• Title/Summary/Keyword: starlike domain

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COEFFICIENT ESTIMATES FOR FUNCTIONS ASSOCIATED WITH VERTICAL STRIP DOMAIN

  • Bulut, Serap
    • 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 fα,β which maps the open unit disc 𝕌 onto the vertical strip domain Ωα,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 Ωα,β. Also, we investigate the Fekete-Szegö type coefficient bounds for functions belonging to these classes.

QUASICONFORMAL EXTENSIONS OF STARLIKE HARMONIC MAPPINGS IN THE UNIT DISC

  • Hamada, Hidetaka;Honda, Tatsuhiro;Shon, Kwang Ho
    • 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 ${\alpha}{\in}(0,1)$.

Univalent Functions Associated with the Symmetric Points and Cardioid-shaped Domain Involving (p,q)-calculus

  • Ahuja, Om;Bohra, Nisha;Cetinkaya, Asena;Kumar, Sushil
    • 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.

MEROMOR0PHIC UNIVALENT HARMONIC FUNCTIONS WITH NEGATIVE COEFFICIENTS

  • Jahangiri, Jay M.;Silverman, Herb
    • 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.

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ON COEFFICIENT PROBLEMS FOR STARLIKE FUNCTIONS RELATED TO VERTICAL STRIP DOMAINS

  • Kwon, Oh Sang;Sim, Young Jae
    • 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.

ON PARTIAL SOLUTIONS TO CONJECTURES FOR RADIUS PROBLEMS INVOLVING LEMNISCATE OF BERNOULLI

  • Gurpreet Kaur
    • 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.

REGIONS OF VARIABILITY FOR GENERALIZED α-CONVEX AND β-STARLIKE FUNCTIONS, AND THEIR EXTREME POINTS

  • Chen, Shaolin;Huang, Aiwu
    • 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.

THE BRIOT-BOUQUET DIFFERENTIAL SUBORDINATION ASSOCIATED WITH VERTICAL STRIP DOMAINS

  • Sim, Young Jae;Kwon, Oh Sang
    • 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 $\mathcal{P}({\alpha},{\beta})$.