Sharp Coefficient and Inverse Problems for Holomorphic Semigroup Generators

TL;DR

This paper investigates extremal coefficient problems for a subclass of holomorphic semigroup generators, providing sharp bounds and explicit extremal functions, thereby unifying and extending classical results in geometric function theory.

Contribution

It introduces new sharp bounds for coefficients of holomorphic semigroup generators and connects these to functions of bounded turning, extending previous work in the field.

Findings

Sharp bounds for initial logarithmic coefficients $oldsymbol{eta}$, inverse coefficients $oldsymbol{A_n}$, and logarithmic inverse coefficients $oldsymbol{\Gamma_n}$.

Explicit extremal functions related to hypergeometric functions demonstrate the sharpness of bounds.

Unified framework linking coefficient problems for bounded turning functions and semigroup dynamics.

Abstract

In this paper, we study extremal problems for coefficient functionals associated with a distinguished subclass of holomorphic semigroup generators, denoted by $\mathcal{A}_{\beta}$ ($0 \le \beta \le 1$), defined on the unit disk $\mathbb{D}$. This class forms a natural filtration of the class $\mathcal{G}_0$ of infinitesimal generators, with the class $\mathcal{R}$ of functions of bounded turning arising as its minimal element. We obtain sharp bounds for the initial logarithmic coefficients $\gamma_n$, the inverse coefficients $A_n$, and the logarithmic inverse coefficients $\Gamma_n$ for $n = 1,2,3$ within the class $\mathcal{A}_{\beta}$. In addition, we address the successive coefficient problem by deriving sharp upper and lower estimates for the differences $|A_{n+1}| - |A_n|$ for $n = 1,2$. Furthermore, we establish sharp bounds for a generalized Fekete--Szeg\"o functional in the class $\mathcal{R}$. The extremality of the obtained results is demonstrated by explicit constructions, including functions related to Gauss hypergeometric functions. Our results unify and extend several earlier contributions in geometric function theory and reveal a structural connection between coefficient problems for functions of bounded turning and the dynamics of holomorphic semigroup generators.