Variability regions for the second derivative of bounded analytic functions

Abstract

Let H be the class of all analytic self-maps of the open unit disk D. Denote by H n f (z) the n-th order hyperbolic derivative of f ∈ H at z ∈ D. For z 0 ∈ D and γIn this paper, we determine the variability region V (z 0 , γ) = {f (n) (z 0 ) : f ∈ H(γ)}, which can be called "the generalized Schwarz-Pick Lemma of n-th derivative". We then apply the generalized Schwarz-Pick Lemma to establish a n-th order Dieudonné's Lemma, which provides an explicit description of the variability region {h (n) (z 0 ) : h ∈ H, h(0) = 0, h(z 0 ) = w 0 , h ′ (z 0 ) = w 1 , . . . , h (n−1) (z 0 ) = w n−1 } for given z 0 , w 0 , w 1 , . . . , w n−1 . Moreover, we determine the form of all extremal functions.