Logarithmic coefficients and a coefficient conjecture for univalent functions

Article Type

Research Article

Publication Title

Monatshefte fur Mathematik

Abstract

Let U(λ) denote the family of analytic functions f(z), f(0) = 0 = f′(0) - 1 , in the unit disk D, which satisfy the condition | (z/ f(z)) 2f′(z) - 1 | OpenSPiltSPi λ for some 0 OpenSPiltSPi λ≤ 1. The logarithmic coefficients γn of f are defined by the formula log(f(z)/z)=2∑n=1∞γnzn. In a recent paper, the present authors proposed a conjecture that if f∈ U(λ) for some 0 OpenSPiltSPi λ≤ 1 , then |an|≤∑k=0n-1λk for n≥ 2 and provided a new proof for the case n= 2. One of the aims of this article is to present a proof of this conjecture for n= 3 , 4 and an elegant proof of the inequality for n= 2 , with equality for f(z) = z/ [ (1 + z) (1 + λz) ]. In addition, the authors prove the following sharp inequality for f∈ U(λ) : (Formula Presented), where Li 2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S.

First Page

489

Last Page

501

DOI

10.1007/s00605-017-1024-3

Publication Date

3-1-2018

Comments

All Open Access, Green

This document is currently not available here.

Share

COinS