Region of Variability for Certain Classes of Univalent Functions Satisfying Differential Inequalities

Abstract. For complex numbers α, β and M ∈ R with 0 < M ≤ |α| and |β| ≤ 1, let B(α, β,M) be the class of analytic and univalent functions f in the unit disk D with f(0) = 0, f (0) = α and f (0) = Mβ satisfying |zf (z)| ≤ M , z ∈ D. Let P(α,M) be the another class of analytic and univalent functions in D with f(0) = 0, f (0) = α satisfying Re (zf (z)) > −M , z ∈ D, where α ∈ C \ {0}, 0 < M ≤ 1/ log 4. For any fixed z0 ∈ D and λ ∈ D we shall determine the region of variability Vj (j = 1, 2) for f ′(z0) when f ranges over the class Sj (j = 1, 2), where