Let $f_{ \bf c}(r)=\sum_{ n=0}^\infty e^{ c_n}r^n$ be an analytic function; ${ \bf c}=(c_n)\in l_\infty$. 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 $\hat{ \lambda}({ \bf c},\phi)=\ln f_{ \bf c}(e^{ \lambda(\phi)})$, where $\lambda(\phi)=\ln r(\phi)$ ($\phi\in L$). In this manner we generalize to the infinite case Theorem 3.1 from \cite{ OZ1}.
Categories: math.FA