On the convolution of convex 2-gons

We study the convolution of functions of the form f alpha ( z) := /I 1+ z \ alpha - 1 1 -z , 2 alpha which map the open unit disk of the complex plane onto polygons of 2 edges when alpha is an element of (0 , 1). Inspired by a work of Cima, we study the limits of convolutions of finitely many f alpha and the convolution of arbitrary unbounded convex mappings. The analysis for the latter is based on the notion of angle at infinity , which provides an estimate for the growth at infinity and determines whether the convolution is bounded or not. A generalization to an arbitrary number of factors shows that the convolution of n randomly chosen unbounded convex mappings has a probability of 1 /n! of remaining unbounded. We provide the precise asymptotic behavior of the coefficients of the functions f alpha . (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).