Asymptotic expansions of integral means and applications to the ratio of gamma functions

Integral means are important class of bivariate means. In this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean. It is based on explicit solving the equation of the form B(A(x))=C(x)B(A(x))=C(x), where asymptotic expansions of B and C are known. The results are illustrated by calculation of some important integral means connected with gamma function.