Variability regions for Schur class

Let S be the class of analytic functions f in the unit disk D with f( D) ⊂ D̄. Fix pairwise distinct points z₁,…,zₙ₊₁∈ D and corresponding interpolation values w₁,…,wₙ₊₁∈ D̄. Suppose that f∈ S and f(zⱼ)=wⱼ, j=1,…,n+1. Then for each fixed z ∈ D ₁,…,zₙ₊₁ \, we obtained a multi-point Schwarz-Pick Lemma, which determines the region of values of $f(z)$. Using an improved Schur algorithm in terms of hyperbolic divided differences, we solve a Schur interpolation problem involving a fixed point together with the hyperbolic derivatives up to a certain order at the point, which leads to a new interpretation to a generalized Rogosinski's Lemma. For each fixed z₀ ∈ D, j=1,2, … n and γ = (γ₀, γ₁ , … , γₙ) ∈ Dⁿ⁺¹, denote by Hʲf(z) the hyperbolic derivative of order j of f at the point z∈ D, let S (γ) = ∈ S : f (z₀) = γ₀,H¹f (z₀) = γ₁,… ,Hⁿf (z₀) = γₙ \. We determine the region of variability V(z, γ ) = \ f(z) : f ∈ S (γ) \ for z∈ D \ z₀ \, which can be called "the generalized Rogosinski-Pick Lemma for higher-order hyperbolic derivatives".