On monotonicity of ratios of some $q$-hypergeometric functions

Khaled Mehrez, Sergei M. Sitnik

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 [S.M. Sitnik, Inequalities for the exponential remainder, preprint, Institute of Automation and Control Process, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok 1993 (in Russian)] and [S.M. Sitnik, Conjectures on Monotonicity of Ratios of Kummer and Gauss Hypergeometric Functions, RGMIA Research Report Collection 17 (2014), Article 107].