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Region of Variability for Exponentially Convex Univalent Functions

Ponnusamy Saminathan
Vasudevarao Allu
M. Vourinen
Journal / Anthology

Complex Analysis and Operator Theory
Year: 2011
Volume: 5
Issue: 3
Page range: 955-966

For α ∈ C\{0} let E(α) denote the class of all univalent functions f in the unit disk D and is given by f (z) = z + a2z2 + a3z3 + , satisfying Re  1 + z f  (z) f (z) + αz f  (z)  >0 inD. For any fixed z0 in the unit disk D and λ ∈ D, we determine the region of variability V(z0, λ) for log f  (z0) + αf (z0) when f ranges over the class Fα(λ) =  f ∈ E(α) : f  (0) = 2λ − α  . We geometrically illustrate the region of variability V(z0, λ) for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.