Turán’s inequality for the Kummer function of the phase shift of two parameters

Abstract

Direct and inverse Turan’s inequalities are proved for the confluent hypergeometric function (the Kummer function) viewed as a function of the phase shift of the upper and lower parameters. The inverse Turan inequality is derived from a stronger result on the log-convexity of a function of sufficiently general form, a particular case of which is the Kummer function. Two conjectures on the log-concavity of the Kummer function are formulated. The paper continues the previous research on the log-convexity and log-concavity of hypergeometric functions of parameters conducted by a number of authors. Bibliography: 18 titles.