Weighted Banach Spaces of Holomorphic Functions With Log-Concave Weight Function

TL;DR

This paper explores the structure of weighted Banach spaces of holomorphic functions with log-concave weights, proving their equivalence to spaces with the smallest log-concave majorant, using convex function theory.

Contribution

It establishes the equivalence of weighted Banach spaces with log-concave weights to those with their minimal log-concave majorants, without relying on associated weight theory.

Findings

H_v(G) and H_{v_0}(G) are equal to H_w(G) and H_{w_0}(G) with w as the smallest log-concave majorant of v

The proof is based on convex function theory and properties of the spaces

No use of associated weights theory in the proofs

Abstract

Some theorems on convex functions are proved and an application of these theorems in the theory of weighted Banach spaces of holomorphic functions is investigated, too. We prove that $H_v(G)$ and $H_{v_0}(G)$ are exactly the same spaces as $H_w(G)$ and $H_{w_0}(G)$ where $w$ is the smallest log-concave majorant of $v$. This investigation is based on the theory of convex functions and some specific properties of the weighted banach spaces of holomorphic functions under considaration and we do not use the theory of associated weights.