The Bloch Constant of Bounded Analytic Functions

Abstract

In this work we solve the extremal problem of characterizing all bounded analytic functions f: Δ → C (where Δ is the open unit disk) for which the Bloch constant βf = sup{(1−|z|2)|f′(z)|: z ε Δ is a bound. Normalizing, we study f: Δ → Δ with βf = 1. We show that these are precisely the conformal automorphisms of Δ together with those functions whose zeros form an infinite sequence (zn)nεN such that lim supn→∞|g(zn)|Πk≠nzn−zk1−znzk=1, where g is the non‐vanishing function such that f/g is a Blaschke product. In particular, non‐vanishing inner functions, finite Blaschke products, and outer functions for the class H∞(Δ) with image contained in Δ are not extremal functions.

Optimal Estimates for the Gradient of Harmonic Functions in the Unit Disk

“…It is well known that harmonic functions from Hardy class can be represented as Poisson integral u(x) = S P (x, ζ)dµ(ζ), x ∈ B n where P (x, ζ) = 1 − |x| 2 |x − ζ| n , x ∈ B n , ζ ∈ S is Poisson kernel and µ is complex Borel measure. In the case p > 1 this measure is absolutely continuous with respect to σ and dµ(ζ) = f (ζ)dσ.…”

“…We refer to [2,Theorem 1] for the proof of (1.4). In the same paper Colonna proved that, if w is a harmonic mapping of the unit disk into itself, then there hold the following sharp inequality (1.5) β w 4 π .…”

The Bloch constant of bounded analytic functions on a multiply connected domain

“…The techniques used here are somewhat different from those of [3]. There, the main result characterizes the extremal functions in terms of the distribution of their zeros, and at the same time provides a recipe for constructing examples.…”

“…In [3] it is observed that the Bloch constant of a bounded analytic function never exceeds its sup-norm, and those functions / for which ßf = suP|zi<i l/(z)l are characterized. Normalizing, we may consider bounded functions as having image in A and the result becomes the following: where g is a nonvanishing factor of fi such that fi/g is a Blaschke product.…”

“…If we are given a class of Lipschitz functions whose Lipschitz numbers are bounded, a further natural problem is to classify those functions in the class whose Lipschitz numbers are the maximum possible. In [3] the problem was solved for bounded analytic functions on the unit disk. Most of this paper is devoted to studying bounded analytic functions on planar domains (with some restrictions), although in §4 we briefly consider analogous results for complex-valued harmonic functions.…”