A Counterexample to a Problem of Pommerenke on Convex Functions in the Class $\Sigma$

Abstract

Let $\Sigma$ denote the class of functions $f(z)=z+b_0+b_1{z}^{-1}+\cdots$ that are analytic and univalent in the exterior unit disk ${\Delta }^{\ast }=\{z\in \mathbb{C}:|z|>1\}$. In 1962, Ch. Pommerenke proved that if $F$ and $G$ are convex functions in $\Sigma$, then every convex linear combination $H=\lambda F+(1-\lambda )G$ ($0<\lambda <1$) remains univalent and belongs to $\Sigma$. In Hayman's problem collection (Research Problems in Function Theory, Problem 6.10), Pommerenke raised the question of whether $H$ is necessarily also a convex function. We resolve this question in the negative by constructing an explicit counterexample in $\Sigma$ with parameters in $\mathbb{Q}(i)$. The argument is self-contained and has been formally certified in the Lean 4 proof assistant.

6 pages, 1 table. Formal verification in Lean 4 available at https://github.com/Theophilus1030/Pommerenke