Nevanlinna theory

The Nevanlinna theory , named after its founder Rolf Nevanlinna , belongs to the mathematical branch of function theory . It makes statements about the value distribution of meromorphic functions .

overview

The basic idea of ​​the Nevanlinna theory (or value distribution theory) is to obtain a quantitative version of Picard's theorem . This theorem says that there is no non-constant meromorphic function for different values from the Riemann sphere . To get a quantitative version of this theorem, consider for and the number of -places of a non-constant, meromorphic function in a closed circle around 0 with a radius . The digits are counted according to the multiplicity. Instead of the function, the integrated number function turns out to be more suitable ${\displaystyle \displaystyle a_{1},a_{2},a_{3}}$ ${\displaystyle \displaystyle {\overline {\mathbb {C} }}}$ ${\displaystyle \displaystyle f:\mathbb {C} \to {\overline {\mathbb {C} }}\setminus \{a_{1},a_{2},a_{3}\}}$ ${\displaystyle \displaystyle r>0}$ ${\displaystyle \displaystyle a\in {\overline {\mathbb {C} }}}$ ${\displaystyle \displaystyle n(r,a,f)}$ ${\displaystyle \displaystyle a}$ ${\displaystyle \displaystyle f}$ ${\displaystyle \displaystyle r}$ ${\displaystyle \displaystyle a}$ ${\displaystyle \displaystyle n(r,a,f)}$

${\displaystyle \displaystyle N(r,a,f)=\int _{0}^{r}{\frac {n(t,a,f)}{t}}dt}$

consider. ( This has to be modified slightly, see below.) Nevanlinna now defined a characteristic function which tends towards infinity and showed that for most of the values ​​of the functions and are of the same order of magnitude. More precisely, its two main theorems state that ${\displaystyle \displaystyle a=f(0)}$ ${\displaystyle \displaystyle T(r,f)}$ ${\displaystyle \displaystyle r}$ ${\displaystyle \displaystyle a}$ ${\displaystyle \displaystyle T(r,f)}$ ${\displaystyle \displaystyle N(r,a,f)}$

${\displaystyle \displaystyle N(r,a,f)\leq T(r,f)+O(1)}$

for everyone and ${\displaystyle \displaystyle a\in {\overline {\mathbb {C} }}}$

${\displaystyle \sum _{j=1}^{q}N(r,a_{j},f)\geq (q-2)T(r,f)-S(r,f)}$

for different ones , with a very small error term compared to . Picard's theorem follows immediately from this. ${\displaystyle \displaystyle a_{1},a_{2},\dots ,a_{q}\in {\overline {\mathbb {C} }}}$ ${\displaystyle \displaystyle T(r,f)}$ ${\displaystyle \displaystyle S(r,f)}$

The Nevanlinna characteristic

So that the integral defining the function also exists for, the number function is defined more precisely than given above by ${\displaystyle \displaystyle N(r,a,f)}$ ${\displaystyle \displaystyle a=f(0)}$

${\displaystyle N(r,a,f)=\int _{0}^{r}{\frac {n(t,a,f)-n(0,a,f)}{t}}dt+n(0,a,f)\log r.}$

Obviously holds true and for . You also write briefly , with what for . Furthermore, the oscillation function is defined through ${\displaystyle \displaystyle n(r,a,f)=n(r,\infty ,1/(f-a))}$ ${\displaystyle \displaystyle N(r,a,f)=N(r,\infty ,1/(f-a))}$ ${\displaystyle \displaystyle a\in \mathbb {C} }$ ${\displaystyle \displaystyle N(r,f)=N(r,\infty ,f)}$ ${\displaystyle \displaystyle N(r,1/(f-a))=N(r,a,f)}$ ${\displaystyle \displaystyle a\in \mathbb {C} }$

${\displaystyle m(r,f)=m(r,\infty ,f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\log ^{+}|f(re^{i\theta })|d\theta ,}$

whereby . For one sets accordingly . The Nevanlinna characteristic is then defined by ${\displaystyle \displaystyle \log ^{+}x=\max\{0,\log x\}}$ ${\displaystyle \displaystyle a\in \mathbb {C} }$ ${\displaystyle \displaystyle m(r,a,f)=m(r,\infty ,1/(f-a))}$ ${\displaystyle \displaystyle T(r,f)}$

${\displaystyle \displaystyle T(r,f)=N(r,f)+m(r,f).}$

It applies to when is not constant. Is transcendent, even applies ${\displaystyle \displaystyle T(r,f)\to \infty }$ ${\displaystyle \displaystyle r\to \infty }$ ${\displaystyle \displaystyle f}$ ${\displaystyle \displaystyle f}$

