Distortion theorem and the radius of convexity for Janowski-Robertson functions

In this note, we consider another family of functions that includes the class of convex functions as a proper subfamily. For - pi/2 < alpha < pi/2, we say that f(z) is an element of C-alpha (Lambda, Beta) if i) f(z) is an element of A ii) f' (z) (sic) 0 is an element of D, e(i alpha) (1 + zf'' (z)/f'(z)) - cos (z) + i sin alpha, where p(z) is analytic in D and satisfies the conditions p(0) = 1, p(z) = 1+A phi(z)/1+B phi(z), -1 <= B < Lambda <= 1, phi(z) analytic in D, and phi(0) = 0, vertical bar phi(z)vertical bar < 1for every z is an element of D. The class of C-alpha (A,B) called Janowski-Robertson class. The aim of this paper is to give a distortion theorem and the radius of convexity for the class C-alpha (A, B).