Region of variability for close-to-convex functions-II

For a complex number alpha with Re alpha > 0 let K-phi (alpha) be the class of analytic functions f in the unit disk D with f (0) = 0 satisfying Re (f'(z)/phi'(z)) > 0 in D; f'(0)/phi'(0) = alpha, for some convex univalent function phi in D. For any fixed z(0) is an element of D, and lambda is an element of (D) over bar we shall determine the region of variability V-phi (z(0), alpha, lambda) for f (z(0)) when f ranges over the class K-phi(alpha,lambda) = {f is an element of K-phi(alpha): d/dz (f'(z)/phi'(z))|(z=0) = 2 lambda(Re alpha)}. In the final section we graphically illustrate the region of variability for several sets of parameters z(0) and alpha. (C) 2009 Elsevier Inc. All rights reserved.