The following inequality relating to the ratio of the gamma functions \alpha log x \le x - \frac{\Gamma(x)}{\Gamma(x+1/x)} ; where \alpha is a suitable constant, is established for every x > 0. This inequality gives a contribution to the recent results, proved by several authors, involving the functions \Gamma (x) and \Gamma(1/x). It also gives an alternative proof of a conjecture formulated by D. Kershaw and recently proved by G.J.O. Jameson and T.P. Jameson [5].
Laforgia, A.I.A., Natalini, P. (2014). On an inequality for the ratio of gamma functions. MATHEMATICAL INEQUALITIES & APPLICATIONS, 17(4), 1591-1599 [10.7153/mia-17-117].
On an inequality for the ratio of gamma functions
LAFORGIA, Andrea Ivo Antonio;NATALINI, PIERPAOLO
2014-01-01
Abstract
The following inequality relating to the ratio of the gamma functions \alpha log x \le x - \frac{\Gamma(x)}{\Gamma(x+1/x)} ; where \alpha is a suitable constant, is established for every x > 0. This inequality gives a contribution to the recent results, proved by several authors, involving the functions \Gamma (x) and \Gamma(1/x). It also gives an alternative proof of a conjecture formulated by D. Kershaw and recently proved by G.J.O. Jameson and T.P. Jameson [5].File in questo prodotto:
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