scholarly journals On Some Complete Monotonicity of Functions Related to Generalized k − Gamma Function

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Hesham Moustafa ◽  
Hanan Almuashi ◽  
Mansour Mahmoud
Keyword(s):  
Lower Bounds ◽  
Gamma Function ◽  
Logarithmic Derivative ◽  
Complete Monotonicity ◽  
Upper And Lower Bounds ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

In this paper, we presented two completely monotonic functions involving the generalized k − gamma function Γ k x and its logarithmic derivative ψ k x , and established some upper and lower bounds for Γ k x in terms of ψ k x .

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

2020 ◽  
pp. 33-33
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Dongkyu Lim ◽  
Bai-Ni Guo
Keyword(s):  
Gamma Function ◽  
Complete Monotonicity ◽  
Bernoulli Trials ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Multivariate Beta ◽  
Multinomial Coefficients

In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.

scholarly journals Complete monotonicity of a difference defined by four derivatives of a function containing trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Integral Inequality ◽  
Sufficient Conditions ◽  
Complete Monotonicity ◽  
Logarithmic Convexity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, logarithmic convexity of the gamma function, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic. Moreover, by virtue of Cebysev integral inequality, the author presents logarithmic convexity of the sequence of polygamma functions.

On the asymptotic expansion of the logarithm of Barnes triple gamma function II

2017 ◽  
Vol 120 (2) ◽  
pp. 291
Author(s):  
Stamatis Koumandos ◽  
Henrik L. Pedersen
Keyword(s):  
Asymptotic Expansion ◽  
Gamma Function ◽  
Positive Order ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

The remainders in an asymptotic expansion of the logarithm of Barnes triple gamma function give rise to completely monotonic functions of positive order.

Completely monotonic functions related to the q-gamma and the q-trigamma functions

2015 ◽  
Vol 13 (02) ◽  
pp. 125-134 ◽  
Author(s):  
Ahmed Salem
Keyword(s):  
Gamma Function ◽  
Infinite Class ◽  
Positive Real ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Monotonic Functions ◽  
Polygamma Function ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

In this paper, two completely monotonic functions involving the q-gamma and the q-trigamma functions where q is a positive real, are introduced and exploited to derive sharp bounds for the q-gamma function in terms of the q-trigamma function. These results, when letting q → 1, are shown to be new. Also, sharp bounds for the q-digamma function in terms of the q-tetragamma function are derived. Furthermore, an infinite class of inequalities for the q-polygamma function is established.

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