A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE GAMMA 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>

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

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.36.2005.101 ◽

2005 ◽

Vol 36 (4) ◽

pp. 303-307 ◽

Cited By ~ 12

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)!!}

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INEQUALITIES ASSOCIATED WITH RATIOS OF GAMMA FUNCTIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000138 ◽

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].

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

Mathematica Slovaca ◽

10.2478/s12175-013-0109-2 ◽

2013 ◽

Vol 63 (3) ◽

Cited By ~ 7

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,∞).

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

Global Journal of Mathematical Analysis ◽

10.14419/gjma.v3i2.4605 ◽

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].

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