Region of variability of two subclasses of univalent functions

Abstract

Let F 1 ( F 2 respectively) denote the class of analytic functions f in the unit disk | z | < 1 with f ( 0 ) = 0 = f ′ ( 0 ) − 1 satisfying the condition Re P f ( z ) < 3 / 2 ( Re P f ( z ) > − 1 / 2 respectively) in | z | < 1 , where P f ( z ) = 1 + z f ″ ( z ) / f ′ ( z ) . For any fixed z 0 in the unit disk and λ ∈ [ 0 , 1 ) , we shall determine the region of variability for log f ′ ( z 0 ) when f ranges over the class { f ∈ F 1 : f ″ ( 0 ) = − λ } and { f ∈ F 2 : f ″ ( 0 ) = 3 λ } , respectively.