• Kyungpook Mathematical Journal
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• 제55권2호
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• pp.383-394
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• 2015

In this paper, we define two new subclass of k-uniformly starlike and convex functions of order ${\alpha}$ type ${\beta}$ with varying argument of coefficients. Further, we obtain coefficient estimates, extreme points, growth and distortion bounds, radii of starlikeness, convexity and results on modified Hadamard products.

• 호남수학학술지
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• 제38권4호
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• pp.701-724
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• 2016

For $f_i$ belonging to various subclasses of univalent functions, we investigate the product given by $h(z)=z{\prod_{i=1}^{n}}(f_i(z)/z)^{{\gamma}_i}$.The largest radius ${\rho}$ is determined such that $h({\rho}z)/{\rho}$ is starlike of order ${\beta}$, $0{\leq}{\beta}$ < 1 or to belong to other subclasses of univalent functions. We also determine the sharp radius of starlikeness of order ${\beta}$and other radius for the convolution f*g of two starlike functions f, g.

• 대한수학회논문집
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• 제22권1호
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• pp.19-26
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• 2007

In this paper we derive certain sufficient conditions for star-likeness and strongly-starlikeness of analytic functions in U, by using the method of differential subordination.

• 호남수학학술지
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• 제23권1호
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• pp.41-50
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• 2001

This paper concerns with the subclass of normalized analytic function f in D = {z : |z| < 1}, namely a ${\delta}$-convex function in a sector. This subclass is denoted by ${\Delta}({\delta})$, where ${\delta}$ is a real positive. Given $0<{\beta}{\leq}1$ then for $z{\in}D$, the exact ${\alpha}({\beta},\;{\delta})$ is found such that $f{\in}{\Delta}({\delta})$ implies $f{\in}S^*({\beta})$, where $S^*({\beta})$ is starlike of order ${\beta}$ in a sector. This work is a more general version of the result of Nunokawa and Thomas [11].

• East Asian mathematical journal
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• 제21권2호
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• pp.233-240
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• 2005

Let $CS^{\alpha}(\beta)$ denote the class of normalized strongly $\alpha$-logarithmic close-to-convex functions of order $\beta$, defined in the open unit disk $\mathbb{U}$ by $$\|arg\{\(\frac{f(z)}{g(z)}\)^{1-\alpha}\(\frac{zf'(z)}{g(z)\)^{\alpha}\}\|\leq\frac{\pi}{2}\beta,\;(\alpha,\beta\geq0)$$ where $g{\in}S^*$ the class of normalized starlike functions. In this paper, we prove sharp $Fekete-Szeg\"{o}$ inequalities for functions $f{\in}CS^{\alpha}(\beta)$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제10권4호
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• pp.265-271
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• 2003

Let $CS_\alpha(\beta)$ denote the class of normalized strongly $\alpha$-close-to-convex functions of order $\beta$, defined in the open unit disk $\cal{U}$ of $\mathbb{C}$${\mid}arg{(1-{\alpha})\frac{f(z)}{g(z)}+{\alpha}\frac{zf'(z)}{g(z)}}{\mid}\;\leq\frac{\pi}{2}{\beta}(\alpha,\beta\geq0)$ such that $g\; \in\;S^{\ask}$, the class of normalized starlike unctions. In this paper, we obtain the sharp Fekete-Szego inequalities for functions belonging to $CS_\alpha(\beta)$.

• 대한수학회보
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• 제56권4호
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• pp.911-927
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• 2019

Let $\mathscr{B}^{{\eta},{\mu}}_{p,n}\;({\alpha});\;({\eta},{\mu}{\in}{\mathbb{R}},\;n,\;p{\in}{\mathbb{N}})$ denote all functions f class in the unit disk ${\mathbb{U}}$ as $f(z)=z^p+\sum_{k=n+p}^{\infty}a_kz^k$ which satisfy: $$\|\[{\frac{f^{\prime}(z)}{pz^{p-1}}}\]^{\eta}\;\[\frac{z^p}{f(z)}\]^{\mu}-1\| <1-{\frac{\alpha}{p}};\;(z{\in}{\mathbb{U}},\;0{\leq}{\alpha}<p)$$. And $\mathscr{M}^{{\eta},{\mu}}_{p,n}\;({\alpha})$ indicates all meromorphic functions h in the punctured unit disk $\mathbb{U}^*$ as $h(z)=z^{-p}+\sum_{k=n-p}^{\infty}b_kz^k$ which satisfy: $$\|\[{\frac{h^{\prime}(z)}{-pz^{-p-1}}}\]^{\eta}\;\[\frac{1}{z^ph(z)}\]^{\mu}-1\|<1-{\frac{\alpha}{p}};\;(z{\in}{\mathbb{U}},\;0{\leq}{\alpha}<p)$$. In this paper several sufficient conditions for some classes of functions are investigated. The authors apply Jack's Lemma, to obtain this conditions. Furthermore, sufficient conditions for strongly starlike and convex p-valent functions of order ${\gamma}$ and type ${\beta}$, are also considered.

• 호남수학학술지
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• 제40권3호
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• pp.391-400
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• 2018

The object of this paper to construct a new class $$A^m_{{\mu},{\lambda},{\delta}}({\alpha},{\beta},{\gamma},t,{\Psi})$$ of bi-univalent functions of complex order defined in the open unit disc. The second and the third coefficients of the Taylor-Maclaurin series for functions in the new subclass are determined. Several special consequences of the results are also indicated.

• East Asian mathematical journal
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• 제5권1호
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• pp.35-47
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• 1989

Let $S_o*({\alpha},{\beta},{\mu})$ denote the class of functions $f(z)=a_1z-{\sum}{\limit}^{\infty}_{n=2}\;a_nz^n$ analytic in the unit disc $U=\{z:{\mid}z{\mid}<1\}$ and satisfying the condition $${\mid}\frac{\frac{zf'(z)}{f(z)}-1}{(1+\mu)\;\beta(\frac{zf'(z)}{f(z)}-\alpha)-(\frac{zf'(z)}{f(z)}-1)}\mid<1$$ for some $\alpha(0{\leq}{\alpha}<1),\;{\beta}(0<{\beta}{\leq}1),\;{\mu}(0{\leq}{\mu}{\leq}1)$ and for all $z{\in}U$. And it is the purpose of this paper to show a necessary and sufficient condition for the class $S_o*({\alpha},{\beta},{\mu})$, some results for the Hadamard products of two functions f(z) and g(z) in the class $S_o*({\alpha},{\beta},{\mu})$, the distortion theorem and the distortion theorems for the fractional calculus.

• 대한수학회보
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• 제42권3호
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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$_{H}$($\alpha$, $\beta$).