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An extension of an inequality for ratios of gamma functions
In this paper, we prove that for and the inequality
{equation*}
\frac{[\Gamma(x+y+1)/\Gamma(y+1)]^{1/x}}{[\Gamma(x+y+2)/\Gamma(y+1)]^{1/(x+1)}}
1x<1\frac12\Gamma(x)$ is the Euler gamma function. This extends the result in [Y. Yu,
\textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl.
\textbf{352} (2009), no.~2, 967\nobreakdash--970.] and resolves an open problem
posed in [B.-N. Guo and F. Qi, \emph{Inequalities and monotonicity for the
ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2,
239\nobreakdash--247.].Comment: 8 page
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