• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.973-983
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• 2018

Fuzzy linear regression model has been widely studied with many successful applications but there have been only a few studies on the fuzzy regression model with monotonic response function as a generalization of the linear response function. In this paper, we propose the fuzzy regression model with the monotonic response function and the algorithm to construct the proposed model by using ${\alpha}-level$ set of fuzzy number and the resolution identity theorem. To estimate parameters of the proposed model, the least squares (LS) method and the least absolute deviation (LAD) method have been used in this paper. In addition, to evaluate the performance of the proposed model, two performance measures of goodness of fit are introduced. The numerical examples indicate that the fuzzy regression model with the monotonic response function is preferable to the fuzzy linear regression model when the fuzzy data represent the non-linear pattern.

• 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 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.

• 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,$\infty$).

• 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.

• Journal of Korea Society of Industrial Information Systems
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• v.15 no.5
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• pp.179-185
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• 2010

In a sensitivity analysis, an uncertainty importance measure is often used to assess how much uncertainty of an output is attributable to the uncertainty of an input, and thus, to identify those inputs whose uncertainties need to be reduced to effectively reduce the uncertainty of output. A function is called monotonic if the output is either increasing or decreasing with respect to any of the inputs. In this paper, for a monotonic function, we propose a method for evaluating the measure which assesses the expected percentage reduction in the variance of output due to ascertaining the value of input. The proposed method can be applied to the case that the output is expressed as linear and nonlinear monotonic functions of inputs, and that the input follows symmetric and asymmetric distributions. In addition, the proposed method provides a stable uncertainty importance of each input by discretizing the distribution of input to the discrete distribution. However, the proposed method is computationally demanding since it is based on Monte Carlo simulation.

• Journal of Korea Society of Digital Industry and Information Management
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• v.7 no.4
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• pp.7-14
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• 2011

Infinite failure NHPP models presented in the literature exhibit either constant, monotonic increasing or monotonic decreasing failure occurrence rate per fault (hazard function). This infinite non-homogeneous Poisson process is model which reflects the possibility of introducing new faults when correcting or modifying the software. In this paper, polynomial hazard function have been proposed, which can efficiency application for software reliability. Algorithm for estimating the parameters used to maximum likelihood estimator and bisection method. Model selection based on mean square error and the coefficient of determination for the sake of efficient model were employed. In numerical example, log power time model of the existing model in this area and the polynomial hazard function model were compared using failure interval time. Because polynomial hazard function model is more efficient in terms of reliability, polynomial hazard function model as an alternative to the existing model also were able to confirm that can use in this area.