Let f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n f c ( r ) = ∑ n = 0 ∞ e c n r n be an analytic function; {\bf c}=(c_n)\in l_\infty c = ( c n ) ∈ l ∞ . We assume that r r is some logarithmically convex and lower semicontinuous functional on a locally convex topological space L 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)})\, λ ^ ( c , φ ) = ln ⁡ f c ( e λ ( φ ) )   where \lambda(\varphi)=\ln r(\varphi) λ ( φ ) = ln ⁡ r ( φ ) ( \varphi\in L φ ∈ 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].