Complex inequalities involving sums of holomorphic selfmaps of the unit disk and some experimental conjectures

Abstract

We prove that for every $p>1$ the set

$$\begin{aligned} D_p=\left\{ c\in {{\mathbb{C}}}\;\big |\; \forall z\in \overline{{{\mathbb{D}}}},\; \left| \frac{1+z}{2}+ c\left( \frac{1-z}{2}\right) ^p\right| \le 1\right\} \end{aligned}$$

is an intersection of closed disks, in particular a closed convex set, and exactly a disk for $p=2$. It is also shown that

$$\begin{aligned} {{\mathbb{D}}}\cup \{1\}\subseteq \bigcup _{p\ge 2} D_p\subseteq \overline{{{\mathbb{D}}}}. \end{aligned}$$