• 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 ${\Delta}=\{z:{\mid}z{\mid}>1\}$.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.433-446
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• 2018

In this present investigation, we introduce a new class of harmonic univalent functions of the form $f=h+{\bar{g}}$ in the open unit disk ${\Delta}$. We get basic properties, like, necessary and sufficient convolution conditions, distortion bounds, compactness and extreme points for these classes of functions.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.887-902
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• 2021

In this paper, we have introduced a generalized class Sⁱ_H (m, n, 𝛾, 𝜙, 𝜓; 𝛼), i ∈ {0, 1} of harmonic univalent functions in unit disc 𝕌, a sufficient coefficient condition for the normalized harmonic function in this class is obtained. It is also shown that this coefficient condition is necessary for its subclass 𝒯 Sⁱ_H (m, n, 𝛾, 𝜙, 𝜓; 𝛼). We further obtained extreme points, bounds and a covering result for the class 𝒯 Sⁱ_H (m, n, 𝛾, 𝜙, 𝜓; 𝛼). Also, show that this class is closed under convolution and convex combination. While proving our results, certain conditions related to the coefficients of 𝜙 and 𝜓 are considered, which lead to various well-known results.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.803-812
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• 2014

Let ${\mathcal{S}}^0_{\mathcal{H}}$ be the class of normalized univalent harmonic mappings in the unit disk. A subclass ${\mathcal{V}}^{\mathcal{H}}(k)$ of ${\mathcal{S}}^0_{\mathcal{H}}$, whose analytic part is function with bounded boundary rotation, is introduced. Some bounds for functionals, specially harmonic pre-Schwarzian derivative, described in ${\mathcal{V}}^{\mathcal{H}}(k)$ are given.

• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.269-278
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• 2021

In this paper, we investigate harmonic univalent functions convex in the direction 𝜃, for 𝜃 ∈ [0, 𝜋). We find bounds for |f_z(z)|, ${\mid}f_{\bar{z}}(z){\mid}$ and |f(z)|, as well as coefficient bounds on the series expansion of functions convex in a given direction.

• Korean Journal of Mathematics
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• v.8 no.2
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• pp.135-138
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• 2000

In this paper, we study harmonic orientation-preserving univalent mappings defined on ${\Delta}=\{z:{\mid}z{\mid}>1\}$ that are typically real.

• 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.

• Kyungpook Mathematical Journal
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• v.62 no.2
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• pp.243-256
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• 2022

In 1914, Bohr proved that there is an r₀ ∈ (0, 1) such that if a power series ∑^∞_m=0 c_mz^m is convergent in the open unit disc and |∑^∞_m=0 c_mz^m| < 1 then, ∑^∞_m=0 |c_mz^m| < 1 for |z| < r₀. The largest value of such r₀ is called the Bohr radius. In this article, we find Bohr radius for some univalent harmonic mappings having different dilatations. We also compute the Bohr radius for functions that are convex in one direction.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.563-569
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• 2005

In [2], Jahangiri studied the harmonic starlike functions of order $\alpha$, and he defined the class T$_{H}$($\alpha$) consisting of functions J = h + $\bar{g}$ where hand g are the analytic and the co-analytic part of the function f, respectively. In this paper, we introduce the class T$_{H}$($\alpha$, $\beta$) of analytic functions and prove various coefficient inequalities, growth and distortion theorems, radius of convexity for the function h, if the function J belongs to the classes T$_{H}$($\alpha$) and T$_{H}$($\alpha$, $\beta$).