Bloch's theorem (complex variables)
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named after André Bloch.
Let f be a holomorphic function in the unit disk |z| ≤ 1 for which
Bloch's Theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72.
If f is a holomorphic function in the unit disk with the property |f′(0)| = 1, then let Lf be the radius of the largest disk contained in the image of f.
Landau's theorem states that there is a constant L defined as the infimum of Lf over all such functions f, and that L ≥ B.
This theorem is named after Edmund Landau.
Bloch's theorem was inspired by the following theorem of Georges Valiron:
Theorem. If f is a non-constant entire function then there exist disks 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.
We first prove the case when f(0) = 0, f′(0) = 1, and |f′(z)| ≤ 2 in the unit disk. 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 disk, 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 disk of radius 1/6 around 0.
Let D(z0, r) denote the open disk 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 disk such that |f′(0)| = 1, and z0 = 0.
- If |f′(z)| ≤ 2|f′(z0)| for |z − z0| < 1/4, then by the first case, the range of f contains a disk of radius |f′(z0)| / 24 = 1/24. Otherwise, there exists z1 such that |z1 − z0| < 1/4 and |f′(z1)| > 2|f′(z0)|.
- If |f′(z)| ≤ 2|f′(z1)| for |z − z1| < 1/8, then by the first case, the range of f contains a disk of radius |f′(z1)| / 48 > |f′(z0)| / 24 = 1/24. Otherwise, there exists z2 such that |z2 − z1| < 1/8 and |f′(z2)| > 2|f′(z1)|.
Repeating this argument, we either find a disk of radius at least 1/24 in the range of f, proving the theorem, or find an infinite sequence (zn) such that |zn − zn−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.
In the proof of Landau's Theorem above, Rouché's theorem implies that not only can we find a disk D of radius at least 1/24 in the range of f, but there is also a small disk D0 inside the unit disk such that for every w ∈ D there is a unique z ∈ D0 with f(z) = w. Thus, f is a bijective analytic function from D0 ∩ f−1(D) to D, so its inverse φ is also analytic by the inverse function theorem.
Bloch's and Landau's constants
The number B is called the Bloch's constant. The lower bound 1/72 in Bloch's theorem is not the best possible. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown.
The best known bounds for B at present are
The similarly defined optimal constant L in Landau's theorem is called the Landau's constant. Its exact value is also unknown, but it is known that
In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of B and L.
For injective holomorphic functions on the unit disk, a constant A can similarly be defined. It is known that
- 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.
- Landau, Edmund (1929), "Über die Blochsche Konstante und zwei verwandte Weltkonstanten", Mathematische Zeitschrift, 30 (1): 608–634, doi:10.1007/BF01187791, S2CID 120877278