• Journal of the Korean Mathematical Society
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• v.36 no.5
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• pp.891-910
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• 1999

We classify analytically surface singularities defined by some weighted homogeneous polynomials which are topologically equivalent to the type {{{{ { z}`_{0 } ^{n } }}}} + {{{{ { z}`_{k } ^{1 } }}}} + {{{{ { z}`_{2 } ^{1 } }}}} = 0.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.439-449
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• 2019

In 2006, S. Lin and W. Lin introduced the definition of weakly weighted-sharing of meromorphic functions which is between "CM" and "IM". In this paper, using the notion of weakly weighted-sharing, we study the uniqueness of nonconstant homogeneous differential polynomials P[f] and P[g] generated by meromorphic functions f and g, respectively. Our results generalize the results due to S. Lin and W. Lin, and H.-Y. Xu and Y. Hu.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.1
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• pp.11-27
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• 2010

Let $\Omega$ be a domain with smooth boundary in ${\mathbb{C}}^{n+1}$ and let $p{\in}{\partial}{\Omega}$. Suppose that $\Omega$ is Kobayashi hyperbolic and p is of Catlin multi-type ${\tau}=({\tau}_0,{\ldots},{\tau}_n)$. In this paper, we show that $\Omega$ admits a $C^{k}$ contraction at p with $k{\geq}\mid{\tau}\mid+1$ if and only if $\Omega$ is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.

• The Pure and Applied Mathematics
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• v.17 no.2
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• pp.157-165
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• 2010

Let (M,p) be a germ of a $C^{\infty}$ CR manifold of CR dimension n and CR codimension d. Suppose (M,p) admits a $C^{\infty}$ contraction at p. In this paper, we show that (M,p) is CR equivalent to a generic submanifold in $\mathbb{C}^{n+d}$ defined by a vector valued weighted homogeneous polynomial.

• Journal of the Korean Mathematical Society
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• v.43 no.3
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• pp.579-591
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• 2006

In this paper we obtain a sufficient and necessary condition for an analytic function f on the unit ball B with Hadamard gaps, that is, for $f(z)\;=\;{\sum}^{\infty}_{k=1}\;P_{nk}(z)$ (the homogeneous polynomial expansion of f) satisfying $n_{k+1}/n_{k}{\ge}{\lambda}>1$ for all $k\;{\in}\;N$, to belong to the weighted Bergman space $$A^p_{\alpha}(B)\;=\;\{f{\mid}{\int}_{B}{\mid}f(z){\mid}^{p}(1-{\mid}z{\mid}^2)^{\alpha}dV(z) < {\infty},\;f{\in}H(B)\}$$. We find a growth estimate for the integral mean $$\({\int}_{{\partial}B}{\mid}f(r{\zeta}){\mid}^pd{\sigma}({\zeta})\)^{1/p}$$, and an estimate for the point evaluations in this class of functions. Similar results on the mixed norm space $H_{p,q,{\alpha}$(B) and weighted Bergman space on polydisc $A^p_{^{\to}_{\alpha}}(U^n)$ are also given.