Turan's inequalities for the Kummer function in a simultaneous shift of the two parameters

Direct and reverse Turan's inequalities are proved for the confluent hypergeometric function (the Kummer function) viewed as a function of a simultaneous shift in the upper and lower parameters. The reverse Turan inequality is derived from a stronger result on the log-convexity of a function of a sufficiently general form, whose particular case is the Kummer function. Two conjectures above the log-concavity of the Kummer function are formulated. The paper continues the research of a number of authors who studied the log-convexity and log-concavity of hypergeometric functions in parameters. Bibl. 18 titles.