scholarly journals A Completely Monotonic Function Used in an Inequality of Alzer

2012 ◽  
Vol 12 (1) ◽  
pp. 329-341 ◽  
Author(s):  
Christian Berg ◽  
L. Henrik
Keyword(s):  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Completely Monotonic

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>

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)!!}

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 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,∞).

A Completely Monotonic Function: 11140

2006 ◽  
Vol 113 (8) ◽  
pp. 764
Author(s):  
Walther Janous ◽  
Rolf Richberg
Keyword(s):  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Completely Monotonic

scholarly journals Integral Representations for Multivariate Logarithmic Polynomials

2017 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Integral Representation ◽  
Uniform Convergence ◽  
Generating Function ◽  
Integral Representations ◽  
Monotonic Function ◽  
Bernstein Function ◽  
Completely Monotonic Function ◽  
Bernstein Functions ◽  
Completely Monotonic ◽  
Integral Formulas

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a L&eacute;vy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

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.

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