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

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