• Honam Mathematical Journal
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• v.10 no.1
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• pp.23-30
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• 1988

Let C[C, D] and $S^{*}[C,\;D]$ denote the classes of functions g, g(0)=1-g'(0)0=0, analytic in the unit disc E such that $\frac{(zg{\prime}(z)){\prime}}{g{\prime}(z)}$ and $\frac{zg{\prime}(z)}{g(z)}$ are subordinate to $\frac{1+Cz}{1+Dz{\prime}}$ $z{\in}E$, respectively. In this paper, the classes K[A,B;C,D] and $C^{*}[A,B;C,D]$, $-1{\leq}B<A{\leq}1$; $-1{\leq}D<C{\leq}1$, are defined. The functions in these classes are close-to-convex. Using the properties of convolution operators, we deal with some problems for our classes.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.429-438
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• 2015

The objective of this paper is to obtain sharp upper bound for the second Hankel functional associated with the $k^{th}$ root transform $[f(z^k)]^{\frac{1}{k}}$ of normalized analytic function f(z) when it belongs to the class of starlike and convex functions with respect to symmetric points, defined on the open unit disc in the complex plane, using Toeplitz determinants.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.957-965
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• 2018

We study the class $C{\mathcal{V}} ({\Omega})$ of analytic functions f in the unit disk ${\mathbb{D}}=\{z{\in}{\mathbb{C}}$ : ${\mid}z{\mid}$ < 1} of the form $f(z)=z+{\sum}_{n=2}^{\infty}a_nz^n$ satisfying $$1+\frac{zf^{{\prime}{\prime}}(z)}{f^{\prime}(z)}{\in}{\Omega},\;z{\in}{\mathbb{D}}$$, where ${\Omega}$ is a convex and proper subdomain of $\mathbb{C}$ with $1{\in}{\Omega}$. Let ${\phi}_{\Omega}$ be the unique conformal mapping of $\mathbb{D}$ onto ${\Omega}$ with ${\phi}_{\Omega}(0)=1$ and ${\phi}^{\prime}_{\Omega}(0)$ > 0 and $$k_{\Omega}(z)={\displaystyle\smashmargin{2}{\int\nolimits_{0}}^z}{\exp}\({\displaystyle\smashmargin{2}{\int\nolimits_{0}}^t}{\zeta}^{-1}({\phi}_{\Omega}({\zeta})-1)d{\zeta}\)dt$$. Let $z_0,z_1{\in}{\mathbb{D}}$ with $z_0{\neq}z_1$. As the first result in this paper we show that the region of variability $\{{\log}\;f^{\prime}(z_1)-{\log}\;f^{\prime}(z_0)\;:\;f{\in}C{\mathcal{V}}({\Omega})\}$ coincides wth the set $\{{\log}\;k^{\prime}_{\Omega}(z_1z)-{\log}\;k^{\prime}_{\Omega}(z_0z)\;:\;{\mid}z{\mid}{\leq}1\}$. The second result deals with the case when ${\Omega}$ is the right half plane ${\mathbb{H}}=\{{\omega}{\in}{\mathbb{C}}$ : Re ${\omega}$ > 0}. In this case $CV({\Omega})$ is identical with the usual normalized class of convex univalent functions on $\mathbb{D}$. And we derive the sharp upper bound for ${\mid}{\log}\;f^{\prime}(z_1)-{\log}\;f^{\prime}(z_0){\mid}$, $f{\in}C{\mathcal{V}}(\mathbb{H})$. The third result concerns how far two functions in $C{\mathcal{V}}({\Omega})$ are from each other. Furthermore we determine all extremal functions explicitly.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.589-598
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• 2006

In the present investigation, sharp upper bounds of $|a3-{\mu}a^2_2|$ for functions $f(z)=z+a_2z^2+a_3z^3+...$ belonging to certain subclasses of starlike and convex functions with respect to symmetric points are obtained. Also certain applications of the main results for subclasses of functions defined by convolution with a normalized analytic function are given. In particular, Fekete-Szego inequalities for certain classes of functions defined through fractional derivatives are obtained.

• The Pure and Applied Mathematics
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• v.10 no.4
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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 $CS_\alpha(\beta)$.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.1-16
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• 1996

Earlier in 1935[12], M. S. Robertson introduced the class of quadrant preserving functions. More precisely, let Q be the class of all functions f(z) analytic in the unit disk $D = {z : $\mid$z$\mid$ < 1}$ such that f(0) = 0, f'(0) = 1, and the range f(z) is in the j-th quadrant whenever z is in the j-th quadrant of D, j = 1,2,3,4. This class Q contains the subclass of normalized, odd univalent functions which have real coefficients. On the other hand, this class Q is contained in the class T of odd typically real functions which was introduced by W. Rogosinski [13]. Clearly, if $f \in Q$, then f(z) is real when z is real and therefore the coefficients of f are all real. Recently, it was observed by Y. Abu-Muhanna and T. H. MacGregor [1] that any function $f \in Q$ is odd. Instead of functions "preserving quadrants", the authors [1] have introduced the notion of "preserving sectors".

• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.571-582
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• 1998

The object of the present paper is to derive some sufficient conditions for p-valently close-to-convexity, p-valently starlikeness and p-valently convexity.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.3
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• pp.433-438
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• 2014

A semi-infinite optimization problem involving a quasi-convex objective function and infinitely many convex constraint functions with data uncertainty is considered. A surrogate duality theorem for the semi-infinite optimization problem is given under a closed and convex cone constraint qualification.

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.671-686
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• 2012

This paper mainly deals with the tilted Carath$\acute{e}$odory class by angle ${\lambda}$ ${\in}$ ($-{\pi}/2$, ${\pi}/2$), denoted by $P{\lambda}$) an element of which maps the unit disc into the tilted right half-plane {<${\omega}$ : Re $e^{i{\lambda}}{\omega}$ > 0}. Firstly we will characterize $P{\lambda}$ from different aspects, for example by subordination and convolution. Then various estimates of functionals over $P{\lambda}$ are deduced by considering these over the extreme points of $P{\lambda}$ or the knowledge of functional analysis. Finally some subsets of analytic functions related to $P{\lambda}$ including close-to-convex functions with argument ${\lambda}$, ${\lambda}$-spirallike functions and analytic functions whose derivative is in $P{\lambda}$ are also considered as applications.

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.393-401
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• 2007

In the present paper we introduce the subclass $S^{\lambda}_{a,c}(p,A,B)$ of analytic functions and then we investigate some interesting properties of functions belonging to this subclass. Our results generalize many results known in the literature and especially generalize some of the results obtained by Ling and Liu [5].