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 A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

2015 ◽  
Vol 3 (4) ◽  
pp. 140 ◽  
Author(s):  
Fang-Fang Liu ◽  
Xiao-Ting Shi ◽  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Catalan Numbers ◽  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Completely Monotonic ◽  
Logarithmically Completely Monotonic Function ◽  
And Function

<p><span>In the paper, the authors find necessary conditions and sufficient conditions for a function involving the gamma function and originating from investigation of properties of the Catalan numbers and function in combinatorics to be logarithmically completely monotonic.</span></p>

INEQUALITIES ASSOCIATED WITH RATIOS OF GAMMA FUNCTIONS

2018 ◽  
Vol 97 (3) ◽  
pp. 453-458
Author(s):  
JENICA CRINGANU
Keyword(s):  
Gamma Function ◽  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Appl Math

We use properties of the gamma function to estimate the products$\prod _{k=1}^{n}(4k-3)/4k$and$\prod _{k=1}^{n}(4k-1)/4k$, motivated by the work of Chen and Qi [‘Completely monotonic function associated with the gamma function and proof of Wallis’ inequality’,Tamkang J. Math.36(4) (2005), 303–307] and Morticiet al.[‘Completely monotonic functions and inequalities associated to some ratio of gamma function’,Appl. Math. Comput.240(2014), 168–174].

scholarly journals A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums

2007 ◽  
Vol 48 (4) ◽  
pp. 523-532 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Divided Difference ◽  
Complete Monotonicity ◽  
Monotonic Function ◽  
Psi Function ◽  
Completely Monotonic Function ◽  
Completely Monotonic ◽  
Equivalent Relation ◽  
Equivalent Inequality

AbstractIn this paper, a function involving the divided difference of the psi function is proved to be completely monotonic, a class of inequalities involving sums is found, and an equivalent relation between complete monotonicity and one of the class of inequalities is established.

scholarly journals COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE DIVIDED DIFFERENCE OF PSI FUNCTIONS

2013 ◽  
Vol 88 (2) ◽  
pp. 309-319 ◽  
Author(s):  
FENG QI ◽  
PIETRO CERONE ◽  
SEVER S. DRAGOMIR
Keyword(s):  
Divided Difference ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Divided Differences ◽  
Psi Function ◽  
Gamma Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

scholarly journals Completely monotonic function associated with the Gamma functions and proof of Wallis' inequality

2005 ◽  
Vol 36 (4) ◽  
pp. 303-307 ◽  
Author(s):  
Chao-Ping Chen ◽  
Feng Qi
Keyword(s):  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Natural Numbers ◽  
Gamma Functions ◽  
Completely Monotonic ◽  
Logarithmically Completely Monotonic Function

We prove: (i) A logarithmically completely monotonic function is completely monotonic. (ii) For $ x>0 $ and $ n=0, 1, 2, \ldots $, then$$ (-1)^{n}\left(\ln \frac{x \Gamma(x)}{\sqrt{x+1/4}\,\Gamma(x+1/2)}\right)^{(n)}>0. $$(iii) For all natural numbers $ n $, then$$ \frac1{\sqrt{\pi(n+4/ \pi-1)}}\leq \frac{(2n-1)!!}{(2n)!!}

scholarly journals A completely monotonic function involving the tri- and tetra-gamma functions

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
Bai-Ni Guo ◽  
Jiao-Lian Zhao ◽  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Gamma Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
The Difference

AbstractThe di-gamma function ψ(x) is defined on (0,∞) by $\psi (x) = \frac{{\Gamma '(x)}} {{\Gamma (x)}} $ and ψ (i)(x) for i ∈ ℕ denote the polygamma functions, where Γ(x) is the classical Euler’s gamma function. In this paper we prove that a function involving the difference between [ψ′(x)]2 + ψ″(x) and a proper fraction of x is completely monotonic on (0,∞).

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 Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function

2021 ◽  
pp. 14-14
Author(s):  
Feng Qi
Keyword(s):  
Exponential Function ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic. These results generalize corresponding known ones.

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.

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