Bloch's theorem (complex variables)

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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 lb 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.

Proof[]

Landau's theorem[]

We first prove the case when f(0) = 0, f′(0) = 1, and |f′(z)| ≤ 2 in the unit disc. By Cauchy's integral formula, we have a bound

where γ is the counterclockwise circle of radius r around z, and 0 < r < 1 − |z|. By Taylor's theorem, for each z in the unit disc, there exists 0 ≤ t ≤ 1 such that f(z) = z + z2f″(tz) / 2. Thus, if |z| = 1/3 and |w| < 1/6, we have

By Rouché's theorem, the range of f contains the disc of radius 1/6 around 0.

Let D(z0, r) denote the open disc of radius r around z0. For an analytic function g : D(z0, r) → C such that g(z0) ≠ 0, the case above applied to (g(z0 + rz) − g(z0)) / (rg′(0)) implies that the range of g contains D(g(z0), |g′(0)|r / 6).

For the general case, let f be an analytic function in the unit disc such that |f′(0)| = 1, and z0 = 0.

  • If |f′(z)| ≤ 2|f′(z0)| for |zz0| < 1/4, then by the first case, the range of f contains a disc of radius |f′(z0)| / 24 = 1/24. Otherwise, there exists z1 such that |z1z0| < 1/4 and |f′(z1)| > 2|f′(z0)|.
  • If |f′(z)| ≤ 2|f′(z1)| for |zz1| < 1/8, then by the first case, the range of f contains a disc of radius |f′(z1)| / 48 > |f′(z0)| / 24 = 1/24. Otherwise, there exists z2 such that |z2z1| < 1/8 and |f′(z2)| > 2|f′(z1)|.

Repeating this argument, we either find a disc of radius at least 1/24 in the range of f, proving the theorem, or find an infinite sequence (zn) such that |znzn−1| < 1/2n+1 and |f′(zn)| > 2|f′(zn−1)|. In the latter case the sequence is in D(0, 1/2), so f′ is unbounded in D(0, 1/2), a contradiction.

Bloch's Theorem[]

In the proof of Landau's Theorem above, Rouché's theorem implies that not only can we find a disc D of radius at least 1/24 in the range of f, but there is also a small disc D0 inside the unit disc such that for every wD there is a unique zD0 with f(z) = w. Thus, f is a bijective analytic function from D0f−1(D) to D, so its inverse φ is also analytic by the inverse function theorem.

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

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[]

  • Ahlfors, Lars Valerian; Grunsky, Helmut (1937). "Über die Blochsche Konstante". Mathematische Zeitschrift. 42 (1): 671–673. doi:10.1007/BF01160101. S2CID 122925005.
  • Baernstein, Albert II; Vinson, Jade P. (1998). "Local minimality results related to the Bloch and Landau constants". Quasiconformal mappings and analysis. Ann Arbor: Springer, New York. pp. 55–89.
  • Bloch, André (1925). "Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation". Annales de la Faculté des Sciences de Toulouse. 17 (3): 1–22. doi:10.5802/afst.335. ISSN 0240-2963.
  • Chen, Huaihui; Gauthier, Paul M. (1996). "On Bloch's constant". Journal d'Analyse Mathématique. 69 (1): 275–291. doi:10.1007/BF02787110. S2CID 123739239.
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