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