• 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$_{H}$($\alpha$, $\beta$).

• 호남수학학술지
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• 제10권1호
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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.

• 호남수학학술지
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• 제19권1호
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• pp.97-105
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• 1997

Let $R_k({\alpha})$, $0{\leq}{\alpha}<1$, $k{\geq}2$ denote certain subclasses of analytic functions in the unit disc E with bounded radius rotation. A function f, analytic in E and given by $f(z)=z+{\sum_{m=2}^{\infty}}a_m{z^m}$, is said to be in the family $R_k(n,{\alpha})n{\in}N_o=\{0,1,2,{\cdots}\}$ and * denotes the Hadamard product. The classes $R_k(n,{\alpha})$ are investigated and same properties are given. It is shown that $R_k(n+1,{\alpha}){\subset}R_k(n,{\alpha})$ for each n. Some integral operators defined on $R_k(n,{\alpha})$ are also studied.

• 대한수학회지
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• 제55권5호
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• pp.1019-1030
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• 2018

In [12,13] the researchers introduced universally convex, universally starlike and universally prestarlike functions in the slit domain ${\mathbb{C}}{\backslash}[1,{\infty})$. These papers extended the corresponding notions from the unit disc to other discs and half-planes containing the origin. In this paper, we introduce universally prestarlike generalized functions of order ${\alpha}$ with ${\alpha}{\leq}1$ and we obtain upper bounds of the second Hankel determinant ${\mid}a_2a_4-a^2_3{\mid}$ for such functions.

• 대한수학회지
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• 제52권6호
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• pp.1139-1148
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• 2015

The estimate of third Hankel determinant $$H_{3,1}(f)=\left|a_1\;a_2\;a_3\\a_2\;a_3\;a_4\\a_3\;a_4\;a_5\right|$$ of the analytic function $f(z)=z+a2z^2+a3z^3+{\cdots}$, for which ${\Re}(1+zf^{{\prime}{\prime}}(z)/f^{\prime}(z))>-1/2$ are investigated. The corrected version of a known results [2, Theorem 3.1 and Theorem 3.3] are also obtained.

• 대한수학회보
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• 제55권3호
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• pp.819-835
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• 2018

Let $f=h+{\bar{g}}$ be a normalized harmonic mapping in the unit disk $\mathbb{D}$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D^{\epsilon}{_f}=zf_z-{\epsilon}{\bar{z}}f_{\bar{z}}({\mid}{\epsilon}{\mid}=1)$ and $F_{\lambda}(z)=(1-{\lambda)f+{\lambda}D^{\epsilon}{_f}(0{\leq}{\lambda}{\leq}1)$ when the coefficients of h and g satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of h and g satisfy the corresponding necessary conditions of the harmonic convex function $f=h+{\bar{g}}$. All results are sharp. Some of the results are motivated by the work of Kalaj et al. [8].

• Korean Journal of Mathematics
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• 제17권2호
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• pp.127-145
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• 2009

In this paper we introduce a new subclass $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ of analytic and multivalent functions with negative coefficients using fractional calculus operators. Connections to the well known and some new subclasses are discussed. A necessary and sufficient condition for a function to be in $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ is obtained. Several distortion inequalities involving fractional integral and fractional derivative operators are also presented. We also give results for radius of starlikeness, convexity and close-to-convexity and inclusion property for functions in the subclass. Modified Hadamard product, application of class preserving integral operator and other interesting properties are also discussed.

• 대한수학회논문집
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• 제33권2호
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• pp.445-458
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• 2018

In this article we study the inclusion properties of the Bessel-Struve kernel functions in the Janowski class. In particular, we find the conditions for which the Bessel-Struve kernel functions maps the unit disk to right half plane. Some open problems with this aspect are also given. The third order differential subordination involving the Bessel-Struve kernel is also considered. The results are derived by defining suitable classes of admissible functions. One of the recurrence relation of the Bessel-Struve kernel play an important role to derive the main results.

• 대한수학회지
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• 제43권2호
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• pp.311-322
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• 2006

For the Briot-Bouquet differential equations of the form given in [1] $${{\mu}(z)+\frac {z{\mu}'(z)}{z\frac {f'(z)}{f(z)}\[\alpha{\mu}(z)+\beta]}=g(z)$$ we can reduce them to $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$ where $$v(z)=\alpha{\mu}(z)+\beta,\;h(z)={\alpha}g(z)+\beta\;and\;F(z)=f(z)/f'(z)$$. In this paper we are going to give conditions in order that if u and v satisfy, respectively, the equations (1) $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$, $${{\mu}(z)+G(z)\frac {v'(z)}{v(z)}=g(z)$$ with certain conditions on the functions F and G applying the concept of strong subordination $g\;\prec\;\prec\;h$ given in [2] by the author, implies that $v\;\prec\;{\mu},\;where\;\prec$ indicates subordination.

• 대한수학회지
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• 제54권1호
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• pp.267-280
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• 2017

In this paper, we study analytic and geometric properties of the solution q(z) to the differential equation q(z) + zq'(z)/q(z) = h(z) with the initial condition q(0) = 1 for a given analytic function h(z) on the unit disk |z| < 1 in the complex plane with h(0) = 1. In particular, we investigate the possible largest constant c > 0 such that the condition |Im [zf"(z)/f'(z)]| < c on |z| < 1 implies starlikeness of an analytic function f(z) on |z| < 1 with f(0) = f'(0) - 1 = 0.