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

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)... Show more