Bloch's Theorem (complex Variables)
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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Bloch's theorem describes the behaviour of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s defined on the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
. 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 ''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.


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 In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
, 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 In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation a ...
, 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 (''zn'') such that , ''zn'' − ''z''''n''−1, < 1/2''n''+1 and , ''f′''(''zn''), > 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 In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
analytic function from ''D''0 ∩ ''f''−1(''D'') to ''D'', so its inverse φ is also analytic by the
inverse function theorem In mathematics, the inverse function theorem is a theorem that asserts that, if a real function ''f'' has a continuous derivative near a point where its derivative is nonzero, then, near this point, ''f'' has an inverse function. The inverse fu ...
.


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 In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
. 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 In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
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

* * * * *


External links

* * {{MathWorld , urlname=LandauConstant, title=Landau Constant Unsolved problems in mathematics Theorems in complex analysis