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

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.

References in corpus (1)

• A short geometric proof of the Zalcman and Bieberbach conjectures