2024/04/15 by Chen, Gangqiang
#30C80 #30F45 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.09965
Let \mathcal S be the class of analytic functions f in the unit disk \mathbb D with f(\mathbb D) ⊂ \mathbb D. Fix pairwise distinct points z1,…,zn+1∈ \mathbbD and corresponding interpolation values w1,…,wn+1∈ \mathbbD. Suppose that f∈\mathcal S and f(zj)=wj, j=1,…,n+1. Then for each fixed z ∈ \mathbb D \backslash \z1,…,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 ∈ \mathbb D, j=1,2, … n and γ= (γ0, γ1 , … , γn) ∈ \mathbb Dn+1, denote by Hjf(z) the hyperbolic derivative of order j of f at the point z∈ \mathbb D, let \mathcal S (γ) = \f ∈ \mathcal S : f (z0) = γ0,H1f (z0) = γ1,… ,Hnf (z0) = γn \. We determine the region of variability V(z, γ) = \ f(z) : f ∈ \mathcal S (γ) \ for z∈ \mathbb D \backslash \ z0 \, which can be called "the generalized Rogosinski-Pick Lemma for higher-order hyperbolic derivatives".