Bounds for the ratio of two gamma functions--From Wendel's limit to Elezović-Giordano-Pečarić's theorem

Abstract

In the survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some known results, including Wendel's, Gurland's, Kazarinoff's, Gautschi's, Watson's, Chu's, Lazarević-Lupaş's, Kershaw's and Elezović-Giordano-Pečarić's inequalities, claim, monotonic and convex properties. On the other hand, we introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years.

16 pages

References in corpus (12)

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Cited by in corpus (18)

• A logarithmically completely monotonic function involving the ratio of gamma functions
• Some best approximation formulas and inequalities for Wallis ratio
• Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function
• Completely monotonic degree of a function involving the tri- and tetra-gamma functions
• From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions
• Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic
• The function $(b^x-a^x)/x$: Ratio's properties
• Some integral representations and properties of Lah numbers
• A completely monotonic function involving the gamma and tri-gamma functions
• Asymptotic formulas and inequalities for gamma function in terms of tri-gamma function
• Some inequalities for the trigamma function in terms of the digamma function
• A Berry-Esseen theorem for Pitman's $\alpha$-diversity
• On the quantum mechanical derivation of the Wallis formula for $\pi$
• De Finetti's theorem: rate of convergence in Kolmogorov distance
• An inequality involving the gamma and digamma functions
• Some new inequalities for the gamma function
• The fastest possible continued fraction approximations of a class of functions
• Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions