An extension of an inequality for ratios of gamma functions

Abstract

In this paper, we prove that for $x+y>0$ and $y+1>0$ the inequality {equation*} \frac{[Γ(x+y+1)/Γ(y+1)]^{1/x}}{[Γ(x+y+2)/Γ(y+1)]^{1/(x+1)}} <\biggl(\frac{x+y}{x+y+1}\biggr)^{1/2} {equation*} is valid if $x>1$ and reversed if $x<1$ and that the power $\frac{1}{2}$ is the best possible, where $\Gamma (x)$ 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.].

8 pages

References in corpus (3)

• Monotonicity and logarithmic convexity relating to the volume of the unit ball
• An Inequality for Ratios of Gamma Functions
• Bounds for the ratio of two gamma functions--From Wendel's and related inequalities to logarithmically completely monotonic functions

Cited by in corpus (8)

• Bounds for the ratio of two gamma functions--From Wendel's limit to Elezović-Giordano-Pečarić's theorem
• Refinements of lower bounds for polygamma functions
• A logarithmically completely monotonic function involving the ratio of gamma functions
• From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions
• Monotonicity and logarithmic convexity relating to the volume of the unit ball
• Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic
• Two monotonic functions involving gamma function and volume of unit ball
• Complete monotonicity of a function involving a ratio of gamma functions and applications