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

Let

be an analytic function with nonnegative coefficients. We derive the Legendre-Fenchel transform of the composition

\ln \circ f \circ \exp ln ⁡ ∘ f ∘ exp ⁡

as a function depending on coefficients of

f 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.