Completely monotonic functions involving the gamma and $q$-gamma functions

Grinshpan, Arcadii Z.; Ismail, Mourad E. H.

doi:10.1090/s0002-9939-05-08050-0

Cited by 88 publications

(45 citation statements)

References 27 publications

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“…In [ 14 ], Grinshpan and Ismail considered the logarithmically complete monotonicity of a more general combination of gamma functions. More precisely, they proved the following theorem.…”

Section: Further Resultsmentioning

confidence: 99%

Complete monotonicity involving some ratios of gamma functions

Yang

Zheng

2017

J Inequal Appl

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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.

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“…In [ 14 ], Grinshpan and Ismail considered the logarithmically complete monotonicity of a more general combination of gamma functions. More precisely, they proved the following theorem.…”

Section: Further Resultsmentioning

confidence: 99%

Complete monotonicity involving some ratios of gamma functions

Yang

Zheng

2017

J Inequal Appl

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“…This result was formally published when revising [35]. Hereafter, this conclusion and its proofs were dug in [9,20,21,48] once and again. Furthermore, in the paper [9], the logarithmically completely monotonic functions on (0, ∞) were characterized as the infinitely divisible completely monotonic functions studied in [24] and all Stieltjes transforms were proved to be logarithmically completely monotonic on (0, ∞), where a function f (x) defined on (0, ∞) is called a Stieltjes transform if it can be of the form…”

Section: Remarksmentioning

confidence: 90%

Complete monotonicity of some functions involving polygamma functions

Guo

2010

Journal of Computational and Applied Mathematics

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In the present paper, we establish necessary and sufficient conditions for the functions $x^\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert$ and $\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert-x\bigl\lvert\psi^{(i+1)}(x+\beta)\bigr\lvert$ respectively to be monotonic and completely monotonic on $(0,\infty)$, where $i\in\mathbb{N}$, $\alpha>0$ and $\beta\ge0$ are scalars, and $\psi^{(i)}(x)$ are polygamma functions.Comment: 14 page

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“…An anonymous referee recommends three papers [7] , [12] , [36] which are said to be highly relevant to the topic of the paper.…”

Section: Four Functions To Be Investigatedmentioning

confidence: 99%

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

Guo

2021

Journal of Mathematical Analysis and Applications

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In this paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums. They establish three new inequalities involving finitely many exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers. They also survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finitely many gamma functions, present complete monotonicity of a linear combination of finitely many trigamma functions, construct a new ratio of finitely many gamma functions, derive monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finitely many gamma functions. Finally, they suggest two linear combinations of finitely many trigamma functions and two ratios of finitely many gamma functions to be investigated.

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Completely monotonic functions involving the gamma and $q$-gamma functions

Cited by 88 publications

References 27 publications

Complete monotonicity involving some ratios of gamma functions

Complete monotonicity involving some ratios of gamma functions

Complete monotonicity of some functions involving polygamma functions

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

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