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 $\Delta^* = \{z \in \mathbb{C} : |z| > 1\}$. 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.