• Korean Journal of Mathematics
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• 제26권2호
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• pp.191-214
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• 2018

The aim of this paper is to investigate the Fekete $Szeg{\ddot{o}}$ inequality for subclass of analytic functions defined by convolution between generalized Al-Oboudi differential operator and Srivastava-Attiya integral operator. Further, application to fractional derivatives are also given.

• Journal of applied mathematics & informatics
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• 제42권4호
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• pp.997-1006
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• 2024

In this paper, making use of symmetric differential operator, we introduce a new class of ℓ-symmetric - mutually adjoint functions. To make this study more comprehensive and versatile, we have used a differential operator involving three-parameter extension of the well-known Mittag-Leffler functions. Mainly we investigated the inclusion relation and subordination conditions which are the main results of the paper. To establish connections or relations with earlier studies, we have presented applications of main results as corollaries.

• 대한수학회지
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• 제52권4호
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• pp.727-749
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• 2015

We obtain the conditions on ${\beta}$ so that $1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3$ implies p(z) ${\prec}$ (2+z)/(2-z), $1+(1-{\alpha})z$, $(1+(1-2{\alpha})z)/(1-z)$, ($0{\leq}{\alpha}$<1), exp(z) or ${\sqrt{1+z}}$. Similar results are obtained by considering the expressions $1+{\beta}zp^{\prime}(z)/p(z)$, $1+{\beta}zp^{\prime}(z)/p^2(z)$ and $p(z)+{\beta}zp^{\prime}(z)/p(z)$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition ${\mid}log(zf^{\prime}(z)/f(z)){\mid}$ < 1 or ${\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}$ < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid $(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1)=0$.

• 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.

• Kyungpook Mathematical Journal
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• 제61권1호
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• pp.99-110
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• 2021

In the present paper, we determine new characterisations of the subclasses ����∗��(α, β; γ) and ������(α, β; γ) of analytic functions associated with Pascal distribution series ${\Phi}^m_q(z)=z-{\sum_{n=2}^{\infty}}(^{n+m-2}_{m-1})q^{n-1}(1-q)^mz^n$. Further, we give necessary and sufficient conditions for an integral operator related to Pascal distribution series ${\mathcal{G}}^m_qf(z)={\int_{0}^{z}}{\frac{{\Phi}^m_q(t)}{t}}dt$ to belong to the above classes. Several corollaries and consequences of the main results are also considered.

• Kyungpook Mathematical Journal
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• 제58권2호
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• pp.307-318
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• 2018

In this paper, we obtain $Fekete-Szeg{\ddot{o}}$ inequalities for certain class of meromorphic functions f(z) for which $-{\frac{(1-{\frac{{\alpha}}{q}})qzD_qf(z)+{\alpha}qzD_q[zD_qf(z)]}{(1-{\frac{{\alpha}}{q}})f(z)+{\alpha}zD_qf(z)}{\prec}{\varphi}(z)$(${\alpha}{\in}{\mathbb{C}}{\backslash}(0,1]$, 0 < q < 1). Sharp bounds for the $Fekete-Szeg{\ddot{o}}$ functional ${\mid}{\alpha}_1-{\mu}{\alpha}^2_0{\mid}$ are obtained.

• 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.

• 호남수학학술지
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• 제44권1호
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• pp.135-145
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• 2022

This paper aims to investigate characterizations on parameters k₁, k₂, k₃, k₄, k₅, l₁, l₂, l₃, and l₄ to find relation between the class of 𝓗(k, l, m, n, o) hypergeometric functions defined by $$5_F_4\[{\array{k_1,\;k_2,\;k_3,\;k_4,\;k_5\\l_1,\;l_2,\;l_3,\;l_4}}\;:\;z\]=\sum\limits_{n=2}^{\infty}\frac{(k_1)_n(k_2)_n(k_3)_n(k_4)_n(k_5)_n}{(l_1)_n(l_2)_n(l_3)_n(l_4)_n(1)_n}z^n$$. We need to find k, l, m and n that lead to the necessary and sufficient condition for the function zF([W]), G = z(2 - F([W])) and $H_1[W]=z^2{\frac{d}{dz}}(ln(z)-h(z))$ to be in 𝓢^*(2^-r), r is a positive integer in the open unit disc 𝒟 = {z : |z| < 1, z ∈ ℂ} with $$h(z)=\sum\limits_{n=0}^{\infty}\frac{(k)_n(l)_n(m)_n(n)_n(1+\frac{k}{2})_n}{(\frac{k}{2})_n(1+k-l)_n(1+k-m)_n(1+k-n)_nn(1)_n}z^n$$ and $$[W]=\[{\array{k,\;1+{\frac{k}{2}},\;l,\;m,\;n\\{\frac{k}{2}},\;1+k-l,\;1+k-m,\;1+k-n}}\;:\;z\]$$.

• Kyungpook Mathematical Journal
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• 제45권3호
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• pp.395-404
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• 2005

Let α be a complex number with 𝕽α > 0. Let the functions f and g be analytic in the unit disc E = {z : |z| < 1} and normalized by the conditions f(0) = g(0) = 0, f'(0) = g'(0) = 1. In the present article, we study the differential subordinations of the forms $${\alpha}{\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}+{\frac{zf^{\prime}(z)}{f(z)}}{\prec}{\alpha}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}}+{\frac{zg^{\prime}(z)}{g(z)}},\;z{\in}E,$$ and $${\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}{\prec}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}},\;z{\in}E.$$ As consequences, we obtain a number of sufficient conditions for star likeness of analytic maps in the unit disc. Here, the symbol ' ${\prec}$ ' stands for subordination

• 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.