A Short Proof of a Conjecture on the Integral Means of the Derivative of a Convex Function

For d > 0 let Dd = {z : |z| < d} with D1 = D and let ∂Dd denote the boundary of Dd. Let S be the standard class of analytic, univalent functions f on D, normalized by f(0) = 0 and f ′(0) = 1 and let K denote the wellknown class of convex functions in S. For 0 ≤ α < 1 let S∗(α) denote the subclass of S of starlike functions of order α, i.e., a function f ∈ S∗(α) if and only if f satisfies the condition Re zf ′(z)/f(z) > α on D. It is well known that K ⊂ S∗(1/2). For F ⊂ S and for 1/4 ≤ d ≤ 1 let