Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions

Abstract: In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from 1993. In this paper we prove these conjectures in the more general setting for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turan--type inequalities.