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Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric \(k\)-function. (English) Zbl 1498.26032

Summary: In the paper, the authors present some inequalities involving the extended gamma function and the Kummer confluent hypergeometric \(k\)-function via some classical inequalities such as Chebychev’s inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent \(k\)-hypergeometric function.

MSC:

26D07 Inequalities involving other types of functions
26D15 Inequalities for sums, series and integrals
33B15 Gamma, beta and polygamma functions
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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