Legendre-Fenchel Transform of the Spectral Exponent of Analytic Functions of Weighted Composition Operators

Let $f$ be an analytic function with nonnegative coefficients. We derive the Legendre-Fenchel transform of the composition $\ln \circ f \circ \exp$ as a function depending on coefficients of $f$. We apply it to obtain the variational principle for the spectral exponent of operators that can be written as analytic functions of the weighted composition operators.