Zalcman and generalized Zalcman conjecture for the class $\mathcal{U}$

TL;DR

This paper proves the Zalcman and generalized Zalcman conjectures for the class of univalent functions , expanding the understanding of their coefficient bounds and geometric properties.

Contribution

The paper establishes the validity of the Zalcman and generalized Zalcman conjectures for the class , a significant step in geometric function theory.

Findings

Confirmed Zalcman conjecture for 

Confirmed generalized Zalcman conjecture for  and specific parameters

Enhanced understanding of coefficient bounds in 

Abstract

Function $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$, normalized, analytic and univalent in the unit disk $\mathbb D=\{z:|z|<1\}$, belongs to the class $\mathcal{U}$. if, and only if, \[ \left| \left(\frac{z}{f(z)}\right)^2 -1\right|<1 \quad\quad (z\in \mathbb D). \] In this paper, we prove the Zalcman and the generalized Zalcman conjecture for the class $\mathcal{U}$ and some values of parameters in the conjectures.