A holomorphic version of Landau's theorem (Q1060315)

For integers r and n, \(0<r<n\), and a smooth function f defined on \({\mathbb{R}}\), it is possible to bound \(f^{(r)}\) in terms of f and \(f^{(n)}\). Furthermore the function which maximizes \(f^{(r)}\) is essentially unique and has interesting properties. Here the author develops analogs of these results of Landau for holomorphic functions on the disk. Suppose K is a compact subset of the unit disk, \(\zeta\) is a point of the disk not interior to the convex hull of K, and r and n are integers \(0<r<n\). For \(1<p\leq \infty\) define \[ \gamma (\sigma,p)=\sup \{| f^{(r)}(\zeta)|:\quad \sup_{K}| f(z)| \leq 1,\quad \| f^{(n)}\|_ p\leq \sigma \}. \] Here \(\| \|_ p\) is the norm in the Hardy space \(H^ p\) of the disk. It is easy to see that there are extremal functions, that are functions for which this supremum is attained. The author shows that, for each p, the extremal function is unique. He then obtains rather delicate results on the behavior of \(\gamma\) (\(\sigma\),p) as a function of \(\sigma\) (smoothness, asymptotics) both for general K and under special restrictions (e.g. K starlike, K finite). In a final section he derives analytic properties of the extremal functions (e.g. conditions on K, \(\zeta\) which insure that F is analytic on a larger disk). The basic framework for the analysis is the theory of dual extremal problems for spaces of analytic functions.