• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.273-287
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• 2008

In this paper, some logarithmically completely monotonic, strongly completely monotonic and completely monotonic functions related to the gamma, digamma and polygamma functions are established. Several inequalities, whose bounds are best possible, are obtained.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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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.

• 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.

• 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.

• 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.

• Journal of the Korean Wood Science and Technology
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• v.30 no.3
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• pp.97-103
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• 2002

This study was carried out to obtain basic information on racking resistance of shear walls and the factors affecting racking resistance of shear walls. Shear walls constructed by larch lumber nominal 50 mm × 100 mm framing and various sheathing materials were tested by applying monotonic and cyclic load functions. Shear walls with various stud spacing such as 305 mm, 406 mm, and 610 mm were tested under both of monotonic and cyclic loads and shear walls with various aspect (height-width) ratios were tested under cyclic load functions. The effect of hold-down connectors in shear walls was also tested under cyclic load functions. Racking resistance of shear walls has very close linear relation with stud spacing and width of shear walls. The ultimate racking strength of shear walls was reached at around or before the displacement of 20 mm. It was proposed in this study that the minimum racking strength and minimum width for shear wall be 500 kgf and 900 mm, respectively. Load-displacement curves obtained by racking tests under monotonic load functions can be represented by three straight line segments. Under cyclic load functions, envelope curves can be divided into three sections that can be represented by straight lines and the third section showed almost constant or decreasing slope.

• 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.

• Proceedings of the Korean Information Science Society Conference
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• 2011.06a
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• pp.344-347
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• 2011

In this paper, we propose the power control problem for cognitive radio networks (CRNs) that maximizes the total utility of the secondary users (SUs). We use the interference temperature constraints to protect the primary users (PUs). The utility functions of SUs can be any increasing functions. We formulate the power control problem as monotonic optimization that can be solved in centralization to achieve the global optimum.

• Bulletin of the Korean Mathematical Society
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• v.44 no.3
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• pp.523-536
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• 2007

Sharp error estimates in approximating the Stieltjes integral with bounded integrands and bounded integrators respectively, are given. Applications for three point quadrature rules of n-time differentiable functions are also provided.

• Journal of Korean Society for Quality Management
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• v.25 no.4
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• pp.99-105
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• 1997

We consider a nonparametric estimation of the renewal function. In this paper, we suggest modified methods for Frees's estimator to enhance the efficiency. The methods are based on a piecewise linearization and on the fact that the bounded monotonic functions converging pointwise to the bounded monotonic continuous function converge uniformly. In a simulation study, we show that the modified methods have the better efficiency than that introduced by Frees.