Conformal Maps And Geometry by Beliaev Dmitry;

Author:Beliaev, Dmitry;
Language: eng
Format: epub
Publisher: World Scientific Publishing Company
Published: 2019-09-15T00:00:00+00:00

For points that are to the left of K, the inequality is reversed.

This lemma has a simple geometric interpretation: the mapping-out function pushes the hull down and it pushes the real line away towards infinity. The proof is a straightforward application of the Schwarz integral formula applied to fK and is left as an exercise.

Exercise 3.19. Use formula (3.21) to prove Lemma 3.30.

Lemma 3.30 can be used to obtain two similar results: one states that gK(x) is monotone in K and the other gives an upper bound on gK(x) − x, given a bound on K.

Lemma 3.31. Suppose that K1 ⊂ K2 are two hulls and let x be a point to the right of K2. Then

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