Extension of Simons' inequality

We prove the following extended version of Simons' inequality and present its applications. Let be a set and be a subset of . Let be a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Let be a bounded function such that the functions are convex for all and whenever 0$">, and Let be a sequence in . Assume that, for every , there exists satisfying . Then

If , then the set in the above inequality can be replaced by .

     

A noncommutative Szegö theorem for subdiagonal subalgebras of von Neumann algebras

For almost forty years now the most frustrating open problem regarding the theory of finite maximal subdiagonal algebras has been the question regarding the universal validity of a non-commutative Szegö theorem and Jensen inequality (Arveson, 1967). These two properties are known to be equivalent. Despite extensive efforts by many authors, their validity has to date only been established in some very special cases. In the present note we solve the general problem in the affirmative by proving the universal validity of Szegö's theorem for finite maximal subdiagonal algebras.

     

Vector Variational Inequalities for Nondifferentiable Convex Vector Optimization Problems

In this paper, we consider a nondifferentiable convex vector optimization problem (VP), and formulate several kinds of vector variational inequalities with subdifferentials. Here we examine relations among solution sets of such vector variational inequalities and (VP). Mathematics Subject classification (2000). 90C25, 90C29, 65K10 This work was supported by the Brain Korea 21Project in 2003. The authors wish to express their appreciation to the anonymous referee for giving valuable comments.     

Global Stability Results for the Weak Vector Variational Inequality

In this paper, we consider the global stability of solutions of a Weak Vector Variational Inequality in a finite-dimensional Euclidean space. Upper semi-continuity of the solution set mapping is established. And by a scalarization method, we derive a sufficient condition that guarantees the lower semi-continuity of the solution set mapping for the Weak Vector Variational Inequality     

The Study of KKM Theorems With Applications to Vector Equilibrium Problems and Implict Vector Variational Inequalities Problems

In this paper, we establish some equivalence relations between coincidence theorems and KKM-type theorems. We also obtain some new coincidence theorems and fixed point theorems. Applying the KKM-type theorems we obtain the existence theorems of generalized vector equilibrium problems. From these results, some existence theorems of generalized vector implicit variational inequality problems are established in this paper.     

On Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, we study the solvability for a class of vector variational inequalities in reflexive Banach spaces. By using Brouwer fixed point theorem, we prove the solvability for this class of vector variational inequalities without monotonicity assumption. The solvability results for this class of vector variational inequalities with monotone mappings are also presented by using the KKM-Fan lemma This paper is dedicated to Professor Franco Giannessi for his 68th birthday     

Harnack’s Inequality for Stable Lévy Processes

We prove Harnacks inequality for harmonic functions of a symmetric stable Lévy process on R^d without the assumption that the density function of its Lévy measure is locally bounded from below. Mathematics Subject Classifications (2000) Primary 60J45, 31C05; Secondary 60G51.Research partially supported by KBN (2P03A 041 22) and RTN (HPRN-CT-2001-00273-HARP).     

Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds

Let (M,g) be a compact Riemannian manifold without boundary, and (N,g) a compact Riemannian manifold with boundary. We will prove in this paper that the and can be attained. Our proof uses the blow-up analysis.     

The best bounds in Wallis' inequality

For all natural numbers , let denote a double factorial. Then

The constants and are the best possible. From this, the well-known Wallis' inequality is improved.

     

An extremal function for the Chang-Marshall inequality over the Beurling functions

S.-Y. A. Chang and D. E. Marshall showed that the functional is bounded on the unit ball of the space of analytic functions in the unit disk with and Dirichlet integral not exceeding one. Andreev and Matheson conjectured that the identity function is a global maximum on for the functional . We prove that attains its maximum at over a subset of determined by kernel functions, which provides a positive answer to a conjecture of Cima and Matheson.