Coefficient inequalities for univalent starlike functions

Abstract

Abstract Let A be the class of analytic functions in the unit disk $$\mathbb{D}$$ with the normalization f(0) = f′(0) − 1 = 0. In this paper the authors discuss necessary and sufficient coefficient conditions for f ∈ A of the form $$\left( {\frac{z} {{f(z)}}} \right)^\mu = 1 + b_1 z + b_2 z^2 + \ldots$$ to be starlike in $$\mathbb{D}$$ and more generally, starlike of some order β, 0 ≤ β < 1. Here µ is a suitable complex number so that the right hand side expression is analytic in $$\mathbb{D}$$ and the power is chosen to be the principal power. A similar problem for the class of convex functions of order β is open.