Bloch's theorem (complex variables)

Bloch's theorem (complex variables) 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.

Contents 1 Statement 2 Landau's theorem 3 Valiron's theorem 4 Proof 4.1 Landau's theorem 4.2 Bloch's Theorem 5 Bloch's and Landau's constants 6 See also 7 References 8 External links 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 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.

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}{2pi i}}oint _{gamma }{frac {f'(w)}{(w-z)^{2}}},mathrm {d} wright|leq {frac {1}{2pi }}cdot 2pi rsup _{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 (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.

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 0.4332approx {frac {sqrt {3}}{4}}+2times 10^{-4}leq Bleq {sqrt {frac {{sqrt {3}}-1}{2}}}cdot {frac {Gamma ({frac {1}{3}})Gamma ({frac {11}{12}})}{Gamma ({frac {1}{4}})}}approx 0.4719,} 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

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