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

Abstract

For a function $f$ starlike of order $\alpha$, $0⩽\alpha <1$, 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.