${\displaystyle \lim _{r\to \infty }{\frac {T(r,f)}{\log r}}=\infty .}$

For entire functions is the maximum amount

${\displaystyle M(r,f)=\max _{|z|\leq r}|f(z)|}$

a measure of the growth of the function. for true ${\displaystyle \displaystyle 1<r<R}$

${\displaystyle T(r,f)\leq \log ^{+}M(r,f)\leq {\dfrac {R+r}{R-r}}T(R,f).}$

The order of a meromorphic function is defined by ${\displaystyle \displaystyle \rho (f)}$ ${\displaystyle \displaystyle f}$

${\displaystyle \rho (f)=\limsup _{r\rightarrow \infty }{\dfrac {\log T(r,f)}{\log r}}.}$

For whole functions, due to the above relationship between Nevanlinna characteristic and maximum amount, one can replace here with . Finite order functions form an important and extensively investigated class of meromorphic functions. ${\displaystyle \displaystyle T(r,f)}$ ${\displaystyle \displaystyle \log M(r,f)}$

As an alternative to the Nevanlinna characteristic, one can also use a variant introduced by Lars Valerian Ahlfors and Shimizu Tatsujirō . The Ahlfors-Shimizu characteristic differs from the Nevanlinna characteristic only by a limited term.

The Nevanlinnaschen main clauses

The first law is that for everyone ${\displaystyle a\in {\overline {\mathbb {C} }}}$

${\displaystyle \displaystyle T(r,f)=N(r,a,f)+m(r,a,f)+O(1)}$

applies. In particular, then

${\displaystyle \displaystyle N(r,a,f)\leq T(r,f)+O(1).}$

The first law is a simple consequence of Jensen's formula .

The second law is much deeper . This says that for different the inequality ${\displaystyle \displaystyle a_{1},a_{2},\dots ,a_{q}\in {\overline {\mathbb {C} }}}$

${\displaystyle \sum _{j=1}^{q}m(r,a_{j},f)\leq 2T(r,f)-N_{1}(r,f)+S(r,f)}$

holds, where

${\displaystyle \displaystyle N_{1}(r,f)=2N(r,f)-N(r,f')+N\left(r,{\dfrac {1}{f'}}\right)\geq 0}$

and is an error term that is too small compared to . More precisely, there is a set of finite measures such that ${\displaystyle \displaystyle S(r,f)}$ ${\displaystyle \displaystyle T(r,f)}$ ${\displaystyle \displaystyle E\subset [1,\infty )}$

${\displaystyle \displaystyle S(r,f)=O(\log T(r,f))+O(\log r)}$

for , . ${\displaystyle \displaystyle r\to \infty }$ ${\displaystyle \displaystyle r\notin E}$

With the help of the first law you can see that the inequality

${\displaystyle (q-2)T(r,f)\leq \sum _{j=1}^{q}N(r,a_{j},f)-N_{1}(r,f)+S(r,f)}$

is an equivalent formulation of the second law.

The term counts the multiple digits of the function. Is called with and the and respective functions, but also multiple -Make be counted only once, we obtain ${\displaystyle \displaystyle N_{1}(r,f)}$ ${\displaystyle \displaystyle {\overline {n}}(r,a,f)}$ ${\displaystyle \displaystyle {\overline {N}}(r,a,f)}$ ${\displaystyle \displaystyle n(r,a,f)}$ ${\displaystyle \displaystyle N(r,a,f)}$ ${\displaystyle \displaystyle a}$

${\displaystyle (q-2)T(r,f)\leq \sum _{j=1}^{q}{\overline {N}}(r,a_{j},f)+S(r,f).}$

The defect relation

One of the essential consequences of the second law is the defect relation . For is called ${\displaystyle \displaystyle a\in {\overline {\mathbb {C} }}}$

${\displaystyle \delta (a,f)=\liminf _{r\rightarrow \infty }{\frac {m(r,a,f)}{T(r,f)}}=1-\limsup _{r\rightarrow \infty }{\dfrac {N(r,a,f)}{T(r,f)}}}$

Nevanlin defect of . The second equal sign applies after the first main clause, because for . (It is always assumed that is not constant.) It follows from the first law that for all . Is called defective value or Nevanlinnaschen exception value if true. According to the second law, the amount of defective values ​​can be counted and the defective relation applies ${\displaystyle \displaystyle a}$ ${\displaystyle \displaystyle T(r,f)\to \infty }$ ${\displaystyle \displaystyle r\to \infty }$ ${\displaystyle \displaystyle f}$ ${\displaystyle \displaystyle 0\leq \delta (a,f)\leq 1}$ ${\displaystyle a\in {\overline {\mathbb {C} }}}$ ${\displaystyle \displaystyle a}$ ${\displaystyle \displaystyle \delta (a,f)>0}$

