The Jacobian Conjecture for the space of all the inner functions

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Abstract

We prove the Jacobian Conjecture for the space of all the inner functions in the unit disc. 1 Known facts; Definition 1.1. Let BF be the set of all the finite Blaschke products defined on the unit disc D = {z ∈ C | |z| < 1}. Theorem A. f(z) ∈ BF ⇔ ∃ n ∈ Z + such that ∀ w ∈ D the equation f(z) = w has exactly n solutions z1, . . . , zn in D, counting multiplicities. That follows from [1] on the bottom of page 1. Theorem B. (BF , ◦) is a semigroup under composition of mappings. That follows by Theorem 1.7 on page 5 of [2]. Theorem C. If f(z) ∈ BF and if f′ (z) 6= 0 ∀ z ∈ D then f(z) = λ z − α 1 − αz for some α ∈ D and some unimodular λ, |λ| = 1, i.e. f ∈ Aut(D). For that we can look at Remark 1.2(b) on page 2, and remark 3.2 on page 14 of [2]. Also we can look at Theorem A on page 3 of [3].
Original languageEnglish GB
JournalarXiv preprint
StatePublished - 2014

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