On monotonicity of ratios of some \(q\)-hypergeometric functions. (English) Zbl 1474.33068

Mehrez, Khaled; Sitnik, Sergei M.

Mat. Vesn. 68, No. 3, 225-231 (2016).

Summary: 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 the main results we apply methods developed for the case of classical Kummer and Gauss hypergeometric functions in [the second author, “Inequalities for the exponential remainder”, Preprint; “Conjectures on monotonicity of ratios of Kummer and Gauss hypergeometric functions”, RGMIA, Res. Group. Math. Inequal. Appl. 17, Paper No. 107, 4 p. (2014)].

MSC:

33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33C05	Classical hypergeometric functions, \({}_2F_1\)

Keywords:

Kummer functions; Gauss hypergeometric functions; \(q\)-Kummer confluent hypergeometric functions; \(q\)-hypergeometric functions; Turán-type inequalities