${\displaystyle \sum _{a}\delta (a,f)\leq 2,}$

whereby the sum is formed over all defective values. The defect relation is a far-reaching generalization of Picard's theorem , because if it is transcendent and only takes the value finitely often, then it holds . Even a tightening of Picard's theorem given by Borel follows easily from the second law. ${\displaystyle \displaystyle f}$ ${\displaystyle \displaystyle f}$ ${\displaystyle \displaystyle a}$ ${\displaystyle \displaystyle \delta (a,f)=1}$

Further results on defects

A central problem of the Nevanlinna theory has long been whether the defect relation and the inequality are the only restrictions for the Nevanlinnadefects of a meromorphic function. This so-called inversion problem of the Nevanlin theory was solved in 1976 by David Drasin . (For whole functions it had previously been solved by Wolfgang Fuchs and Walter Hayman .) For functions of finite order, however, there are various other restrictions. For example, if the defect relation is equal, it follows with a natural number . This was suspected by Rolf Nevanlinna's brother Frithiof and was proven by Drasin in 1987. Another result on Nevanlinne defects of meromorphic functions of finite order is a result by Allen Weitsman, who showed in 1972 that for such functions ${\displaystyle \displaystyle 0\leq \delta (a,f)\leq 1}$ ${\displaystyle \displaystyle \rho (f)=n/2}$ ${\displaystyle n\geq 2}$

${\displaystyle \sum _{a}\delta (a,f)^{1/3}<\infty }$

applies.

Many more results on Nevanlinne defects can be found in the books below, the book by Goldberg and Ostrovskii containing an appendix by A. Eremenko and JK Langley, in which more recent developments are also presented.

Applications

Nevanlinna theory has found applications in various fields. It has proven to be an essential aid in the investigation of differential equations and functional equations in the complex, see for example the books by Jank-Volkmann and Laine.

Nevanlinna proved as one of the first applications of his theory the following uniqueness theorem : If the -places of two meromorphic functions and for 5 values match, then the following applies . This sentence was the starting point for many other sentences of this type. ${\displaystyle \displaystyle a}$ ${\displaystyle \displaystyle f}$ ${\displaystyle \displaystyle g}$ ${\displaystyle \displaystyle a\in {\overline {\mathbb {C} }}}$ ${\displaystyle \displaystyle f=g}$

More recently, analogies found by Paul Vojta between Nevanlinna theory and Diophantine approximation have met with great interest, cf. the book of Ru.

Generalizations

This article is limited to the classical theory in a complex variable. There are various generalizations, for example on algebroid functions, holomorphic curves, functions of several complex variables and quasi-regular mappings.

literature

• AA Goldberg , IV Ostrovskii : Distribution of values ​​of meromorphic functions. American Mathematical Society, 2008; (Translation: Russian original 1970).
• WK Hayman : Meromorphic functions. Oxford University Press, 1964.
• G. Jank, L. Volkmann: Introduction to the theory of whole and meromorphic functions with applications to differential equations. Birkhäuser, Basel / Boston / Stuttgart 1985.
• I. Laine: Nevanlinna theory and complex differential equations. Walter de Gruyter, New York 1993.
• R. Nevanlinna: Le théorème de Picard-Borel et la théorie des fonctions méromorphes. Gauthier-Villars, Paris 1929.
• R. Nevanlinna: Unique analytic functions. Springer, Berlin 1953.
• Min Ru : Nevanlinna theory and its relation to Diophantine approximation. World Scientific, River Edge, NJ, 2001.

Individual evidence

1. ^ R. Nevanlinna: On the theory of meromorphic functions. In: Acta Mathematica. Volume 46, 1925, pp. 1-99.
2. ↑ D. Drasin: The inverse problem in Nevanlinna theory. In: Acta Mathematica. Volume 138, 1976, pp. 83-151. Updated in: D. Drasin: On Nevanlinnas inverse problem. In: Complex Variables Theory Application. Volume 37, 1998, pp. 123-143.
3. ^ D. Drasin: Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two. In: Acta Mathematica. 158, 1987, pp. 1-94.
4. ↑ A. Weitsman: A theorem on Nevanlinna deficiencies. In: Acta Mathematica. Volume 128, 1972, pp. 41-52.
5. ^ R. Nevanlinna: Some uniqueness theorems in the theory of the meromorphic functions. In: Acta Mathematica. Volume 48, 1926, pp. 367-391.
6. ^ H. Weyl: Meromorphic functions and analytic curves. Princeton University Press, 1943.
7. ↑ S. Rickman: Quasi Regular mappings. Springer-Verlag, Berlin 1993.