The Third Logarithmic Coefficient For The Subclasses Of Close-To-Convex Functions

TL;DR

This paper investigates the upper bounds of the third logarithmic coefficient for certain subclasses of close-to-convex functions, extending understanding of their coefficient behavior in complex analysis.

Contribution

It provides new bounds for the third logarithmic coefficient specifically within subclasses of close-to-convex functions, considering the general case of $f''(0)$.

Findings

Derived upper bounds for the third logarithmic coefficient

Extended results to subclasses of close-to-convex functions

Analyzed the impact of $f''(0)$ on coefficient bounds

Abstract

Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z| < 1\}$ normalized by $f (0) = 0$ and $f'(0) = 1.$ The logarithmic coefficients $\gamma_n$ of $f \in \mathcal{A}$ are defined by $ \log f(z)/z =2 \sum_{n=1}^{\infty}\gamma_{n}z^{n}.$ In the present paper, the upper bound of the third logarithmic coefficient in general case of $f''(0)$ was computed when $f$ belongs to some familiar subclasses of close-to-convex functions.