arXiv subjects

Somya Malik

Publications and source records attributed to Somya Malik.

4 recordsLinked to original sources

arXiv2022

Starlikeness of a product of starlike functions with non-vanishing polynomials

For a function $f$ starlike of order $α$, $0\leqslant α<1$, a non-constant polynomial $Q$ of degree $n$ which is non-vanishing in the unit disc $\mathbb{D}$ and $β>0$, we consider the function $F:\mathbb{D}\to\mathbb{C}$ defined by $F(z)=f(z) (Q(z))^{β/n}$ and find the largest value of $r\in (0,1]$ such that $r^{-1} F(rz)$ lies in various known subclasses of starlike functions such as the class of starlike functions of order $λ$, the classes of starlike functions associated with the exponential function, cardioid, a rational function, nephroid domain and modified sigmoid function. Our radii results are sharp. We also discuss the correlation with known radii results as special cases.

math.CV↗

arXiv2022

On functions starlike with respect to $n$-ply symmetric, conjugate and symmetric conjugate points

For given non-negative real numbers $α_k$ with $ \sum_{k=1}^{m}α_k =1$ and normalized analytic functions $f_k$, $k=1,\dotsc,m$, defined on the open unit disc, let the functions $F$ and $F_n$ be defined by $ F(z):=\sum_{k=1}^{m}α_k f_k (z)$, and $F_{n}(z):=n^{-1}\sum_{j=0}^{n-1} e^{-2jπi/n} F(e^{2jπi/n} z)$. This paper studies the functions $f_k$ satisfying the subordination $zf'_{k} (z)/F_{n} (z) \prec h(z)$ where the function $h$ is a convex univalent function with positive real part. We also consider the analogues of the classes of starlike functions with respect to symmetric, conjugate, and symmetric conjugate points. Inclusion and convolution results are proved for these and related classes. Our classes generalize several well-known classes and connection with the previous works are indicated.

math.CV↗

arXiv2022

The Booth Lemniscate Starlikeness Radius for Janowski Starlike Functions

The function $G_α(z)=1+ z/(1-αz^2)$, \, $0\leq α<1$, maps the open unit disc $\mathbb{D}$ onto the interior of a domain known as the Booth lemniscate. Associated with this function $G_α$ is the recently introduced class $\mathcal{BS}(α)$ consisting of normalized analytic functions $f$ on $\mathbb{D}$ satisfying the subordination $zf'(z)/f(z) \prec G_α(z)$. Of interest is its connection with known classes $\mathcal{M}$ of functions in the sense $g(z)=(1/r)f(rz)$ belongs to $\mathcal{BS}(α)$ for some $r$ in $(0,1)$ and all $f \in \mathcal{M}$. We find the largest radius $r$ for different classes $\mathcal{M}$, particularly when $\mathcal{M}$ is the class of starlike functions of order $β$, or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disc contained in $G_α(\mathbb{D})$ and centered at a certain point $a \in \mathbb{R}$.

math.CV↗

arXiv2020

Radius of Starlikeness for Bloch Functions

For normalised analytic functions $f$ defined on the open unit disc $\mathbb{D}$ satisfying the condition $\sup_{z\in \mathbb{D}}(1-|z^2|) |f'(z)|\leq 1$, known as Bloch functions, we determine various starlikeness radii.

math.CV↗