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

TL;DR

This paper investigates the starlikeness of a product of a starlike function and a non-vanishing polynomial raised to a power, determining sharp radii for various subclasses of starlike functions.

Contribution

It introduces new sharp radius results for the starlikeness of functions formed by multiplying a starlike function with a non-vanishing polynomial raised to a fractional power.

Findings

Derived sharp radii for starlikeness in various subclasses.

Established correlation with existing radii results as special cases.

Extended known results to broader classes of functions.

Abstract

For a function $f$ starlike of order $\alpha$, $0\leqslant \alpha <1$, a non-constant polynomial $Q$ of degree $n$ which is non-vanishing in the unit disc $\mathbb{D}$ and $\beta>0$, we consider the function $F:\mathbb{D}\to\mathbb{C}$ defined by $F(z)=f(z) (Q(z))^{\beta /n}$ 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 $\lambda$, 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.