Subordination by convex functions

The following theorem is proven: Let F(z) be convex and univalent.in A = lz: zl 0, -7 _Z 7 -1 ft r) d ' n + 1. Another main tool is the convolution theorem of T. Sheil-Small and the second author. We recall that the Hadamard convolution of two functions f(z) = a azn and g(z) = b zn is denoted by / * g(z) and defined by / * g(z) = a b zn. In [6] the authors prove that the convolution of two convex univalent functions is convex and univalent. A function f(z) analytic in A with f(O) = 0, f '(O) = 1, is said to be convex of order a, .0 < a< 1, Received by the editors August 19, 1974. AMS (MOS) subject classifications (1970). Primary 30A32, 30A40.