On a class of analytic functions with missing coefficients

Let T-n(A, B, alpha) denote the class of functions of the form: f(z) = z + Sigma(infinity)(k=n+1) a(k)z(k) (n is an element of N = {1, 2, 3, ... }), which are analytic in the open unit disk U and satisfy the following subordination condition: f'(z) + alpha zf ''(z) < 1 + Az/1 + Bz (z is an element of U; -1 <= B < 1; B < A; alpha > 0). In this paper, we find the sharp lower bound on |z| = r < 1 for the functional Re{f'(z) + alpha zf ''(z)} over the class: T-n(A, B, 0) = { f(z) = z + Sigma(infinity)(k=n+1) a(k)z(k) f'(z) < 1 + Az/1 + Bz (n is an element of N = {1, 2, 3, ... }; z is an element of U)}. By applying this result, several interesting consequences are given. (C) 2009 Elsevier Inc. All rights reserved.