Weak subordination for convex univalent harmonic functions

Abstract

For two complex-valued harmonic functions f and F defined in the open unit disk Δ with f ( 0 ) = F ( 0 ) = 0 , we say f is weakly subordinate to F if f ( Δ ) ⊂ F ( Δ ) . Furthermore, if we let E be a possibly infinite interval, a function f : Δ × E → C with f ( ⋅ , t ) harmonic in Δ and f ( 0 , t ) = 0 for each t ∈ E is said to be a weak subordination chain if f ( Δ , t 1 ) ⊂ f ( Δ , t 2 ) whenever t 1 , t 2 ∈ E and t 1 < t 2 . In this paper, we construct a weak subordination chain of convex univalent harmonic functions using a harmonic de la Vallée Poussin mean and a modified form of Pommerenke's criterion for a subordination chain of analytic functions.