Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function

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.