Convex conjugates of analytic functions of logarithmically convex functionals

Abstract

Let $f_{\mathbf{c}}(r)={\sum }_{n=0}^{\infty }{e}^{c_n}{r}^n$ be an analytic function; $\mathbf{c}=(c_n)\in l_{\infty }$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this paper we derive a formula on the Legendre-Fenchel transform of a functional $\hat{\lambda }(\mathbf{c},\varphi )=\ln f_{\mathbf{c}}(e^{\lambda (\varphi )})$, where $\lambda (\varphi )=\ln r(\varphi )$ ($\varphi \in L$). In this manner we generalize to the infinite case Theorem 3.1 from \cite{OZ1}.