On Coefficient problems for classes $\mathcal{S}_e^{\ast}$ and $\mathcal{C}_e$

TL;DR

This paper investigates sharp bounds of logarithmic coefficients, Hermitian-Toeplitz determinants, and examines conjectures like Zalcman and Fekete-Szeg"o for specific classes of univalent functions defined via exponential subordination.

Contribution

It provides new sharp bounds and confirms conjectures for the classes es and c, extending the understanding of their coefficient properties.

Findings

Sharp bounds for logarithmic coefficients established.

Hermitian-Toeplitz determinants computed for the classes.

Conjectures like Zalcman and Fekete-Szegf3 are verified as sharp.

Abstract

Logarithmic coefficients play a crucial role in the theory of univalent functions. In this study,we focus on the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ of starlike and convex functions, respectively, \begin{align*} \mathcal{S}_e^\ast := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec e^z, \ z \in \mathbb{D} \right\}, \end{align*} and \begin{align*} \mathcal{C}_e := \left\{ f \in \mathcal{S} : 1 + \frac{z f''(z)}{f'(z)} \prec e^z, \ z \in \mathbb{D} \right\}. \end{align*} This paper investigates the sharp bounds of the logarithmic coefficients and the Hermitian-Toeplitz determinant of these coefficients for the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$. Additionally, we examine the generalized Zalcman conjecture and the generalized Fekete-Szeg\"o inequality for these classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ and show that the inequalities are sharp.