• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.687-704
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• 2002

We show that the class of inner chordarc domains is properly contained in the class of exterior quasiconvex domains. We also show that the class of exterior quasiconvex domains is properly contained in the class of John disks. We give the conditions which make the converses of the above results be true. Next , we show that an exterior quasiconvex domain satisfies certain growth conditions for the exterior Riemann mapping. From the results we show that the domain satisfies the Bernstein inequality and the integrated version of it. Finally, we assume that f is a function which is continuous in the closure of a domain D and analytic in D. We show connections between the smoothness of f and the rate at which it can be approximated by polynomials on an exterior quasiconvex domain and a $Lip_\alpha$-extension domain.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.251-261
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• 2008

In this paper second order cone convex, second order cone pseudoconvex, second order strongly cone pseudoconvex and second order cone quasiconvex functions are introduced and their interrelations are discussed. Further a MondWeir Type second order dual is associated with the Vector Minimization Problem and the weak and strong duality theorems are established under these new generalized convexity assumptions.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.599-610
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• 2006

The aim of this paper is to give an existence theorem for a strong solution of generalized vector quasi-equilibrium problems of the weak type II due to Hou et al. using the equilibrium existence theorem for 1-person game, and as an application, we shall give a generalized quasivariational inequality.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.357-367
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• 2001

Let $Q(\beta)$ be the class of normalized strongly quasiconvex functions of order $\beta$ in the open unit disk. Sharp Fekete-Szego inequalities are obtained for functions belonging to the class $Q(\beta)$. We also consider the integral preserving properties in a sector.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.719-730
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• 1997

From a minimax inequality related to admissible multimaps, we deduce generalized versions of lopsided saddle point theorems, fixed point theorems, existence of maximizable linear functionals, the Walras excess demand theorem, and the Gale-Nikaido-Debreu theorem.

• Honam Mathematical Journal
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• v.10 no.1
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• pp.23-30
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• 1988

Let C[C, D] and $S^{*}[C,\;D]$ denote the classes of functions g, g(0)=1-g'(0)0=0, analytic in the unit disc E such that $\frac{(zg{\prime}(z)){\prime}}{g{\prime}(z)}$ and $\frac{zg{\prime}(z)}{g(z)}$ are subordinate to $\frac{1+Cz}{1+Dz{\prime}}$ $z{\in}E$, respectively. In this paper, the classes K[A,B;C,D] and $C^{*}[A,B;C,D]$, $-1{\leq}B<A{\leq}1$; $-1{\leq}D<C{\leq}1$, are defined. The functions in these classes are close-to-convex. Using the properties of convolution operators, we deal with some problems for our classes.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.273-285
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• 1997

The aim of this paper is to present theorems on the exitence of zeros for mappings defined on convex subsets of topological vector spaces with values in a vector space. In addition to natural assumptions of continuity, convexity, and compactness, the mappings are subject to some geometric conditions. In the first theorem, the mapping satisfies a "Darboux-type" property expressed in terms of an auxiliary numerical function. Typically, this functions is, in this case, related to an order structure on the target space. We derive an existence theorem for "obtuse" quasiconvex mappings with values in an ordered vector space. In the second theorem, we prove the existence of a "common zero" for an arbitrary (not necessarily countable) family of mappings satisfying a general "inwardness" condition againg expressed in terms of numerical functions (these numerical functions could be duality pairings (more generally, bilinear forms)). Our inwardness condition encompasses classical inwardness conditions of Leray-Schauder, Altman, or Bergman-Halpern types.