Logarithmic coefficients for certain subclasses of close-to-convex functions

TL;DR

This paper establishes sharp upper bounds for the first three logarithmic coefficients of functions within certain subclasses of close-to-convex functions, expanding understanding of their geometric properties.

Contribution

It provides the first precise bounds for the initial logarithmic coefficients in specific subclasses of close-to-convex functions.

Findings

Sharp bounds for |1|2|3| of functions in subclasses

Enhanced understanding of geometric function theory

Extension of coefficient estimates to close-to-convex functions

Abstract

Let $\mathcal{S}$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}$ normalized by $f(0)=0=f'(0)-1$. The logarithmic coefficients $\gamma_n$ of $f\in\mathcal{S}$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} \gamma_n z^n.$ In the present paper, we determine the sharp upper bounds for $|\gamma_1|$, $|\gamma_2|$ and $|\gamma_3|$ when $f$ belongs to some familiar subclasses of close-to-convex functions.