• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.993-1010
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• 2022

In this paper, we establish some radius results and inclusion relations for starlike functions associated with a petal-shaped domain.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.537-549
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• 2022

In this paper, we consider a convex univalent function f_α,β which maps the open unit disc 𝕌 onto the vertical strip domain Ω_α,β = {w ∈ ℂ : α < ℜ < (w) < β} and introduce new subclasses of both close-to-convex and bi-close-to-convex functions with respect to an odd starlike function associated with Ω_α,β. Also, we investigate the Fekete-Szegö type coefficient bounds for functions belonging to these classes.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1377-1387
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• 2013

Let $f$ be a harmonic mapping on the unit disc ${\Delta}$ in $\mathbb{C}$. We give some condition for $f$ to be a quasiconformal homeomorphism on ${\Delta}$ and to have a quasiconformal extension to the whole plane $\bar{\mathbb{C}}$. We also obtain quasiconformal extension results for starlike harmonic mappings of order ${\alpha}{\in}(0,1)$.

• Honam Mathematical Journal
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• v.43 no.4
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• pp.627-639
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• 2021

Numerical techniques are used to determine the radius of convexity of the starlike functions related to cardioid shaped bounded domain. In addition, radius constants of certain starlikeness associated with right half plane of various starlike functions are computed.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.75-98
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• 2021

In this paper, we introduce new classes of post-quantum or (p, q)-starlike and convex functions with respect to symmetric points associated with a cardiod-shaped domain. We obtain (p, q)-Fekete-Szegö inequalities for functions in these classes. We also obtain estimates of initial (p, q)-logarithmic coefficients. In addition, we get q-Bieberbachde-Branges type inequalities for the special case of our classes when p = 1. Moreover, we also discuss some special cases of the obtained results.

• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.763-770
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• 1999

The purpose of this paper is to give sufficient coefficient conditions for a class of univalent harmonic functions that map each $$\mid$z$\mid$$ = r >1 onto a curve that bounds a domain that is starlike with respect to origin. Furthermore, it is shown that these conditions are also necessary when the coefficients are negative. Extreme points for these classes are also determined. Finally, comparable results are given for the convex analgo.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.451-464
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• 2019

In the present paper, we find the sharp bound for the fourth coefficient of starlike functions f which are normalized by f(0) = 0 = f'(0) - 1 and satisfy the following two-sided inequality: $$1+{\frac{{\gamma}-{\pi}}{2\;{\sin}\;{\gamma}}}\;<\;{\Re}\{{\frac{zf^{\prime}(z)}{f(z)}}\}\;<\;1+{\frac{{\gamma}}{2\;{\sin}\;{\gamma}}},\;z{\in}{\mathbb{D}}$$, where ${\mathbb{D}}:=\{z{\in}{\mathbb{C}}:{\left|z\right|}<1\}$ is the unit disk and ${\gamma}$ is a real number such that ${\pi}/2{\leq}{\gamma}<{\pi}$. Moreover, the sharp bound for the fifth coefficient of f defined above with ${\gamma}$ in a subset of [${\pi}/2,{\pi}$) also will be found.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.433-444
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• 2023

Given a function f analytic in open disk centred at origin of radius unity and satisfying the condition |f(z)/g(z) - 1| < 1 for a analytic function g with certain prescribed conditions in the unit disk, radii constants R are determined for the values of Rzf'(Rz)/f(Rz) to lie inside the domain enclosed by the curve |w² - 1| = 1 (lemniscate of Bernoulli). This, in turn, provides a partial solution to the conjectures and problems for determination of sharp bounds R for such functions f.

• Communications of the Korean Mathematical Society
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• v.25 no.4
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• pp.557-569
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• 2010

Suppose that n is a positive integer. For any real number $\alpha$($\beta$ resp.) with $\alpha$ < 1 ($\beta$ > 1 resp.), let $K^{(n)}(\alpha)$ ($K^{(n)}(\beta)$ resp.) be the class of analytic functions in the unit disk $\mathbb{D}$ with f(0) = f'(0) = $\cdots$ = $f^{(n-1)}(0)$ = $f^{(n)}(0)-1\;=\;0$, Re($\frac{zf^{n+1}(z)}{f^{(n)}(z)}+1$) > $\alpha$ (Re($\frac{zf^{n+1}(z)}{f^{(n)}(z)}+1$) < $\beta$ resp.) in $\mathbb{D}$, and for any ${\lambda}\;{\in}\;\bar{\mathbb{D}}$, let $K^{(n)}({\alpha},\;{\lambda})$ $K^{(n)}({\beta},\;{\lambda})$ resp.) denote a subclass of $K^{(n)}(\alpha)$ ($K^{(n)}(\beta)$ resp.) whose elements satisfy some condition about derivatives. For any fixed $z_0\;{\in}\;\mathbb{D}$, we shall determine the two regions of variability $V^{(n)}(z_0,\;{\alpha})$, ($V^{(n)}(z_0,\;{\beta})$ resp.) and $V^{(n)}(z_0,\;{\alpha},\;{\lambda})$ ($V^{(n)}(z_0,\;{\beta},\;{\lambda})$ resp.). Also we shall determine the extreme points of the families of analytic functions which satisfy $f(\mathbb{D})\;{\subset}\;V^{(n)}(z_0,\;{\alpha})$ ($f(\mathbb{D})\;{\subset}\;V^{(n)}(z_0,\;{\beta})$ resp.) when f ranges over the classes $K^{(n)}(\alpha)$ ($K^{(n)(\beta)$ resp.) and $K^{(n)}({\alpha},\;{\lambda})$ ($K^{(n)}({\beta},\;{\lambda})$ resp.), respectively.

• Honam Mathematical Journal
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• v.39 no.4
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• pp.503-514
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• 2017

For real parameters ${\alpha}$ and ${\beta}$ such that ${\alpha}$ < 1 < ${\beta}$, we denote by $\mathcal{P}({\alpha},{\beta})$ the class of analytic functions p, which satisfy p(0) = 1 and ${\alpha}$ < ${\Re}\{p(z)\}$ < ${\beta}$ in ${\mathbb{D}}$, where ${\mathbb{D}}$ denotes the open unit disk. Let ${\mathcal{A}}$ be the class of analytic functions in ${\mathbb{D}}$ such that f(0) = 0 = f'(0) - 1. For $f{\in}{\mathcal{A}}$, ${\mu}{\in}{\mathbb{C}}{\backslash}\{0\}$ and ${\nu}{\in}{\mathbb{C}}$, let $I_{{\mu},{\nu}:{\mathcal{A}}{\rightarrow}{\mathcal{A}}$ be an integral operator defined by $$I_{{\mu},{\nu}[f](z)}=\({\frac{{\mu}+{\nu}}{z^{\nu}}}{\int}^z_0f^{\mu}(t)t^{{\nu}-1}dt\)^{1/{\mu}}$$. In this paper, we find some sufficient conditions on functions to be in the class $\mathcal{P}({\alpha},{\beta})$. One of these results is applied to the integral operator $I_{{\mu},{\nu}}$ of two classes of starlike functions which are related to the class $\mathcal{P}({\alpha},{\beta})$.