Weak subordination for convex univalent harmonic functions

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, t1) ⊂ f (Δ, t2) whenever t1, t2 ∈ E and t1 < t2. 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