• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.493-507
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• 1995

The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.133-150
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• 2007

In a development of efficient primal-dual interior-points algorithms for self-scaled convex programming problems, one of the important properties of such cones is the existence and uniqueness of "scaling points". In this paper through the identification of scaling points with the notion of "(metric) geometric means" on symmetric cones, we extend several well-known matrix inequalities (the classical L$\ddot{o}$wner-Heinz inequality, Ando inequality, Jensen inequality, Furuta inequality) to symmetric cones. We also develop a theory of spectral geometric means on symmetric cones which has recently appeared in matrix theory and in the linear monotone complementarity problem for domains associated to symmetric cones. We derive Nesterov-Todd inequality using the spectral property of spectral geometric means on symmetric cones.

• Honam Mathematical Journal
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• v.44 no.4
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• pp.513-520
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• 2022

The Jensen and Mercer inequalities are very important inequalities in information theory. The article provides the generalization of Mercer's inequality for convex functions on the line segments. This result contains Mercer's inequality as a particular case. Also, we investigate bounds for Shannon's entropy and give some new applications in zeta function and analysis.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.51-57
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• 2022

The Jensen, Simic and Mercer inequalities are very important inequalities in theory of inequalities and some results are devoted to this inequalities. In this paper, firstly, we establish extension of Jensen-Simic-Mercer inequality. After that, we investigate bounds for Shannons entropy of a probability distribution. Finally, We give some new applications in analysis.

• The Transactions of the Korean Institute of Electrical Engineers A
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• v.48 no.5
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• pp.560-570
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• 1999

This paper formulates the suboptimal robust Kalman filtering problem into two coupled Linear Matrix Inequality (LMI) problems by applying Lyapunov theory to the augmented system which is composed of the state equation in the uncertain linear system and the estimation error dynamics. This formulations not only provide the sufficient conditions for the existence of the desired filter, but also construct the suboptimal robust Kalman filter. The proposed filter can guarantee the optimized upper bound of the estimation error variance for uncertain systems with parametric uncertainties in both the state and measurement matrices. In addition, this paper shows how the problem of finding the minimizing solution subject to Quadratic Matrix Inequality (QMI), which cannot be easily transformed into LMI using the usual Schur complement formula, can be successfully modified into a generic LMI problem.

• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.95-100
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• 2016

In this paper, we establish a new refinement of the arithmetic-geometric mean inequality. Applying this result in information theory, we obtain a more precise upper bound for Shannon's entropy.

• Journal of Applied and Pure Mathematics
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• v.6 no.3_4
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• pp.191-200
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• 2024

Motivated by the previously reported results, this work attempts to provide fresh refinements to both vector and numerical radius inequalities by providing a refinement to the well known Buzano's inequality which as a consequence yielded another refinement of the Cauchy-Schwartz (CS) inequality. Utilizing the new refinements of the Buzano's and Cauchy-Schwartz inequalities, we proceed to obtain various vector and numerical radius type inequalities. Methods used in the paper are standard for the operator theory inequality topics.

• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.45-60
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• 2002

Some error estimates in terms of the p-norms of the fourth derivative for the remainder in a perturbed trapezoid formula are given. Applications for the expectation of a random variable and the Hermite-Hadamard divergence in Information Theory are also pointed out.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.153-162
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• 1995

Ky Fan's minimax inequality is an important tool in nonlinear functional analysis and its applications, e.g. game theory and economic theory. Since Fan gave his minimax inequality in [2], various extensions of this interesting result have been obtained (see [4,11] and the references therein). Using Fan's minimax inequality, Ha [6] obtained a non-compact version of Sion's minimax theorem in topological vector spaces, and next Geraghty-Lin [3], Granas-Liu [4], Shih-Tan [11], Simons [12], Lin-Quan [10], Park-Bae-Kang [17], Bae-Kim-Tan [1] further generalize Fan's minimax theorem in more general settings. In [9], using the concept of submaximum, Komiya proved a topological minimax theorem which also generalized Sion's minimax theorem and another minimax theorem of Ha in [5] without using linear structures. And next Lin-Quan [10] further generalizes his result to two function versions and non-compact topological settings.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.527-535
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• 2008

Let $I{\in}C^{1,1}$ be a strongly indefinite functional defined on a Hilbert space H. We investigate the number of the critical points of I when I satisfies one pair of Torus-Sphere variational linking inequality. We show that I has at least two critical points when I satisfies one pair of Torus-Sphere variational linking inequality with $(P.S.)^*_c$ condition. We prove this result by use of the limit relative category and critical point theory on the manifold with boundary.