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On the \(q\)-analogue of gamma functions and related inequalities. (English) Zbl 1080.33014
Let \(0<q<1,\;a\geq{1}\) and \(x\in{[0,1]}.\) Then the following double inequality for the Jackson \(\Gamma_{q}\) function obtains: \[ \frac{1}{\Gamma_{q}(1+a)}\leq\frac{(\Gamma_{q}(1+x))^{a}}{\Gamma_{q}(1+ax)}\leq{1}. \] The very elegant proof uses a series expansion for the logarithmic derivative of \(\Gamma_{q}\) known from [M. Ismail and M. Muldoon, Int. Ser. Numer. Math. 119, 309–323 (1994; Zbl 0819.33001)]. The convexity of the \(\Gamma_{q}\) function has also been discussed in [N. Elezovic, C. Giordano and J. Pecaric, Rend. Circ. Mat. Palermo, II. Ser. 48, No. 2, 285–298 (1999; Zbl 0940.33008)].

MSC:
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
33B15 Gamma, beta and polygamma functions
26D20 Other analytical inequalities
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