# 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.

## Statement

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

- [math]\displaystyle{ |f'(0)|=1 }[/math]

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

- [math]\displaystyle{ |f''(z)|=\left|\frac{1}{2\pi i}\oint_\gamma\frac{f'(w)}{(w-z)^2}\,\mathrm{d}w\right|\le\frac{1}{2\pi}\cdot2\pi r\sup_{w=\gamma(t)}\frac{|f'(w)|}{|w-z|^2}\le\frac{2}{r}, }[/math]

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* + *z*^{2}*f″*(*tz*) / 2.

Thus, if |*z*| = 1/3 and |*w*| < 1/6, we have

- [math]\displaystyle{ |(f(z)-w)-(z-w)|=\frac12|z|^2|f''(tz)|\le\frac{|z|^2}{1-t|z|}\le\frac{|z|^2}{1-|z|}=\frac16\lt |z|-|w|\le|z-w|. }[/math]

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

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

For the general case, let *f* be an analytic function in the unit disk such that |*f′*(0)| = 1, and *z*_{0} = 0.

- If |
*f′*(*z*)| ≤ 2|*f′*(*z*_{0})| for |*z*−*z*_{0}| < 1/4, then by the first case, the range of*f*contains a disk of radius |*f′*(z_{0})| / 24 = 1/24. - Otherwise, there exists
*z*_{1}such that |*z*_{1}−*z*_{0}| < 1/4 and |*f′*(*z*_{1})| > 2|*f′*(*z*_{0})|. - If |
*f′*(*z*)| ≤ 2|*f′*(*z*_{1})| for |*z*−*z*_{1}| < 1/8, then by the first case, the range of*f*contains a disk of radius |*f′*(*z*_{1})| / 48 > |*f′*(z_{0})| / 24 = 1/24. - Otherwise, there exists
*z*_{2}such that |*z*_{2}−*z*_{1}| < 1/8 and |*f′*(*z*_{2})| > 2|*f′*(*z*_{1})|.

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}| < 1/2

^{n+1}and |

*f′*(

*z*)| > 2|

_{n}*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 *D*_{0} inside the unit disk such that for every *w* ∈ *D* there is a unique *z* ∈ *D*_{0} with *f*(*z*) = *w*. Thus, *f* is a bijective analytic function from *D*_{0} ∩ *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

- [math]\displaystyle{ 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 0.47186, }[/math]

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

- [math]\displaystyle{ 0.5 \lt L \le \frac{\Gamma(\frac{1}{3})\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{6})} = 0.543258965342... \,\! }[/math] (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

- [math]\displaystyle{ 0.5 \lt A \le 0.7853 }[/math]

## See also

## References

- Ahlfors, Lars Valerian; Grunsky, Helmut (1937). "Über die Blochsche Konstante".
*Mathematische Zeitschrift***42**(1): 671–673. doi:10.1007/BF01160101. - Baernstein, Albert II; Vinson, Jade P. (1998). "Local minimality results related to the Bloch and Landau constants". 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. http://www.numdam.org/item/AFST_1925_3_17__1_0.pdf. - Chen, Huaihui; Gauthier, Paul M. (1996). "On Bloch's constant".
*Journal d'Analyse Mathématique***69**(1): 275–291. doi:10.1007/BF02787110. - Landau, Edmund (1929), "Über die Blochsche Konstante und zwei verwandte Weltkonstanten",
*Mathematische Zeitschrift***30**(1): 608–634, doi:10.1007/BF01187791

## External links

- Weisstein, Eric W.. "Bloch Constant". http://mathworld.wolfram.com/BlochConstant.html.
- Weisstein, Eric W.. "Landau Constant". http://mathworld.wolfram.com/LandauConstant.html.

Original source: https://en.wikipedia.org/wiki/Bloch's theorem (complex variables).
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