Bloch's theorem (complex analysis)

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

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

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.

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.

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

:::$|f\text{'}\text{'}(z)|=\backslash left|\backslash frac\{1\}\{2\backslash pi\; i\}\backslash oint\_\backslash gamma\backslash frac\{f\text{'}(w)\}\{(w-z)^2\}\backslash ,\backslash mathrm\{d\}w\backslash right|\backslash le\backslash frac\{1\}\{2\backslash pi\}\backslash cdot2\backslash pi\; r\backslash sup\_\{w=\backslash gamma(t)\}\backslash frac$

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.

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.

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

:$0.4332\backslash approx\backslash frac\{\backslash sqrt\{3\}\}\{4\}+2\backslash times10^\{-14\}\backslash leq\; B\backslash leq\; \backslash sqrt\{\backslash frac\{\backslash sqrt\{3\}-1\}\{2\}\}\; \backslash cdot\; \backslash frac\{\backslash Gamma(\backslash frac\{1\}\{3\})\backslash Gamma(\backslash frac\{11\}\{12\})\}\{\backslash Gamma(\backslash frac\{1\}\{4\})\}\backslash approx\; 0.47186,$

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

: {{OEIS|id=A081760}}

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

:

• Table of selected mathematical constants

• {{cite journal|first1=Lars Valerian|last1=Ahlfors|author-link=Lars Ahlfors|last2=Grunsky|first2=Helmut|author2-link=Helmut Grunsky|title=Über die Blochsche Konstante|journal=Mathematische Zeitschrift|year=1937|volume=42|number=1|pages=671–673|doi=10.1007/BF01160101|s2cid=122925005}}
• {{cite conference|first=Albert II|last=Baernstein|author2=Vinson, Jade P.|title=Local minimality results related to the Bloch and Landau constants|book-title=Quasiconformal mappings and analysis|place=Ann Arbor|publisher=Springer, New York|year=1998|pages=55–89}}
• {{cite journal|first=André|last=Bloch|title=Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation|journal=Annales de la Faculté des Sciences de Toulouse|number=3|volume=17|year=1925|pages=1–22|doi=10.5802/afst.335|issn=0240-2963|url=http://www.numdam.org/item/AFST_1925_3_17__1_0.pdf}}
• {{ cite journal | first=Huaihui | last=Chen |author2=Gauthier, Paul M. | title=On Bloch's constant|journal=Journal d'Analyse Mathématique|year=1996|volume=69|number=1|pages=275–291|doi=10.1007/BF02787110|doi-access=| s2cid=123739239 }}
• {{Citation | last1=Landau | first1=Edmund | title=Über die Blochsche Konstante und zwei verwandte Weltkonstanten | doi=10.1007/BF01187791 | year=1929 | journal= Mathematische Zeitschrift | volume=30 | issue=1| pages=608–634 | s2cid=120877278 }}

• {{MathWorld | urlname=BlochConstant| title=Bloch Constant}}
• {{MathWorld | urlname=LandauConstant| title=Landau Constant}}

Category:Unsolved problems in mathematics

Category:Theorems in complex analysis