Variability regions for certain families of harmonic univalent mappings

Let A be the class of analytic functions h in the unit disk D. Fix h ∈ A with h(0) = h′(0) - 1 = 0 satisfying Re h′(z) > 0 in D. For b ∈ D, we consider the sets B(h, b) = {g ∈ A with g(0) = g′(0) - b = 0 satisfying {pipe}g′(z){pipe} < Re h′(z) in D}. For fixed z0 ∈ D \ {0} and b ∈ D, we study in this article the regions of variabilities of the sets V0(z0, b) = {g(z0): g ∈ B(h, b)} and V1(z0, b) = {g′(z0): g ∈ B(h, b)}. The problem originated from the class of harmonic univalent mappings of the unit disk. © 2013 Copyright Taylor and Francis Group, LLC.