Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

Qi, Feng; Niu, Da-Wei; Lim, Dongkyu; Guo, Bai‐Ni

doi:10.2298/aadm191111033q

Cited by 21 publications

(20 citation statements)

References 28 publications

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“…In this paper, we use the notation For more information and recent developments of the gamma function Γ(z) and its logarithmic derivatives ψ (n) (z) for n ≥ 0, please refer to [1,Chapter 6], [25,Chapter 3], or recently published papers [14,18,20,21,31] and closely related references therein.…”

Section: Motivations and Main Resultsmentioning

confidence: 99%

“…Recall from [11, Chapter XIII], [23, Chapter 1], [30, Chapter IV], and recently published papers [14,18,20,21] that (1) a function q(x) is said to be completely monotonic on an interval I if it is infinitely differentiable and (−1) n q (n) (x) ≥ 0 for n ≥ 0 on I. (2) a positive function q(x) is said to be logarithmically completely monotonic on an interval I ⊆ R if it is infinitely differentiable and its logarithm ln f (x) satisfies (−1) k [ln q(x)] (k) ≥ 0 for k ∈ N on I.…”

Section: Lemmasmentioning

confidence: 99%

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Increasing property and logarithmic convexity of functions involving Riemann zeta function

Guo¹,

Qi²

2022

Preprint

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Let α > 0 be a constant, let ℓ ≥ 0 be an integer, and let Γ(z) denote the classical Euler gamma function. With the help of the integral representation for the Riemann zeta function ζ(z), by virtue of a monotonicity rule for the ratio of two integrals with a parameter, and by means of complete monotonicity and another property of the function 1 e t −1 and its derivatives, the authors present that,(1) for ℓ ≥ 0, the function 2010

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Section: Motivations and Main Resultsmentioning

confidence: 99%

Section: Lemmasmentioning

confidence: 99%

Increasing property and logarithmic convexity of functions involving Riemann zeta function

Guo¹,

Qi²

2022

Preprint

Self Cite

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show abstract

“…There are a number of papers and mathematicians dedicated to investigation of complete monotonicity of some functions involving the gamma and polygamma functions. For more information and details, please refer to the papers [2,9,11,15] and closely related references therein.…”

Section: Motivationsmentioning

confidence: 99%

Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

Han

Yin

2020

Preprint

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In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the author (1) presents the decreasing monotonicity of a ratio constituted via three derivatives of a function involving trigamma function; (2) discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. These results conform previous guesses posed by the author.

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“…Motivated by the papers [15,28] and related texts in the survey article [19], by establishing the inequality…”

Section: Motivationmentioning

confidence: 99%

A ratio of many gamma functions and its properties with applications

Qi,

Li,

et al. 2019

Preprint

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In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.

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Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

Cited by 21 publications

References 28 publications

Increasing property and logarithmic convexity of functions involving Riemann zeta function

Increasing property and logarithmic convexity of functions involving Riemann zeta function

Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

A ratio of many gamma functions and its properties with applications

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