Abstract
In the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions {Gamma }_{q}(x+1)/{Gamma }_{q}(x+s) for all real number s and 0< qneq1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all 0< s<1 when letting qto 1 and a new inequality for all s>1.
Highlights
There exist several published articles providing different upper and lower bounds for the ratio of two gamma functions (x + 1)/ (x + s), x > 0, s ∈ (0, 1) where is the gamma function; we refer to [3, 4] and the references given therein
We introduce sharp upper and lower bounds to the ratio of two q-gamma functions q(x + 1)/ q(x + s) for all real number s and 0 < q = 1 in terms of the q-digamma function
Ismail and Muldoon [1] presented the q-analogue of the right hand side of (1.1) for the q-gamma function: Let 0 < q < 1 and 0 < s < 1
Summary
There exist several published articles providing different upper and lower bounds for the ratio of two gamma functions (x + 1)/ (x + s), x > 0, s ∈ (0, 1) where is the gamma function; we refer to [3, 4] and the references given therein. For the classical gamma function, our results give a Kershaw inequality for all 0 < s < 1 when letting q → 1 and a new inequality for all s > 1. Q(x + s) hold true for all x > –s where q is the q-gamma function defined for all positive real x as [6, 7]
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