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

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.