The class of alpha-convex functions

Let f be analytic in D = {z : |z| < 1} with f ( z ) = z + ∑ n = 2 ∞ a n z n . . For any real number α, M(α) is the class of alpha-convex, or so-called Ma-Minda functions. By used the definitions of Ma-Minda functions, we get two results. The first, if f is starlike and there is α >1 such that f ∈ M(α), then f is convex. The second, if f is starlike and not convex, then f ∈ M(α) for all α < 0.