Complex Analysis, Riemann Surfaces and Integrable Systems by Sergey M. Natanzon

Author:Sergey M. Natanzon
Language: eng
Format: epub, pdf
ISBN: 9783030346409
Publisher: Springer International Publishing

with equality holding exactly for parabolic automorphisms.

Exercise 5.20

Show that the last inequality is equivalent to the inequality

which, in turn, holds only for

Fig. 5.14

Now let {C 1, C 2, C 3} be a sequential set and be the roots of (5.2). Then (see Fig. 5.14), and, by Vieta’s formulas,

whence α(λ 1λ 2 − 1) > β(λ 1 + λ 2 − 2). Besides, λ 1λ 2 − 1 > λ 1 + λ 2 − 2, since λ i > 1. Therefore,

Thus,

Now let us prove the converse. Let

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