Schwarzian derivative and convexity of order α

For 0 <= alpha <1, a normalised analytic function f defined on the unit disc D is convex of order alpha if Re Q(CV) (f) > alpha, where Q(CV) (f) := 1 + zf ''(z)/f'(z). We find numerous sufficient conditions for the function f to be alpha convex function of order a in terms of Q(CV)(f) and Q(SD)(f) := z(2){f, z}, where {f, z} := (f ''(z)/f'(z))' - (f ''(z)/f'(z))(2)/2 is the Schwarzian derivative of f We obtain these conditions by using the theory of second order differential subordination.