Bounded p-valent Robertson functions defined by using a differential operator

Let F(M)(lambda, alpha, p, q)(M>1/2,vertical bar lambda vertical bar<pi/2,0 <=alpha<p-q,p is an element of N,q is an element of N(0),p > q) denote the class of functions f(z) = z(p) + Sigma(infinity)(k=p)(+1) a(k)z(k) analytic in U ={z : |z| < 1} which satisfy for fixed M, z = re(i theta) is an element of U and |e(i lambda)zf((1+q))(z)/f((q))(z)-alpha cos lambda-i(p-q)sin lambda/(p-q-alpha)cos lambda-M| < M. Also let G(M)(lambda, alpha, p, q)(M>1/2,vertical bar lambda vertical bar<pi/2,0 <=alpha<p-q,p is an element of N,q is an element of N(0),p > q) denote the class of bounded p-valent lambda-Robertson functions of order alpha, 0 <= alpha < p-q, that is g((q))(z) is an element of G(M)(lambda, alpha, p, q) if and only if (zg((1+q))(z)/(p-q)) is an element of F(M)(lambda, alpha, p, q). In this paper we investigate certain properties of the above-mentioned classes. (C) 2010 The Franklin Institute. Published Elsevier Ltd. All rights reserved.