Convexity of the embedding parameter sets of some analytic function spaces

Abstract

In this note, we study the geometric structure of the parameter sets governing continuous embeddings between weighted Bergman-Orlicz spaces. First, for a fixed pair of growth functions, we show that the set of admissible weight exponents $(\alpha, \beta)$ is convex, provided the growth functions satisfy specific log-convexity and log-concavity conditions of the inverses. Second, we consider the dual problem where the weight exponents are fixed. We prove that the collection of growth function pairs that yield such an embedding is log-convex under a natural interpolation of their inverses. We then obtain interpolated embeddings between Bergman-Orlicz spaces.