8.5 The Great Picard, Bloch, Landau, and Schottky Theorems
Recall the Casorati–Weierstrass Theorem, one of the earliest results on the value distribution near essential singularities:
We will now prove a more advanced characterization of this distribution by methods of differential geometry.
Theorem 8.5.1 (Picard's Great Theorem).
Suppose is holomorphic on a punctured neighborhood of . If is an essential singularity of , then omits at most one value of .
Proof.
Without loss of generality, assume and that omits the values and (otherwise, consider , where and are the omitted values). Define the family
of holomorphic functions on . Since omits and , each element of does as well. By the Fundamental Normality Test (Theorem 8.4.2), is spherically normal. Thus, there exists a subsequence that converges locally uniformly on in the spherical metric. By Proposition 8.4.2, this subsequence converges locally uniformly either to a holomorphic function on or to thereon.
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Suppose converges locally uniformly to a holomorphic function on . Then is uniformly bounded on . Hence, there exists such that
In other words, is bounded by on every circle for . By the Maximum Modulus Principle (Theorem 3.4.1), is then bounded by on each annulus for . As
it follows that is bounded on . By Riemann’s Removable Singularity Theorem (Theorem 3.2.6), therefore extends holomorphically to .
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Suppose converges locally uniformly to on . Then, for every , there exists such that, for all ,
By the same reasoning as in the previous case, on
Thus, by the definition of the limit, , so has a pole at .
In either case, we have derived a meromorphic continuation of to , contradicting the assumption that is an essential singularity of .
Corollary 8.5.1.
Suppose that is meromorphic on a punctured neighborhood of . If omits at least three different values of , then has a meromorphic continuation to .
Proof.
A linear fractional transformation maps the omitted values to , , and , mapping so that it exhibits holomorphy. Similar to Corollary 8.4.1, the preceding result is preserved under the inverse linear fractional transformation.
Remark.
An accumulation point of poles is an essential singularity on the Riemann sphere.
Picard’s Great Theorem is also a generalization of Picard’s Little Theorem (Theorem 8.3.4):
Proof.
Let with an isolated singularity at and a removable singularity at . By Picard’s Great Theorem (Theorem 8.5.1), has a meromorphic extension to . If is removable, by virtue of Proposition 4.3.1 and Theorem 3.2.3, the constancy of and follows.
If instead is a pole of , then is a pole of , and hence is a polynomial. Assume, for the sake of contradiction that is non-constant. Then , the Fundamental Theorem of Algebra (Theorem 3.3.1) gives the existence of some such that . Hence, attains every value . This contradicts the statement and hence is constant.
The efforts of many mathematicians resulted in several alternative proofs following that of Picard; the geometric realization of Ahlfors (Theorem 8.3.1) was followed by results discovered by R. M. Robinson. Other approaches from Nevanlinna theory appeared later in the 20th century.
Picard’s original proof, providing an advanced characterization of the value distribution at essential singularities, relied primarily on the properties of the elliptic modular function (as a “covering map”). From this, Picard deduced that the function would necessarily extend holomorphically across the singularity, contradicting its essential nature. Thus, his proof established that near an essential singularity, a holomorphic function attains every complex value, with at most one exception, infinitely often.
More importantly, we have shown the utility of even seemingly fundamental differential geometry, which can also be used in the proof of many other important results.
The methods of differential geometry can also be used to prove the statements of the following theorems (which can also be independently used to prove the Picard theorems), but is made meaningful with the notion of Riemann surfaces.
Theorem 8.5.2 (Bloch).
Let be holomorphic such that . Then there is a region on which is univalent such that contains a disk with a radius of at least (known as a schlicht disk).
Remark.
Bloch’s constant is defined as the supremum of the radii of such disks that can be contained in for any holomorphic function satisfying .
The precise value of remains unknown to this day. In 1937, H. Grunsky and L. Ahlfors established the bound
where denotes the Gamma function (as in @ eq:gammafunction). Later the lower bound of was given, then to be refined to by M. Bonk, which was further improved to in 1996 by H. Chen and P. M. Gauthier.
Grunsky and Ahlfors actually conjectured that the upper bound in their inequality is exact – that is, .
Theorem 8.5.3 (Landau).
The image of any holomorphic function in satisfying and contains a disk with radius of at least .
Remark.
Similarly, the estimate is not optimal. It was established that the corresponding Landau’s constant lies between and .
Without Riemann surfaces, the proof of the two aforesaid results are rather difficult, as a distinction must be established for a point which two values map to. Otherwise, when we describe a schlicht disk at a point in the image, we may be talking about different “sheets” or “branches,” although each fixed sheet may describe perfectly well-defined analytic functions, although they describe different “copies.” More details may be found in [2].
Hence, for simplicity, we entertain a much simpler case without algebraic branch points.
Theorem 8.5.4.
Let be univalent such that . Then contains a disk with a radius of at least .
Proof.
For , let denote the radius of the largest schlicht disk in centered at (it is mapped to univalently by on some subdomain). Trivially, is and vanishes toward the boundary of .
Define the metric
for and , where is a constant. We may assume that is finite, since otherwise the theorem is already proved for .
For every point , the bounding circle corresponding to passes through a (at least one) boundary point, denoted by . Let and let
By the definition of , we have everywhere in this neighborhood. Since is the pullback of the hyperbolic metric in (8.5.4), the metric has the constant negative curvature of .
Our goal is to construct so that it is the function of a supporting metric for (satisfies the criteria for Theorem 8.3.3). For to be satisfied, we consider
In particular, we want
to be increasing on for arbitrary . The function itself can be calculated to be increasing for by elementary methods (using derivative tests). Therefore, the conditions for a supporting metric are satisfied if . Without loss of generality we let and thus, under the condition that , Theorem 8.3.3 gives that
Let , , so that by the theorem conditions, , and therefore
By the previous assumptions on , the corresponding function on the right-hand side is increasing, and since , we have
As ( was chosen arbitarily, so this is valid), it follows that
It is however notable that the proof follows similarly for general functions, but instead we consider functions , where is a Riemann surface and the “singularities” are not only boundary points but also algebraic branch points (where ).
Theorem 8.5.5 (Landau–Carathéodory).
Let such that and is holomorphic on . If omits and , then dependent only on and such that .
Theorem 8.5.6 (Schottky).
Suppose that is holomorphic and omits and . Then
where (common notation in Nevanlinna theory).
Proof.
Consider conformal map , which extends to , , and continuously such that , , . Explicitly, we have the relationship
as an affine transformation of the inverse Joukowski transform (inverse of ). The solution is given by
where the branch cut of the square root is taken to be the negative real axis, which maps to in terms of (and with the principal branch logarithm). Moreover, this explicit map maps to , since for , the term involving the square root is purely imaginary thus . Because as , , here is a valid conformal map.