• 대한수학회지
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• 제58권1호
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• pp.133-147
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• 2021

Our aim in this paper is to derive several new monotonicity properties and functional inequalities of some functions involving the q-gamma, q-digamma and q-polygamma functions. More precisely, some classes of functions involving the q-gamma function are proved to be logarithmically completely monotonic and a class of functions involving the q-digamma function is showed to be completely monotonic. As applications of these, we offer upper and lower bounds for this special functions and new sharp upper and lower bounds for the q-analogue harmonic number harmonic are derived. Moreover, a number of two-sided exponential bounding inequalities are given for the q-digamma function and two-sided exponential bounding inequalities are then obtained for the q-tetragamma function.

• 대한수학회논문집
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• 제34권4호
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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.

• 대한수학회지
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• 제47권6호
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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.

• 대한수학회지
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• 제45권2호
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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.

• 대한수학회지
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• 제48권3호
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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.

• 대한수학회보
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• 제47권1호
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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,$\infty$).

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제21권2호
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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, ${\infty}$).