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

Abstract

For a function f starlike of order α, 0 α < 1, a non-constant polynomial Q of degree n which is non-vanishing in the unit disc D and β > 0, we consider the functionand find the largest value of r ∈ (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.