Region of variability of two subclasses of univalent functions

Let F-1 (F-2 respectively) denote the class of analytic functions f in the unit disk vertical bar z vertical bar < I with f (0) 0 = f'(0) - 1 satisfying the condition Re P-f (z) < 3/2 (Re P-f (z) > -1/2 respectively) in vertical bar z vertical bar < 1, where P-f (z) = 1 +zf ''(z)/f'(z). For any fixed z(0) in the unit disk and lambda is an element of [0, 1]),we shall determine the region of variability for log f'(z(0)) when f ranges over the class {f is an element of F-1: f ''(0)= -lambda} and {f is an element of F-2: f ''(0) = 3 lambda}, respectively. (C) 2006 Elsevier Inc. All rights reserved.