 2 for n≥ 3. The functional of the form Φ λ(n, f) is indeed related to Fekete–Szegő functional of the n-th root transform of the corresponding function in S. This conjecture has been verified for a certain special geometric subclasses of S but it remains open for f∈ S and for n> 6. In the present paper, we prove sharp bounds on | Φ λ(n, f) | for f∈ F(α) and for all n≥ 3 , in the case that λ is a positive real parameter, where F(α) denotes the family of all functions f∈ S satisfying the condition Re(1+zf′′(z)f′(z))>αforz∈D, where - 1 / 2 ≤ α< 1. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order α, α∈ [ - 1 / 2 , 1).

First Page

234

Last Page

246

DOI

10.1007/s10474-016-0639-5

Publication Date

10-1-2016

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