Editing Nevanlinna theory (section)

=== 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]].