Sufficient conditions for starlike and convex functions

For n >= 1, let A denote the class of all analytic functions f in the unit disk Delta of the form f (z) = z + Sigma(infinity)(k=2) a(k)z(k). For Re alpha < 2 and gamma > 0 given, let P(gamma, alpha) denote the class of all functions f E A satisfying the condition vertical bar f'(z) - alpha f(z)/z + alpha - 1 vertical bar <= gamma, z is an element of Delta. We find sufficient conditions for functions in P(gamma, alpha) to be starlike of order beta. A generalization of this result along with some convolution results is also obtained.