On Bounded Analytic Functions

The objective of this paper is to give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [l] and Garabedian [2].1 While it is admitted that the main idea to be used is more in the nature of a lucky guess than of a method, it will be found that the gain in brevity and simplicity of the argument is considerable. As a by-product, we shall also obtain a number of hitherto unknown identities between various domain functions. The basic problem treated in the above-mentioned papers is the following generalization of the classical Schwarz lemma: Given a finite schlicht domain D of connectivity n (w=l) in the complex zplane and a point f in D, to find a function F(z) with the following properties: (a) F(z) belongs to the family B of analytic functions /(z) which are single-valued and regular in D and satisfy there |/(z) | = 1; (b) I F'(f)| ^ |/'(f)|, where/(z) is any function in B. Evidently, it is sufficient to solve this problem for any domain D' which is conformally equivalent to D. In particular, we may therefore assume, without restricting the generality of what follows, that D is bounded by analytic curves. It was shown by Ahlfors that F(z) yields a (1, n) conformal mapping of D onto the interior of the unit circle and that n — 1 of the n zeros of F(z) coincide with the zeros of a single-valued function h(z) which is regular in D with the exception of a simple pole at z = f and satisfies — ih(z)dz>0 on the boundary r of D\ the nth zero of F{z) is located at z = f. It was subsequently noticed by Garabedian that the function ä(z) can be written in the form h{z) = F{z)q{z) where q(z) — (z — f)-2 is regular in D and that the extremal property of F(z) can be deduced in a very elegant manner from the resulting inequality