Abstract
In this paper, by using the properties of an auxiliary function, we mainly present the necessary and sufficient conditions for various ratios constructed by gamma functions to be respectively completely and logarithmically completely monotonic. As consequences, these not only unify and improve certain known results including Qi’s and Ismail’s conclusions, but also can generate some new inequalities.
Highlights
1 Introduction It is well known that the classical Euler gamma function is defined by
Various bounds concerning certain ratios of gamma functions have been researched by many mathematicians
5 Conclusions In this paper, we investigate the complete monotonicity of various ratios structured by gamma functions including
Summary
It is well known that the classical Euler gamma function is defined by ∞. ). for x > , and its logarithmic derivative ψ(x) = (x)/ (x) is known as the psi or digamma function, while ψ , ψ , . Various bounds concerning certain ratios of gamma functions have been researched by many mathematicians. Wendel [ ] showed that, for s ∈ ( , ) and x > , the following double inequalities hold:. Based on a different motivation from Wendel [ ], Gautschi [ ] in independently got the two double inequalities: for n ∈ N and ≤ s ≤ , one has e(s– )ψ(n+ ) < (n + s) < ns– , (n + ) –s (n + s) –s < n+ (n + ) n
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