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Journal of Convex Analysis 20 (2013), No. 1, 243--252
Copyright Heldermann Verlag 2013

Convex Conjugates of Analytic Functions of Logarithmically Convex Functionals

Krzysztof Zajkowski
Institute of Mathematics, University of Bialystok, Akademicka 2, 15-267 Bialystok, Poland


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].

Keywords: Legendre-Fenchel transform, logarithmic convexity, log-exponential function, entropy function, spectral radius, weighted composition operators.

MSC: 44A15, 47A10, 47B37

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