PreprintCC BY 4.0arXiv
A Counterexample to a Problem of Pommerenke on Convex Functions in the Class Σ
Abstract
Let Σ denote the class of functions f(z) = z + b0 + b1 z−1 + ⋯ that are analytic and univalent in the exterior unit disk Δ* = {z ∈ ℂ : |z| > 1}. In 1962, Ch. Pommerenke proved that if F and G are convex functions in Σ, then every convex linear combination H = λF + (1−λ)G (0 < λ< 1) remains univalent and belongs to Σ. 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 Σ with parameters in ℚ(i). The argument is self-contained and has been formally certified in the Lean 4 proof assistant.
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BibTeX
@misc{guo2026counterexample,
title = {{A Counterexample to a Problem of Pommerenke on Convex Functions in the Class $Σ$}},
author = {Yuankai Guo and Xiaozhe Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04279},
eprint = {2609.04279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04279},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Counterexample to a Problem of Pommerenke on Convex Functions in the Class Σ
AU - Yuankai Guo
AU - Xiaozhe Hu
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04279
UR - https://arxiv.org/abs/2609.04279
AB - Let Σ denote the class of functions f(z) = z + b_0 + b_1 z^−1 + ⋯ that are analytic and univalent in the exterior unit disk Δ^* = {z ∈ ℂ : |z| > 1}. In 1962, Ch. Pommerenke proved that if F and G are convex functions in Σ, then every convex linear combination H = λF + (1−λ)G (0 < λ< 1) remains univalent and belongs to Σ. 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 Σ with parameters in ℚ(i). The argument is self-contained and has been formally certified in the Lean 4 proof assistant.
ER -