Abstract
Let varGamma (x) denote the classical Euler gamma function. The logarithmic derivative psi (x)=[ln varGamma (x)]'=frac{varGamma '(x)}{ varGamma (x)}, psi '(x), and psi ''(x) are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function [psi '(x)]^{2}+ psi ''(x), its q-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.
Highlights
Qi and Agarwal Journal of Inequalities and Applications (2019) 2019:36 converges for x ∈ [0, ∞)
Speaking, a function f (x) is completely monotonic on [0, ∞) if and only if it is a Laplace transform of a bounded and non-decreasing measure α(t). This is one of many reasons why mathematicians have been studying the class of completely monotonic functions for so many years
6.1 Monotonicity and convexity In [34, Theorem 2], by virtue of the inequality (2.6), it was proved that the function eψ(x+1) – x is strictly decreasing and strictly convex on (–1, ∞)
Summary
In [83, 108], the authors alternatively generalized the function δs,t(x) in (5.5) as δs,t;λ. On (–α, ∞) for s, t, λ ∈ R and α = min{s, t}, and discovered the following necessary and sufficient conditions of complete monotonicity: 1. 5.10 Complete monotonicity of q-analogs In [67, 105, 135], the second function in (5.1) was generalized to the q-analogs (1 – q)qx 1 (1 – q)qx 2 fq(x) = ψq(x) –. + (1 – q)(3 – q) 2 on (0, ∞) for 0 < q < 1 and these two q-analogs were proved to be completely monotonic on (0, ∞). 10 – 5q + q2 are completely monotonic with respect to x ∈ (0, ∞)
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