Robert B. Burckel

Area, capacity and diameter versions of Schwarz’s Lemma

The now canonical proof of Schwarz’s Lemma appeared in a 1907 paper of Caratheodory, who attributed it to Erhard Schmidt. Since then, Schwarz’s Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc.

Curves, Connectedness and Convexity

We want to develop here as expeditiously as possible some routine facts about the topology of the plane centering on connectedness; these are essential in all that follows. A deeper exploration of the topology of the plane occurs in Chapter IV, after the exponential function has been developed (in Chapter III) and the useful concept of the index of a curve is available.

Convergent Sequences of Holomorphic Functions

We have already seen three amazing results on the convergence of sequences of holomorphic functions [5.38(iv), 5.45(iii) and 5.74] which have no analog for differentiate functions on the line. In this chapter we want to explore this theme more thoroughly. We naturally start with a convenient definition.

Simple and Double Connectivity

We come now to the climax of much of our previous work. To state the result in its most awesome form we introduce some convenient definitions.

Complex) Derivative and (Curvilinear) Integrals

This short chapter is comprised of very basic and very easy material, probably familiar to most readers in one form or other. Even the reader with only modest experience can probably supply his own proofs for most of these results as quickly as he can read mine! We record them here for later reference and to fix notation and terminology.

Schwarz’ Lemma and its Many Applications

This chapter is fairly tightly unified around the following simple result: Theorem 6.1 (Schwarz’ Lemma) Let f be holomorphic and bounded by 1 in D = D(0, 1) and f(0) = 0. Then (i) ∣f′(0)∣ ≤ 1 (ii) ∣f(z)∣ ≤ ∣z∣ ∀z ∈ D and equality in (i) or in (ii) for some non-zero z occurs if and only if (iii) f(z) = cz for some unimodular complex constant c.

Polynomial and Rational Approximation—Runge Theory

The power of “integral representations” was demonstrated in 3.8: they lead to local power series expansions. If the (Riemann) integrals involved are approximated by Riemann sums, we get global rational approximations to the function. (See 8.8 below.) If further the domains are properly disposed in ℂ, the “poles” in these rational functions can be “shoved to infinity” and the rational functions thereby approximated by polynomials. Global polynomial approximants are thus produced. They are a very powerful tool for investigating holomorphic functions: often a theorem is easy for polynomials and remains valid under local uniform convergence, hence its validity passes over to holomorphic unctions generally.

Omitted Values and Normal Families

The center-piece of this chapter is the following remarkable result due to Friedrich Schottky: if f is holomorphic in D(0, 1) and assumes neither of the values 0 or 1, then f is bounded in D(0, r) by a bound depending on r and ∣f(0)∣ only, for each r 0) does not assume 1.] The principal application is the almost immediate fact that a family of holomorphic functions on a common domain, none of which has 0 or 1 in its range, if bounded at a point, is uniformly bounded in a neighborhood of that point. The compactness theorems of Chapter VII then come into play with astounding consequences. Of course “0” and “1” here are convenient normalizations: any two distinct complex numbers would serve as well.

Consequences of the Cauchy—Goursat Theorem—Maximum Principles and the Local Theory

Of great utility in the differential calculus of functions on ℝ are primitives or antiderivatives. The Fundamental Theorem here affirms that integration always produces such primitives. In particular, every continuous function on R has a primitive. [We will see that this is not so for continuous functions on regions in ℂ.] Naturally we look to integration to produce primitives in the plane too. But now we must face the fact that integration can be carried out over many paths from z0 to z, whereas if the integrand f has a primitive F, then the integral over any such path must equal F(z) - F(z0) (see 2.10) and be consequently independent of the path chosen. This independence of path is equivalent to the integral over any closed path being 0. Conversely, when the integral is path independent, a primitive can be manufactured by using an “indefinite” integral (see the proof of 2.11). Thus the corresponding fundamental theorem(s) in the plane affirm that certain integrals over closed paths are 0.

The Index and Some Plane Topology

In this chapter we complete the preliminary investigations of Chapter I on the topology of the plane. The chief tools are the index and deformation and extension techniques. The former rests squarely on the theory of the exponential function developed in the last chapter.