Editing Nevanlinna theory (section)

== Nevanlinna characteristic ==

=== Nevanlinna's original definition ===

Let ''f'' be a meromorphic function. For every ''r''&nbsp;≥&nbsp;0, let ''n''(''r'',''f'') be the number of poles, counting multiplicity, of the meromorphic function ''f'' in the disc |''z''| ≤ ''r''. Then define the '''Nevanlinna counting function''' by

:<math>  N(r,f) = \int\limits_0^r\left( n(t,f) - n(0,f) \right)\dfrac{dt}{t} + n(0,f)\log r.\,</math>

This quantity measures the growth of the number of poles in the discs |''z''| ≤ ''r'', as
''r'' increases. Explicitly, let ''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;...,&nbsp;''a''<sub>''n''</sub> be the poles of ''ƒ'' in the punctured disc 0 < |''z''| ≤ ''r'' repeated according to multiplicity. Then ''n'' = ''n''(''r'',''f'') - ''n''(0,''f''), and

:<math>  N(r,f) = \sum_{k=1}^{n} \log \left( \frac{r}{|a_k|}\right) + n(0,f)\log r .\,</math>

Let log<sup>+</sup>''x''&nbsp;=&nbsp;max(log&nbsp;''x'',&nbsp;0). Then the '''proximity function''' is defined by

:<math> m(r,f)=\frac{1}{2\pi}\int_{0}^{2\pi}\log^+ \left| f(re^{i\theta})\right| d\theta. \,</math>

Finally, define the '''Nevanlinna characteristic''' by (cf. [[Jensen's formula]] for meromorphic functions)

: <math>T(r,f) = m(r,f) + N(r,f).\,</math>

=== Ahlfors&ndash;Shimizu version ===
A second method of defining the Nevanlinna characteristic is based on the formula

:<math> \int_0^r\frac{dt}{t}\left(\frac{1}{\pi}\int_{|z|\leq t}\frac{|f'|^2}{(1+|f|^2)^2}dm\right)=T(r,f)+O(1), \,</math>

where ''dm'' is the area element in the plane. The expression in the left hand side is called the
Ahlfors&ndash;Shimizu characteristic. The bounded term ''O''(1) is not important in most questions.

The geometric meaning of the Ahlfors—Shimizu characteristic is the following. The inner integral ''dm'' is the spherical area of the image of the disc |''z''| ≤ ''t'', counting multiplicity (that is, the parts of the [[Riemann sphere]] covered ''k'' times are counted ''k'' times). This area is divided by  {{pi}} which is the area of the whole Riemann sphere. The result can be interpreted as the average number of sheets in the covering of the Riemann sphere by the disc |''z''| ≤ ''t''. Then this average covering number is integrated with respect to ''t'' with weight 1/''t''.

=== Properties ===

The role of the characteristic function in the theory of meromorphic functions in the plane is similar to that of

:<math>\log M(r, f) = \log \max_{|z|\leq r} |f(z)| \,</math>

in the theory of [[entire function]]s. In fact, it is possible to directly compare ''T''(''r'',''f'') and ''M''(''r'',''f'') for an entire function:

:<math>T(r,f) \leq \log^+ M(r,f) \,</math>

and

:<math>\log M(r,f) \leq \left(\dfrac{R+r}{R-r}\right)T(R,f),\,</math>

for any ''R''&nbsp;>&nbsp;''r''.

If ''f'' is a [[rational function]] of degree ''d'', then ''T''(''r'',''f'')&nbsp;~&nbsp;''d''&nbsp;log&nbsp;''r''; in fact, ''T''(''r'',''f'')&nbsp;=&nbsp;''O''(log&nbsp;''r'') if and only if ''f'' is a rational function.

The '''order''' of a meromorphic function is defined by

:<math>\rho(f) = \limsup_{r \rightarrow \infty} \dfrac{\log^+ T(r,f)}{\log r}.</math>

Functions of finite order constitute an important subclass which was much studied.

When the radius ''R'' of the disc |''z''| ≤ ''R'', in which the meromorphic function is defined, is finite, the Nevanlinna characteristic may be bounded. Functions in a disc with bounded characteristic, also known as functions of [[bounded type (mathematics)|bounded type]], are exactly those functions that are ratios of bounded analytic functions. Functions of bounded type may also be so defined for another domain such as the [[upper half-plane]].