• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.3
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• pp.279-282
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• 2014

In this paper, we report a new sharp inequality of interpolation type in $\mathbb{R}^n$. This inequality is for controlling weighted average of a function via $L^n$ norm of the gradient of a function together with its' certain exponential norm.

• Kyungpook Mathematical Journal
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• v.50 no.2
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• pp.213-228
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• 2010

A weighted version of the geometric mean of k ($\geq\;3$) positive invertible operators is given. For operators $A_1,{\ldots},A_k$ and for nonnegative numbers ${\alpha}_1,\ldots,{\alpha}_k$ such that $\sum_\limits_{i=1}^k\;\alpha_i=1$, we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to $A_1^{\alpha_1}{\cdots}A_k^{{\alpha}_k}$ if $A_1,{\ldots},A_k$ commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

• The Pure and Applied Mathematics
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• v.26 no.3
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• pp.189-197
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• 2019

In this paper we obtain some new inequalities for the weighted chaotically geometric mean of two positive operators on a complex Hilbert space.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.325-338
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• 2023

In this paper, we derive a Reilly-type inequality for the Laplacian with density on weighted manifolds with boundary. As its applications, we obtain some new Poincaré-type inequalities not only on weighted manifolds, but more interestingly, also on their boundary. Furthermore, some mean-curvature type inequalities on the boundary are also given.

• Journal of applied mathematics & informatics
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• v.39 no.1_2
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• pp.117-124
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• 2021

We obtain a new matrix generalization result dealing with weighted mean summability of infinite series by using a new general class of power increasing sequences obtained by Sulaiman [9]. This theorem also includes some new and known results dealing with some basic summability methods.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.769-780
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• 2016

In this paper, we first prove some Liouville type theorems for elliptic inequalities on weighted manifolds which support a weighted Sobolev-type inequality. Secondly, applying the Liouville type theorems to self-shrinkers, we obtain some global rigidity theorems.

• Kyungpook Mathematical Journal
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• v.47 no.4
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• pp.595-600
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• 2007

In this paper we consider weighted arithmetic and geometric means of several positive definite operators proposed by Sagae and Tanabe and we establish a reverse inequality of the arithmetic and geometric means via Specht ratio and the Thompson metric on the convex cone of positive definite operators.

• Kyungpook Mathematical Journal
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• v.49 no.1
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• pp.167-181
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• 2009

Based on Ricatti equation $XA^{-1}X=B$ for two (positive invertible) operators A and B which has the geometric mean $A{\sharp}B$ as its solution, we consider a cubic equation $X(A{\sharp}B)^{-1}X(A{\sharp}B)^{-1}X=C$ for A, B and C. The solution X = $(A{\sharp}B){\sharp}_{\frac{1}{3}}C$ is a candidate of the geometric mean of the three operators. However, this solution is not invariant under permutation unlike the geometric mean of two operators. To supply the lack of the property, we adopt a limiting process due to Ando-Li-Mathias. We define reasonable geometric means of k operators for all integers $k{\geq}2$ by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.

• The Journal of Industrial Distribution & Business
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• v.8 no.6
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• pp.87-95
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• 2017

Purpose - The structural changes of Korean agriculture are complex due to heterogeneous production processes and farms' features. This study analyzed trends of dualism in Korean agriculture over the period 2000-15 based on farm-level data to clarify the specific trends of dualism in terms of farm income, farm-size, and farm operators' age. From the results of this study, we would be able to understand the features of structural changes in Korean agriculture more profoundly. Research design, data, and methodology - We incorporated farm-level data in South Korea: Agricultural census and Farm household economy survey. As measures of inequality, we used size-weighted quantiles, and normalized Gini coefficients as well as mean and conventional quantiles. The size-weighted quantiles are more robust to changes in the number of small farms, but they are more sensitive to changes in the distribution of farm-size. Thus, they would be more useful to identify trends of dualism of Korean agriculture. Results - The results show that the farmland distribution of crop farms became more skewed and dispersed. However, the herd distribution of livestock farms became more concentrated. To be specific, their mean and 1st quantile increases more rapidly than their size-weighted 2nd quantile and size-weighted 3rd quantile. Gini coefficients of livestock farms regarding their herd distribution decreased by 0.1 on average. In the case of income distribution, the results indicate that the polarization regarding farm household/agricultural/non-agricultural income became more severe. However, we also found that the distribution of transfer income became concentrated continuously. The results imply that transfer income including subsidies would decrease farm income polarization. Lastly, during the study periods, Korean farms were aging over time, and age distribution of them more concentrated. Conclusions - The structure of Korean agriculture has been changing, even though the absolute size of it decreased over time. Land (herd) distribution became more dispersed (concentrated). Inequality regarding agricultural income became more severe, and it made farm household income more polarized even though transfer income would decrease income gaps among farms. Lastly, farms continue to age regardless of farm types and this might affect the structural changes in Korean agriculture in the future.