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Comparison of a pair of upper bounds for a ratio of gamma functions. (English) Zbl 1045.33004
For the ratio $\Gamma(x+1)/\Gamma(x+s)$, $0< s< 1$, {\it J. D. Kečkič} and {\it P. M. Vasič} [Publ. Inst. Math., Nouv. Sér. 11(25), 107--114 (1971; Zbl 0222.26011)] have established the upper bound $(x+1)^{x+1/2}/ (x+s)^{x+s-1/2}$, $x> 1/2$. For the same ratio, {\it D. Kershaw} [Math. Comp. 41, 607--611 (1983; Zbl 0536.33002)] have proved that $\exp[(1-s) \psi(s+ (s+1)/2)]$ is an upper bound. He noted further that numerical evidence supports the view that the latter bound is closer than the one due to Kečkič and Vasič when $x>.07$. Here this is proved and some supplementary results furnished.
33B15Gamma, beta and polygamma functions