Zalcman Conjecture for certain analytic and univalent functions

TL;DR

This paper proves the Zalcman and generalized Zalcman conjectures for specific classes of univalent functions using extreme point theory, advancing understanding of coefficient bounds in geometric function theory.

Contribution

The paper establishes the Zalcman and generalized Zalcman conjectures for classes  and , providing new results in the coefficient bounds of univalent functions.

Findings

Proved Zalcman conjecture for class .

Proved generalized Zalcman conjecture for class .

Confirmed Zalcman conjecture for initial coefficients in class .

Abstract

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ and $\mathcal{S}$ denote the class of functions $f\in\mathcal{A}$ which are univalent ({\it i.e.}, one-to-one). In 1960s, L. Zalcman conjectured that $|a_n^2-a_{2n-1}|\le (n-1)^2$ for $n\ge 2$, which implies the famous Bieberbach conjecture $|a_n|\le n$ for $n\ge 2$. For $f\in \mathcal{S}$, Ma \cite{Ma-1999} proposed a generalized Zalcman conjecture $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1) $$ for $n\ge 2, m\ge 2$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying $$ \left|f'(z)\left(\frac{z}{f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{D}. $$ and $\mathcal{F}$ denote the class of functions $f\in \mathcal{A}$ satisfying ${\rm Re\,}(1-z)^{2}f'(z)>0$ in $\mathbb{D}$. In the present paper, we prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{U}$ using extreme point theory. We also prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{F}$ for the initial coefficients.