Koneri Chandrasekran | Searcharxiv

arXiv2019

Geometric properties of Clausen's Hypergeometric Function $_3F_2(a,b,c;d,e;z)$

The Clausen's Hypergeometric Function is given by $${}_3F_2(a,b,c;d,e;z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n(c)_n}{(d)_n(e)_n(1)_n}z^n\, ; \ a,b,c,d,e\in \mathbb{C}$$ provided $d,\, e\, \neq 0,-1,-2,\cdots$ in unit disc $\mathbb{D} =\{z\in \mathbb{C} \,:\, |z|<1\}$. In this paper, an operator $\mathcal{I}_{a,b,c}(f)(z)$ involving Clausen's Hypergeometric Function by means of Hadamard Product is introduced. Geometric properties of $\mathcal{I}_{a,b,c}(f)(z)$ are obtained based on its Taylor's coefficient.

math.CV↗

arXiv2019

Stable Functions of Janowski Type

A function $f\in \mathcal{A}_1$ is said to be stable with respect to $g\in \mathcal{A}_1 $ if \begin{align*} \frac{s_n(f(z))}{f(z)} \prec \frac{1}{g(z)}, \qquad z\in\mathbb{D}, \end{align*} holds for all $n \in \mathbb{N}$ where $\mathcal{A}_1$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D} =\{z\in \mathbb{C}: |z|<1 \}$ normalized by $f(0)=1$. Here $s_n(f(z))$, the $n^{th}$ partial sum of $f(z)=\displaystyle\sum_{k=0}^{\infty} a_kz^k$ is given by $s_n(f(z)) = \displaystyle\sum_{k=0}^{n} a_kz^k, \ n\in \mathbb{N} \cup \{0\}$. In this work, we consider the following function \begin{align*} v_λ(A,B,z)=\left(\frac{1+Az}{1+Bz}\right)^λ \end{align*} for $-1\leq B < A \leq 1$ and $0\leq λ\leq 1 $ for our investigation. The main purpose of this paper is to prove that $v_λ(A,B,z)$ is stable with respect to $\displaystyle v_λ(0,B,z)= \frac{1}{(1+Bz)^λ}$ for $0 < λ\leq 1 $ and $-1\leq B < A \leq 0$. Further, we prove that $v_λ(A,B,z)$ is not stable with respect to itself, when $0 < λ\leq 1 $ and $-1\leq B < A <0$. \end{abstract}

math.CV↗