On products of starlike functions. I

Theorem 1. The radius of starlikeness of the class F(M) is R∗= 1/(2M + 1). The result is sharp for the function F(z) = z(1 −z)2M. P r o o f. Fix m ∈(0, M] and a ∈[−m, m]. Consider the subclass F(m) a = n F ∈F(M) : n X j=1 (1 −αj) Re aj = a o .

Theorem 2. The radius of convexity of the class G(M) is RC = 1 2M + 1. The result is sharp for the function eG(z) = R z 0 (1 −u)2M du. P r o o f. Let G ∈G(M). Since 1 + zG′′(z)/G′(z) = zF ′(z)/F(z) for some F ∈F(M), the proof is a repetition of the proof of Theorem 1. Special cases. Let M > 0. For a fixed A ∈[−M, M] consider the class FA = n F ∈F(M) : A = n X