Convex Conjugates of Analytic Functions of Logarithmically Convex Functionals

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 $$\widehat{\lambda}({\bf c},\varphi)= \ln f_{\bf c}(e^{\lambda(\varphi)})\,$$ where $\lambda(\varphi)=\ln r(\varphi)$ ($\varphi\in L$). In this manner we generalize to the infinite case Theorem 3.1 of the paper of U. Ostaszewska and K. Zajkowski ["Legendre-Fenchel transform of the spectral exponent of polynomials of weighted composition operators", Positivity, DOI 10.1007/s11117-009-0023-6].