• Communications of the Korean Mathematical Society
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• v.23 no.1
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• pp.125-131
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• 2008

We introduce and study the concepts of strong generalized neighborhood systems, SGNS and sg-open sets. We also introduce and investigate the concept of (${\psi},\;{\psi}^'$)-continuity and sg-continuity on SGNS's.

• Kyungpook Mathematical Journal
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• v.52 no.4
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• pp.405-412
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• 2012

The purpose of the present paper is towards working out a unified version of the study of certain weak forms of generalized open sets and their neighbouring forms, as are already available in the literature. In terms of an operation, as initiated by $\acute{A}$. Cs$\acute{a}$sz$\acute{a}$r, we introduce unified definitions of ${\wedge}_{\psi}$-sets, ${\vee}_{\psi}$-sets, $g{\cdot}{\wedge}_{\psi}$-sets and $g{\cdot}{\vee}_{\psi}$-sets and derive results concerning them.

• Communications of the Korean Mathematical Society
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• v.26 no.3
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• pp.455-462
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• 2011

An analytic self-map ${\phi}$ of the open unit disk $\mathbb{D}$ in the complex plane and an analytic map ${\psi}$ on $\mathbb{D}$ induce the so-called weighted composition operator $C_{{\phi},{\psi}}$: $H(\mathbb{D})\;{\rightarrow}\;H(\mathbb{D})$, $f{\mapsto} \;{\psi}\;(f\;o\;{\phi})$, where H($\mathbb{D}$) denotes the set of all analytic functions on $\mathbb{D}$. We study when such an operator acting between different weighted Bergman spaces is bounded, compact and Schatten class.

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.787-811
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• 1999

Let M be the maximal ideal space of the Banach algebra $H^{\infty}$ of bounded analytic functions on the open unit disc $\triangle$. For a positive singular measure ${\mu}\;on\;{\partial\triangle},\;let\;{L_{+}}^1(\mu)$ be the set of measures v with $0\;{\leq}\;{\nu}\;{\ll}\;{\mu}\;and\;{{\psi}_{\nu}}$ the associated singular inner functions. Let $R(\mu)\;and\;R_0(\mu)$ be the union sets of $\{$\mid$\psiv$\mid$\;<\;1\}\;and\;\{$\mid${\psi}_{\nu}$\mid$\;<\;0\}\;in\;M\;{\setminus}\;{\triangle},\;{\nu}\;\in\;{L_{+}}^1(\mu)$, respectively. It is proved that if $S(\mu)\;=\;{\partial\triangle}$, where $S(\mu)$ is the closed support set of $\mu$, then $R(\mu)\;=\;R0(\mu)\;=\;M{\setminus}({\triangle}\;{\cup}\;M(L^{\infty}(\partial\triangle)))$ is generated by $H^{\infty}\;and\;\overline{\psi_{\nu}},\;{\nu}\;{\in}\;{L_1}^{+}(\mu)$. It is proved that %d{\theta}(S(\mu))\;=\;0$ if and only if there exists as Blaschke product b with zeros $\{Zn\}_n$ such that $R(\mu)\;{\subset}\;{$\mid$b$\mid$\;<\;1}\;and\;S(\mu)$ coincides with the set of cluster points of $\{Zn\}_n$. While, we proved that $\mu$ is a sum of finitely many point measure such that $R(\mu)\;{\subset}\;\{$\mid${\psi}_{\lambda}$\mid$\;<\;1}\;and\;S(\lambda)\;=\;S(\mu)$. Also it is studied conditions on \mu for which $R(\mu)\;=\;R0(\mu)$.

• Korean Journal of Mathematics
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• v.24 no.3
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• pp.545-565
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• 2016

In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form: $$T_{\lambda}(f;x)={\large\int_{\Omega}}K_{\lambda}(t,x,f(t))dt,\;x{\in}{\Psi},\;{\lambda}{\in}{\Lambda}$$, where ${\Psi}$ = and ${\Omega}$ = stand for arbitrary closed, semi-closed or open bounded intervals in ${\mathbb{R}}$ or these set notations denote $\mathbb{R}$, and ${\Lambda}$ is a set of non-negative numbers, to the function $f{\in}L_{p,{\omega}}({\Omega})$, where $L_{p,{\omega}}({\Omega})$ denotes the space of all measurable functions f for which $\|{\frac{f}{\omega}}\|^p$ (1 ${\leq}$ p < ${\infty}$) is integrable on ${\Omega}$, and ${\omega}:{\mathbb{R}}{\rightarrow}\mathbb{R}^+$ is a weight function satisfying some conditions.