Direct and inverse Turán’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 Turán 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.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 383, 2010, pp. 110–125.
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Karp, D.B. Turán’s inequality for the Kummer function of the phase shift of two parameters. J Math Sci 178, 178–186 (2011). https://doi.org/10.1007/s10958-011-0537-x
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DOI: https://doi.org/10.1007/s10958-011-0537-x