• Title, Summary, Keyword: completely monotonic function

### INEQUALITIES AND COMPLETE MONOTONICITY FOR THE GAMMA AND RELATED FUNCTIONS

• Chen, Chao-Ping;Choi, Junesang
• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1261-1278
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• 2019
• It is well-known that if ${\phi}^{{\prime}{\prime}}$ > 0 for all x, ${\phi}(0)=0$, and ${\phi}/x$ is interpreted as ${\phi}^{\prime}(0)$ for x = 0, then ${\phi}/x$ increases for all x. This has been extended in [Complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions, J. Math. Anal. Appl. 336 (2007), 812-822]. In this paper, we extend the above result to the very general cases, and then use it to prove some (logarithmically) completely monotonic functions related to the gamma function. We also establish some inequalities for the gamma function and generalize some known results.

### SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS RELATED TO THE GAMMA FUNCTION

• Qi, Feng;Guo, Bai-Ni
• Journal of the Korean Mathematical Society
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• v.47 no.6
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• pp.1283-1297
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• 2010
• In this article, the logarithmically complete monotonicity of some functions such as $\frac{1}{[\Gamma(x+1)]^{1/x}$, $\frac{[\Gamma(x+1)]^{1/x}}{x^\alpha}$, $\frac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}$ and $\frac{[\Gamma(x+\alpha+1)]^{1/(x+\alpha})}{[\Gamma(x+1)^{1/x}}$ for $\alpha{\in}\mathbb{R}$ on ($-1,\infty$) or ($0,\infty$) are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.

### FOUR LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS INVOLVING GAMMA FUNCTION

• Qi, Feng;Niu, Da-Wei;Cao, Jian;Chen, Shou-Xin
• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.559-573
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• 2008
• In this paper, two classes of functions, involving a parameter and the classical Euler gamma function, and two functions, involving the classical Euler gamma function, are verified to be logarithmically completely monotonic in $(-\frac{1}{2},\infty)$ or $(0,\infty)$; some inequalities involving the classical Euler gamma function are deduced and compared with those originating from certain problems of traffic flow, due to J. Wendel and A. Laforgia, and relating to the well known Stirling's formula.

### A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

• Guo, Bai-Ni;Qi, Feng
• Journal of the Korean Mathematical Society
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• v.48 no.3
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• pp.655-667
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• 2011
• A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

### TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION

• Guo, Bai-Ni;Qi, Feng
• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.103-111
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• 2010
• In the present paper, we give two new proofs for the necessary and sufficient condition $\alpha\leq1$ such that the function $x^{\alpha}[lnx-\psi(x)]$ is completely monotonic on (0,$\infty$).

### COMPLETE MONOTONICITY OF A DIFFERENCE BETWEEN THE EXPONENTIAL AND TRIGAMMA FUNCTIONS

• Qi, Feng;Zhang, Xiao-Jing
• The Pure and Applied Mathematics
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• v.21 no.2
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• pp.141-145
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• 2014
• In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function $e^{1/t}$ and the trigamma function ${\psi}^{\prime}(t)$ on (0, ${\infty}$).