Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions

TL;DR

This paper proves the generalized Zalcman conjecture for specific initial coefficients of univalent functions, confirming the conjecture for certain cases using advanced complex analysis techniques.

Contribution

The paper establishes the validity of the generalized Zalcman conjecture for the cases (n=2, m=3) and (n=2, m=4) using properties of holomorphic motion and Krushkal's Surgery Lemma.

Findings

Proves the conjecture for (n=2, m=3)

Proves the conjecture for (n=2, m=4)

Confirms the conjecture's validity in these cases

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 $f\in \mathcal{S}$, 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 this paper using the properties of holomorphic motion and Krushkal's Surgery Lemma \cite{Krushkal-1995}, we prove the generalized Zalcman conjecture when $n=2$, $m=3$ and $n=2$, $m=4$.