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

For a complex number α with Re α > 0 let Kφ{symbol} (α) be the class of analytic functions f in the unit disk D with f (0) = 0 satisfying Re (f′ (z) / φ{symbol}′ (z)) > 0 in D, f′ (0) / φ{symbol}′ (0) = α, for some convex univalent function φ{symbol} in D. For any fixed z0 ∈ D, and λ ∈ over(D, -) we shall determine the region of variability Vφ{symbol} (z0, α, λ) for f (z0) when f ranges over the class. Kφ{symbol} (α, λ) = fenced(f ∈ Kφ{symbol} (α) : fenced(frac(d, dz) fenced(frac(f′ (z), φ{symbol}′ (z))))z = 0 = 2 λ (Re α)) . In the final section we graphically illustrate the region of variability for several sets of parameters z0 and α. © 2009 Elsevier Inc. All rights reserved.