Logarithmic coefficients of close-to-convex functions

TL;DR

This paper establishes the sharp upper bound for the third logarithmic coefficient of close-to-convex functions, confirming a recent conjecture and advancing understanding of their coefficient bounds.

Contribution

It determines the exact maximum of |3| for close-to-convex functions, resolving a conjecture posed by the authors.

Findings

Proved the sharp upper bound of |3| for close-to-convex functions.

Confirmed the conjecture of the authors regarding logarithmic coefficients.

Enhanced the understanding of coefficient bounds in geometric function theory.

Abstract

For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $\gamma_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} \gamma_n z^n$. In the present paper, we consider the class of close-to-convex functions (with argument $0$), and determine the sharp upper bound of $|\gamma_3|$ for such functions $f$, which proves a recent conjecture of the first and third authors [1].