On the generalized Zalcman conjecture

TL;DR

This paper investigates the generalized Zalcman conjecture for univalent functions, proving specific cases and establishing geometric conditions for extremal functions, advancing understanding of coefficient inequalities in complex analysis.

Contribution

The paper proves the generalized Zalcman conjecture for specific parameters and provides geometric conditions for extremal functions, contributing new results to the theory of univalent functions.

Findings

Proved the inequality for λ=3, n=2, m=3.

Established geometric conditions for extremal functions at λ=2, n=2, m=3.

Enhanced understanding of coefficient bounds in univalent function theory.

Abstract

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $ f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, In 1999, Ma proposed the generalized Zalcman conjecture that $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\,\mbox{ for } n\ge2,\, m\ge 2,$$ with equality only for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In the same paper, Ma \cite{Ma-1999} asked for what positive real values of $\lambda$ does the following inequality hold? \begin{equation}\label{conjecture} |\lambda a_na_m-a_{n+m-1}|\le \lambda nm -n-m+1 \,\,\,\,\, (n\ge 2, \,m\ge3). \end{equation} Clearly equality holds for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In this paper, we prove the inequality (\ref{conjecture}) for $\lambda=3, n=2, m=3$. Further, we provide a geometric condition on extremal function maximizing (\ref{conjecture}) for $\lambda=2,n=2, m=3$.