Monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turán types inequalities

Abstract

In this paper we prove monotonicity of some ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions. The results are also closely connected with Turán type inequalities. In order to obtain main results we apply methods developed for the case of classical Kummer and Gauss hypergeometric functions in [1]-[2].

Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions

“…Recently the authors also proved generalizations of above monotonicity properties and Turán type inequalities for the case of q-hypergeometric functions [26]- [27].…”

Turán Type Inequalities for The $q$-exponential functions

“…Motivated by these applications, the Turán type inequalities have recently come under the spotlight once again and it has been shown that, for example, the classical Gauss and Kummer hypergeometric functions, as well the generalized hypergeometric functions, satisfy naturally some Turán type inequalities [8,9,10]. For deep study about this subject we refer to [2,3,4,11].…”