Bloch's theorem (complex analysis)

For the quantum physics theorem, see Bloch's theorem.

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.

Statement

Let f be a holomorphic function in the unit disk |z| ≤ 1 for which

${\displaystyle |f'(0)|=1}$

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.

Landau's theorem

If f is a holomorphic function in the unit disk with the property |f′(0)| = 1, then let L_f 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 L_f over all such functions f, and that L is greater than Bloch's constant L ≥ B.

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 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.

Proof

Landau's theorem

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

${\displaystyle |f''(z)|=\left|{\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f'(w)}{(w-z)^{2}}}\,\mathrm {d} w\right|\leq {\frac {1}{2\pi }}\cdot 2\pi r\sup _{w=\gamma (t)}{\frac {|f'(w)|}{|w-z|^{2}}}\leq {\frac {2}{r}},}$

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

${\displaystyle |(f(z)-w)-(z-w)|={\frac {1}{2}}|z|^{2}|f''(tz)|\leq {\frac {|z|^{2}}{1-t|z|}}\leq {\frac {|z|^{2}}{1-|z|}}={\frac {1}{6}}<|z|-|w|\leq |z-w|.}$

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 (z_n) such that |z_n − z_n−1| < 1/2ⁿ⁺¹ and |f′(z_n)| > 2|f′(z_n−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 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

${\displaystyle 0.4332\approx {\frac {\sqrt {3}}{4}}+2\times 10^{-14}\leq B\leq {\sqrt {\frac {{\sqrt {3}}-1}{2}}}\cdot {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {11}{12}})}{\Gamma ({\frac {1}{4}})}}\approx 0.47186,}$

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.

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

${\displaystyle 0.5<L\leq {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {5}{6}})}{\Gamma ({\frac {1}{6}})}}=0.543258965342...\,\!}$ (sequence A081760 in the OEIS)

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

${\displaystyle 0.5<A\leq 0.7853}$

See also

• Table of selected mathematical constants

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" (PDF). 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

External links

• Weisstein, Eric W. "Bloch Constant". MathWorld.
• Weisstein, Eric W. "Landau Constant". MathWorld.