scholarly journals Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Tie-Hong Zhao ◽  
Yu-Ming Chu ◽  
Hua Wang
Keyword(s):  
Gamma Function ◽  
Complete Monotonicity ◽  
Completely Monotonic ◽  
Monotonicity Properties

We prove that the functionfα,β(x)=Γβ(x+α)/xαΓ(βx)is strictly logarithmically completely monotonic on(0,∞)if(α,β)∈{( α,β):1/α≤β≤1, α≠1}∪{(α,β):0<β≤1,φ1(α,β)≥0,φ2(α,β)≥0}and[fα,β(x)]-1is strictly logarithmically completely monotonic on(0,∞)if(α,β)∈{(α,β):0<α≤1/2,0<β≤1}∪{(α,β):1≤β≤1/α≤2,α≠1}∪{(α,β):1/2≤α<1,β≥1/(1-α)}, whereφ1(α,β)=(α2+α-1)β2+(2α2-3α+1)β-αandφ2(α,β)=(α-1)β2+(2α2-5α+2)β-1.

Some complete monotonicity properties for the -gamma function

2013 ◽  
Vol 219 (21) ◽  
pp. 10538-10547 ◽  
Author(s):  
V.B. Krasniqi ◽  
H.M. Srivastava ◽  
S.S. Dragomir
Keyword(s):  
Gamma Function ◽  
Complete Monotonicity ◽  
Monotonicity Properties

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 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.

scholarly journals Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions

2015 ◽  
Vol 3 (2) ◽  
pp. 77
Author(s):  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Exponential Function ◽  
Gamma Function ◽  
Unit Interval ◽  
Complete Monotonicity ◽  
Psi Function ◽  
Completely Monotonic Function ◽  
Stieltjes Function ◽  
Completely Monotonic ◽  
Logarithmically Completely Monotonic Function ◽  
Appl Math

Let \(\Gamma\) and \(\psi=\frac{\Gamma'}{\Gamma}\) be respectively the classical Euler gamma function and the psi function and let \(\gamma=-\psi(1)=0.57721566\dotsc\) stand for the Euler-Mascheroni constant. In the paper, the authors simply confirm the logarithmically complete monotonicity of the power-exponential function \(q(t)=t^{t[\psi(t)-\ln t]-\gamma}\) on the unit interval \((0,1)\), concisely deny that \(q(t)\) is a Stieltjes function, surely point out fatal errors appeared in the paper [V. Krasniqi and A. Sh. Shabani, On a conjecture of a logarithmically completely monotonic function, Aust. J. Math. Anal. Appl. 11 (2014), no.1, Art.5, 5 pages; Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v11n1/V11I1P5.tex], and partially solve a conjecture posed in the article [B.-N. Guo, Y.-J. Zhang, and F. Qi, Refinements and sharpenings of some double inequalities for bounding the gamma function, J. Inequal. Pure Appl. Math. 9 (2008), no.1, Art.17; Available online at http://www.emis.de/journals/JIPAM/article953.html].

scholarly journals Short Remarks on Complete Monotonicity of Some Functions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 537 ◽  
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Gamma Function ◽  
Logarithmic Derivative ◽  
Complete Monotonicity ◽  
Completely Monotonic ◽  
Β Function

In this paper, we show that the functions x m | β ( m ) ( x ) | are not completely monotonic on ( 0 , ∞ ) for all m ∈ N , where β ( x ) is the Nielsen’s β -function and we prove the functions x m − 1 | β ( m ) ( x ) | and x m − 1 | ψ ( m ) ( x ) | are completely monotonic on ( 0 , ∞ ) for all m ∈ N , m > 2 , where ψ ( x ) denotes the logarithmic derivative of Euler’s gamma function.

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 Logarithmically complete monotonicity properties and characterizations of the gamma function

2007 ◽  
Vol 38 (4) ◽  
pp. 313-322
Author(s):  
Ai-Jun Li ◽  
Chao-Ping Chen
Keyword(s):  
Gamma Function ◽  
Complete Monotonicity ◽  
Monotonicity Properties ◽  
Monotonic Properties

n this paper, the logarithmically complete monotonic properties of the functions $ \prod_{i=1}^{n}\frac{\Gamma(x-a_i)}{\Gamma(x-b_i)} $ ,$ \Gamma(x)^{\alpha}\Gamma\Big(x-\sum_{i=1}^n a_i\Big)/\prod_{i=1}^{n}\Gamma(x-a_i) $, and $ x^r(e/x)^x \Gamma(x) $ are obtained. Some characterizations of the gamma function are deduced.

Complete monotonicity of entropy production in chemical reaction

1981 ◽  
Vol 46 (2) ◽  
pp. 452-456
Author(s):  
Milan Šolc
Keyword(s):  
Entropy Production ◽  
Equilibrium Constant ◽  
Chemical Reaction ◽  
Relative Entropy ◽  
Order Reaction ◽  
Complete Monotonicity ◽  
First Order Reaction ◽  
First Order ◽  
Completely Monotonic ◽  
Derivatives Of

The successive time derivatives of relative entropy and entropy production for a system with a reversible first-order reaction alternate in sign. It is proved that the relative entropy for reactions with an equilibrium constant smaller than or equal to one is completely monotonic in the whole definition interval, and for reactions with an equilibrium constant larger than one this function is completely monotonic at the beginning of the reaction and near to equilibrium.

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