For the spectral radius of weighted composition operators with positive weight e φ T α , $${\varphi\in C(X)}$$ , acting in the spaces L p (X, μ) the following variational principle holds $$\ln r(e^\varphi T_\alpha)=\max_{\nu\in M^1_\alpha} \left\{\int\limits_X\varphi d\nu-\frac{\tau_\alpha(\nu)}{p}\right\},$$ where X is a Hausdorff compact space, $${\alpha:X\mapsto X}$$ is a continuous mapping and τ α some convex and lower semicontinuous functional defined on the set $${M^1_\alpha}$$ of all Borel probability and α-invariant measures on X. In other words $${\frac{\tau_\alpha}{p}}$$ is the Legendre– Fenchel conjugate of ln r(e φ T α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form $${A_{\varphi, {\bf c}}= \sum_{k=0}^n e^{c_k} (e^{\varphi} T_{\alpha})^k}$$ , $${{\bf c}=(c_k)\in {\Bbb R}^{n+1}}$$ . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r(A φ,c ) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ, by fixing c.