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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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