An exact estimate of the third Hankel determinants for functions inverse to convex functions

Q3 Mathematics
B. Rath, K. S. Kumar, D. V. Krishna
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引用次数: 0

Abstract

Invesigation of bounds for Hankel determinat of analytic univalent functions is prominent intrest of many researcher from early twenth century to study geometric properties. Many authors obtained non sharp upper bound of third Hankel determinat for different subclasses of analytic univalent functions until Kwon et al. obtained exact estimation of the fourth coefficeient of Caratheodory class. Recently authors made use of an exact estimation of the fourth coefficient, well known second and third coefficient of Caratheodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions. Let $w=f(z)=z+a_{2}z^{2}+\cdots$ be analytic in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$, and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0)=0$, and $f'(0)=1$. Let $z=f^{-1}$ be the inverse function of $f$, given by $f^{-1}(w)=w+t_2w^2+\cdots$ for some $|w|
对逆凸函数的第三个汉克尔行列式的精确估计
解析一元函数的汉克尔行列式界的研究是20世纪初以来许多学者对解析一元函数几何性质研究的一个突出兴趣。许多作者得到了解析一元函数不同子类的第三汉克尔行列式的非尖锐上界,直到Kwon等人得到了卡拉多类第四系数的精确估计。最近作者利用第四系数的精确估计,得到了与解析一元函数子类相关的第三汉克尔行列式的锐界。让 $w=f(z)=z+a_{2}z^{2}+\cdots$ 在单位圆盘上解析 $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$,和 $\mathcal{S}$ 的归一化一元函数的子类 $f(0)=0$,和 $f'(0)=1$. 让 $z=f^{-1}$ 是的反函数 $f$,由 $f^{-1}(w)=w+t_2w^2+\cdots$ 对一些人来说 $|w|<r_o(f)$. 让 $\mathcal{S}^c\subset\mathcal{S}$ 中的凸函数的子集 $\mathbb{D}$. 在本文中,我们估计了逆函数的第三汉克尔行列式的最佳可能上界 $z=f^{-1}$ 什么时候 $f\in \mathcal{S}^c$.让 $\mathcal{S}^c$ 是一类凸函数。我们证明了下列命题(定理) $f\in$ $\mathcal{S}^c$那么,\begin{equation*}\big|H_{3,1}(f^{-1})\big| \leq \frac{1}{36}\end{equation*} 达到不平等是为了 $p_0(z)=(1+z^3)/(1-z^3).$
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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