Variability regions for the fourth derivative of bounded analytic functions

Abstract

Let z0 and w0 be given points in the open unit disk D with |w0| < |z0|, and H0 be the class of all analytic self-maps f of D normalized by f (0) = 0. In this paper, we establish the fourth-order Dieudonné's Lemma and apply it to determine the variability region {f (4) (z0) : f ∈ H0, f (z0) = w0, f ′ (z0) = w1, f ′′ (z0) = w2} for given z0, w0, w1, w2 and give the form of all the extremal functions.