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
:$, f'(0), =1$

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
:::$, f''(z), =\left, \frac\oint_\gamma\frac\,\mathrmw\\le\frac\cdot2\pi r\sup_\frac\le\frac,$
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
:::$, (f(z)-w)-(z-w), =\frac12, z, ^2, f''(tz), \le\frac\le\frac=\frac16<, z, -, w, \le, z-w, .$
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′''(z₀), / 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′''(z₀), / 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_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 ''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
:$0.4332\approx\frac+2\times10^\leq B\leq \sqrt \cdot \frac\approx 0.47186,$
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
:$0.5 < L \le \frac = 0.543258965342... \,\!$

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
:$0.5 < A \le 0.7853$

See also

* Table of selected mathematical constants

References

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External links

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* {{MathWorld , urlname=LandauConstant, title=Landau Constant

Unsolved problems in mathematics
Theorems in complex analysis