On Coefficient problems for classes $\mathcal{S}_e^{\ast}$ and $\mathcal{C}_e$
Sujoy Majumder, Nabadwip Sarkar, Molla Basir Ahamed
TL;DR
The paper investigates coefficient problems for the exponential-subordinated classes $\mathcal{S}_e^{\ast}$ and $\mathcal{C}_e$, focusing on sharp bounds for logarithmic coefficients, the second-order Hermitian-Toeplitz determinant, and generalized Zalcman and Fekete–Szegö functionals. It leverages subordination to $e^z$ and Carathéodory machinery to derive explicit coefficient relations and establish sharp inequalities up to the fourth logarithmic coefficient, with conjectures regarding the general case and explicit extremals. The results include tight bounds for the Hermitian-Toeplitz determinant and complete sharp analyses of the generalized Zalcman and Fekete–Szegö inequalities across parameter regimes, contributing precise benchmarks for nonclassical starlike/convex classes. Overall, the work advances the coefficient theory for exponential-type subclasses and provides a framework for further extremal problems in geometric function theory.
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-Szegö inequality for these classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ and show that the inequalities are sharp.