### On Schwarz's Lemma and Inner Functions

Stephen D. Fisher
1969 Transactions of the American Mathematical Society
Introduction. Let D be an arbitrary domain on the Riemann sphere (that is, D is an open connected set on the sphere) and let p be a point of D. Let HX(D) denote the space of bounded analytic functions on D. We consider the problem of maximizing \f'(p)\ for /in Hm{D) with |/(z)| S 1 for all z in D, and/(/?)=0. In the case when D is the unit disc U and p=0, this is Schwarz's lemma and the solution is classical: the maximum is one and if |/'(0)| = 1, then /(z) = ewz for some real constant 6. In
more » ... general case, the theory of normal families assures us that there is at least one/, in HX(D) with ||/0|^1 and f¿(p)^ \f\p)\ for all /in HX(D) bounded by 1 which vanish at p. We call any such function extremal. Theorem 1 of this paper provides an elementary proof that there is only one extremal function. We thus add to our problem the question of investigating the properties of the extremal function. In the case when D is bounded by a finite number of disjoint analytic simple closed curves, the properties of the extremal function have been studied extensively (see  and ). In this case, the extremal function is known to be analytic over the boundary and to have modulus one there. We show that this is a local property and carries over to all points in the boundary of D where the boundary is sufficiently well behaved, no matter what the remainder of the boundary of D is like. In §1 we also show that the extremal function is closely related to the problem of removable singularities for bounded analytic functions. In §2 we discuss a related extremal problem and show that the properties of its solution imply that on many planar domains D, each function analytic and bounded by 1 on D may be approximated uniformly on compact subsets of D by a sequence of inner functions. We also briefly discuss extensions of the results to open Riemann surfaces. Finally, some remarks on previous work on Schwarz's lemma are in order. Carleson [3, pp. 78-82] and Havinson [8, Theorem 9] have previously shown that the extremal function is unique. Our proof differs substantially from each of theirs and is considerably more elementary. Furthermore, it does not make use of any of the results from the finitely-connected case. Also, our Theorems 2 and 3 and a weaker version of Theorem 4 have previously been proved by Havinson [8, Theorems 22, 28, and 20]; however, as in the case of Theorem 1, the proofs given here are appreciably simpler and more elementary. Indeed, part of the motivation for this paper was to find simpler and more transparent proofs of these results. In particular, it is only in the proof of Theorem 5, which describes the behavior of the extremal function at "nice" boundary points, that we make use of any of the results for the finitely-connected case; otherwise the proofs are equally valid for all domains.