A Counterexample to a Problem of Pommerenke on Convex Functions in the Class Σ

Abstract

A Counterexample to a Problem of Pommerenke on Convex Functions in the Class Σ This preprint studies Problem 6.10 in W. K. Hayman and E. F. Lingham's Research Problems in Function Theory. The paper presents an explicit construction involving the functions \[ F(z)=z+\alpha z^{-1}, \qquad \alpha=\frac{8}{9}+\frac{4}{9}i, \] \[ G'(z)=\left(1-\beta z^{-3}\right)^{2/3}, \] \[ H(z)=\frac{3}{5}F(z)+\frac{2}{5}G(z). \] The construction satisfies the required univalence properties, while H fails the relevant convexity condition at z0 = 400/399. More precisely, \[ \operatorname{Re}\left( 1+\frac{z_0H''(z_0)}{H'(z_0)} \right) = -\frac{54160961609}{69013985609}<0. \] The accompanying Lean 4 formalization is available at https://github.com/Theophilus1030/Pommerenke . The formalization uses Lean 4 and Mathlib.