An inequality for the modulus of the ratio of two complex gamma functions

Abstract

The Euler gamma function is closely connected with the theory of zeta-functions. We prove a new inequality for the modulus of the ratio of two complex gamma functions Γ(s) /Γ(2 − s), arising in problems of the size of Selberg zeta-functions at places symmetric with respect to the critical line. This inequality, used together with technics of estimation, allows us in a different way re-prove and extend the result of R. Garunkštis and A. Grigutis for the modified Selberg zeta-function.