# Value regions of univalent self-maps with two boundary fixed points

## TL;DR

This paper determines the precise set of possible values for the evaluation of univalent self-maps of the unit disk with specific boundary fixed points and derivative conditions, advancing understanding of boundary behavior in complex analysis.

## Contribution

It provides the exact value region for the point evaluation functional for a class of univalent self-maps with boundary fixed points and specified derivative conditions.

## Key findings

- Exact value region $\\mathcal V(z_0,T)$ characterized.
- Results apply to holomorphic injective self-maps with boundary fixed points.
- Enhances understanding of boundary behavior of univalent functions.

## Abstract

In this paper we find the exact value region $\mathcal V(z_0,T)$ of the point evaluation functional $f\mapsto f(z_0)$ over the class of all holomorphic injective self-maps $f:\mathbb D\to\mathbb D$ of the unit disk $\mathbb D$ having a boundary regular fixed point at $\sigma=-1$ with $f'(-1)=e^{T}$ and the Denjoy - Wolff point at $\tau=1$.