Bloch's theorem (complex variables)

For the quantum physics theorem, see Bloch wave.

In complex analysis, a field within mathematics, Bloch's theorem gives a lower bound on the size of a disc in which an inverse to a holomorphic function exists. It is named after André Bloch.

Statement

Let f be a holomorphic function in the unit disk |z| ≤ 1. Suppose that |f′(0)| = 1. Then there exists a disc of radius b and an analytic function φ in this disc, such that f(φ(z)) = z for all z in this disc. Here b > 1/72 is an absolute constant.

Landau's theorem

If f is a holomorphic function in the unit disc with the property |f′(0)| = 1, then the image of f contains a disc of radius l, where l ≥ b is an absolute constant.

This theorem is named after Edmund Landau.

Valiron's theorem

Bloch's theorem was inspired by the following theorem of Georges Valiron:

Theorem. If f is a non-constant entire function then there exist discs D of arbitrarily large radius and analytic functions φ in D such that f(φ(z)) = z for z in D.

Bloch's theorem corresponds to Valiron's theorem via the so-called Bloch's Principle.

Bloch's and Landau's constants

The lower bound 1/72 in Bloch's theorem is not the best possible. The number B defined as the supremum of all b for which this theorem holds, is called the Bloch's constant. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown.

The similarly defined optimal constant L in Landau's theorem is called the Landau's constant. Its exact value is also unknown.

The best known bounds for B at present are

0.4332\approx\frac{\sqrt{3}}{4}+2\times10^{-4}\leq B\leq \sqrt{\frac{\sqrt{3}-1}{2}} \cdot \frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})}\approx 0.4719,

where Γ is the Gamma function. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to Ahlfors and Grunsky. They also gave an upper bound for the Landau constant.

In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of B and L.

References

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