THE SUBORDINATION PRINCIPLE AND ITS APPLICATION TO THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR

This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties. First, we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination. As applications, we obtain that if beta is an element of [0, 1], gamma is an element of [0, 1/r] and beta + gamma <= 1, then the Roper-Suffridge extension operator Phi(beta, gamma)(f)(z) = (f(z(1)), (f(z(1))/z(1))(beta) (f'(z(1)))(gamma)w), z is an element of Omega(p, r) preserves an almost starlike mapping of complex order lambda on Omega(p, r) = {z = (z(1), w) is an element of C x X : vertical bar z(1)vertical bar(p) + parallel to w parallel to(r)(X) < 1}, where 1 <= p <= 2, r >= 1 and X is a complex Banach space. Second, by applying the principle of subordination, we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order lambda. Finally, we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator. This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator, and enables us to generalize many known results from p = 2 to 1 <= p <= 2.