
Journal of Convex Analysis 20 (2013), No. 1, 243252 Copyright Heldermann Verlag 2013 Convex Conjugates of Analytic Functions of Logarithmically Convex Functionals Krzysztof Zajkowski Institute of Mathematics, University of Bialystok, Akademicka 2, 15267 Bialystok, Poland kryza@math.uwb.edu.pl [Abstractpdf] 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 LegendreFenchel 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 ["LegendreFenchel transform of the spectral exponent of polynomials of weighted composition operators", Positivity, DOI 10.1007/s1111700900236]. Keywords: LegendreFenchel transform, logarithmic convexity, logexponential function, entropy function, spectral radius, weighted composition operators. MSC: 44A15, 47A10, 47B37 [ Fulltextpdf (134 KB)] for subscribers only. 