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

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 $\tau=1$.