• Korean Journal of Mathematics
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• v.10 no.1
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• pp.1-3
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• 2002

In this paper, we obtain some coefficient bounds of harmonic, orientation-preserving, univalent mappings defined on ${\Delta}=\{z:{\mid}z{\mid}>1\}$.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.841-854
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• 2019

In this article, a subclass of univalent harmonic mapping is introduced by restricting its analytic part to lie in the class $S^{\delta}[{\alpha}]$, $0{\leq}{\alpha}<1$, $-{\infty}<{\delta}<{\infty}$ which has been introduced and studied by Kumar [17] (see also [20], [21], [22], [23]). Coefficient estimations, growth and distortion properties, area theorem and covering estimates of functions in the newly defined class have been established. Furthermore, we also found bound for the Bloch's constant for all functions in that family.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.803-814
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• 1995

Let $\Sigma$ be the class of all complex-valued, harmonic, orientation-preserving, univalent mappings defined on $\Delta = {z : $\mid$z$\mid$ > 1}$ that map $\infty$ to $\infty$.

• Korean Journal of Mathematics
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• v.20 no.4
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• pp.433-439
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• 2012

In this paper, we study harmonic, orientation-preserving, univalent mappings defined on ${\Delta}$ = {z : |z| > 1} that have real coefficients or starlike analytic functions and obtain some coefficients bounds.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.679-684
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• 2015

In this paper we discuss bounds for the total curvature of nonparametric minimal surfaces by using the properties of planar harmonic mappings.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.285-289
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• 1999

In this paper, we estimate the total curvature of non-parametric minimal surfaces by using the properties of univalent harmonic mappings defined on ${\Delta}=\{z:{\mid}z:{\mid}>1\}$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.1925-1935
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• 2015

In this paper, we study right half-plane harmonic mappings $f_0$ and f, where $f_0$ is fIxed and f is such that its dilatation of a conformal automorphism of the unit disk. We obtain a sufficient condition for the convolution of such mappings to be convex in the direction of the real axis. The result of the paper is a generalization of the result of by Li and Ponnusamy [11], which itself originates from a problem posed by Dorff et al. in [7].

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.105-123
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• 2015

Let $f_{\beta}=h_{\beta}+\bar{g}_{\beta}$ and $F_a=H_a+\bar{G}_a$ be harmonic mappings obtained by shearing of analytic mappings $h_{\beta}+g_{\beta}=1/(2isin{\beta})log\((1+ze^{i{\beta}})/(1+ze^{-i{\beta}})\)$, 0 < ${\beta}$ < ${\pi}$ and $H_a+G_a=z/(1-z)$, respectively. Kumar et al. [7] conjectured that if ${\omega}(z)=e^{i{\theta}}z^n({\theta}{\in}\mathbb{R},n{\in}\mathbb{N})$ and ${\omega}_a(z)=(a-z)/(1-az)$, $a{\in}(-1,1)$ are dilatations of $f_{\beta}$ and $F_a$, respectively, then $F_a\tilde{\ast}f_{\beta}{\in}S^0_H$ and is convex in the direction of the real axis, provided $a{\in}[(n-2)/(n+2),1)$. They claimed to have verified the result for n = 1, 2, 3 and 4 only. In the present paper, we settle the above conjecture, in the affirmative, for ${\beta}={\pi}/2$ and for all $n{\in}\mathbb{N}$.