• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.833-851
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• 2018

In this article, we prove some normality criteria for a family of meromorphic functions having zeros with some multiplicity. Our main result involves sharing of a holomorphic function by certain differential polynomials. Our results generalize some of the results of Fang and Zalcman [4] and Chen et al. [2] to a great extent.

• Journal of the Korean Mathematical Society
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• v.59 no.3
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• pp.449-468
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• 2022

In this paper we determine explicitly the kernels 𝕜_α,β associated with new Bergman spaces A²_α,β(𝔻) considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when α ∈ ℕ where the zeros are given by the zeros of a real polynomial Q_α,β. Some numerical results are given throughout the paper.

• The Pure and Applied Mathematics
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• v.24 no.3
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• pp.129-145
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• 2017

In this paper, we establish lower estimates for the modulus of the non-tangential derivative of the holomorphic functionf(z) at the boundary of the unit disc. Also, we shall give an estimate below |f''(b)| according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and $z_0{\neq}0$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.325-334
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• 2016

In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$ holomorphic in the unit disc and $\|\frac{f(z)}{{\lambda}f(z)+(1-{\lambda})z}-{\alpha}\|$ < ${\alpha}$ for ${\mid}z{\mid}$ < 1, where $\frac{1}{2}$ < ${\alpha}$ ${\leq}{\frac{1}{1+{\lambda}}}$, $0{\leq}{\lambda}$ < 1. If we know the second and the third coefficient in the expansion of the function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account $c_{p+1}$, $c_{p+2}$ and zeros of f(z) - z. We obtain a sharp lower bound of ${\mid}f^{\prime}(b){\mid}$ at the point b, where ${\mid}b{\mid}=1$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.993-1002
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• 2017

Let ${\Omega}$ be a bounded, uniformly totally pseudoconvex domain in ${\mathbb{C}}^2$ with the smooth boundary b${\Omega}$. Assuming that ${\Omega}$ satisfies the negative ${\bar{\partial}}$ property. Let M be a positive, finite area divisor of ${\Omega}$. In this paper, we will prove that: if ${\Omega}$ admits a maximal type F and the ${\check{C}}eck$ cohomology class of the second order vanishes in ${\Omega}$, there is a bounded holomorphic function in ${\Omega}$ such that its zero set is M. The proof is based on the method given by Shaw [27].