Wiman-Valiron theory explained

Wiman-Valiron theory is a mathematical theory invented by Anders Wiman as a tool to study the behavior ofarbitrary entire functions. After the work of Wiman, the theory was developed by other mathematicians,and extended tomore general classes of analytic functions. The main result of the theory is an asymptotic formula for the functionand its derivatives near the point where the maximum modulus of this function is attained.

Maximal term and central index

By definition, an entire function can be represented by a power series which is convergent for all complex

z :f(z) = \sum ^{inf ty} _{n =0} a ^{n.} _{nz}

The terms of this series tend to 0 as

n\to inf ty , so for each z there is a term of maximal modulus.This term depends on r: = |z| .Its modulus is called the maximal term of the series:\mu(r,f) =max _{k} |a ^{n,} _{n|r} r\geq 0 .

Here

n is the exponent for which the maximum is attained; if there are several maximal terms, we define n as thelargest exponent of them. This number n depends on r , it is denoted by n(r,f) and is called the central index.

Let

M(r,f) =max \{ |f(z)|: |z|\leq r \}

be the maximum modulus of the function

f . Cauchy's inequality implies that \mu(r,f)\leq M(r,f) for all r\geq 0 .The converse estimate M(r,f)\leq (\mu(r,f)) ^{1+\epsilon} was first proved by Borel, anda more precise estimate due to Wiman reads^[1] M(r,f)\leq \mu(r,f)\left( log \mu(r,f)\right) ^{1/2+\epsilon} ,

in the sense that for every

\epsilon> 0 there exist arbitrarily large values of r for which thisinequality holds. In fact, it was shown by Valiron that the above relation holds for "most" values of r : the exceptional set E for which it does not hold has finite logarithmic measure:\int _{E (dr)/(r)} < inf ty.

Improvements of these inequality were subject of much research in the 20th century.^[2]

The main asymptotic formula

The following result of Wiman ^[3] is fundamental for various applications: let

z _{r} be the point for which the maximum inthe definition of M(r,f) is attained; by the Maximum Principle we have |z _{r|=r} . It turnsout that f(z) behaves near the point z _{r} like a monomial: there are arbitrarily large values of r such that the formulaf(z) = ( 1+ o( 1 ))\left( (z)/(z _{r}) \right) ^{n(r,f)} f(z _{r),}

holds in the disk

|z - z _{r|< (r)/(\left(n(r)\right) ^{1/2+\epsilon})} .

Here

\epsilon> 0 is an arbitrary positive number, and the o(1) refers to r\to inf ty, r\not\in E ,where E is the exceptional set described above. This disk is usually called the Wiman-Valiron disk.

Applications

The formula for

f(z) for z near z _{r} can be differentiated so we have an asymptotic relationf ^{(m)} (z) = ( 1+ o( 1 ))\left( (n(r))/(z) \right) ^{m\left( (r)/(z _{r})} \right) ^{n(r)} f(z _{r).}

This is useful for studies of entire solutions of differential equations.

Another important application is due to Valiron^[4] who noticed that the image of the Wiman-Valiron disk contains a "large" annulus (

\{ z:r _{1<|z|<r} _{2\}} where both r _{1} and r _{2/r} _{1} are arbitrarily large). This implies the important theorem of Valiron that there are arbitrarily large discs inthe plane in which the inverse branches of an entire function can be defined. A quantitative version of this statementis known as the Bloch theorem.

This theorem of Valiron has further applications inholomorphic dynamics: it is used in the proof of the fact that the escaping set of an entire function is not empty.

Later development

In 1938, Macintyre ^[5] found that one can get rid of the central index and of power series itself in this theory.Macintyre replaced the central index by the quantity

a(r,f): = r (M'(r,f))/(M(r,f))

and proved the main relation in the form

f ^{(m)} (z) = ( 1+ o( 1 ))\left( (a(r,f))/(z) \right) ^{m\left( (z)/(z _{r})} \right) ^{a(r,f)} f(z _{r) for |z-z} ^{r|\leq (r)/((a(r,f)) ^{1/2+\epsilon})} .

This statement does not mention the power series, but the assumption that

f is entire was used by Macintyre.

The final generalization was achieved byBergweiler, Rippon and Stallard^[6] who showed that this relation persists for every unbounded analytic function

f defined in an arbitrary unbounded regionD in the complex plane, under the only assumption that |f(z)| is bounded for z\in\partial D .The key statement which makes this generalization possible is that the Wiman-Valiron disk is actually contained in D for all non-exceptional r .

References

1. A.. Wiman. Anders Wiman. Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Gliede der zugehörigen taylor'schen Reihe. Acta Mathematica. 1914. 37. 305–326 (German). 10.1007/BF02401837. free. 121155803.
2. W.. Hayman. Walter Hayman. The local growth of power series: a survey of the Wiman-Valiron method. Canadian Mathematical Bulletin. 17. 1974. 3. 317–358. 10.4153/CMB-1974-064-0. free.
3. A.. Wiman. Anders Wiman. Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Betrage bei gegebenem Argumente der Funktion. Acta Mathematica. 1916. 41. 1–28 (German). 10.1007/BF02422938. free. 122491610.
4. Book: Valiron, G.. Georges Valiron

   . Georges Valiron. Lectures on the general theory of integral functions. 1949. Chelsea, reprint of the 1923 ed.. NY.

5. A.. Macintyre. Wiman's method and the "flat regions" of integral functions. Quarterly Journal of Mathematics. 1938. 81–88. 10.1093/qmath/os-9.1.81.
6. W.. Bergweiler. Ph.. Rippon. G.. Stallard. Dynamics of meromorphic functions with direct or logarithmic singularities. Proceedings of the London Mathematical Society. 2008. 97. 2. 368–400. 10.1112/plms/pdn007. 0704.2712. 16873707.