Growth order of a meromorphic function ($\rho (f)=\mathrm{limsup}{\log }^{+}{T}_f(r)/\log r$)

The growth order of a meromorphic function is the upper power exponent of its Nevanlinna characteristic, with constants assigned order zero. It is different from the local signed order at a zero or pole. A Nevanlinna integrated counting function has an analogous growth order. The Nevanlinna first main theorem bounds every counting order by the function order.

At most two values can have counting order smaller than a positive finite function order. Otherwise the Nevanlinna second main theorem with three such values bounds the characteristic by a smaller power outside a finite-length exceptional set. Monotonicity extends this bound to all large radii by choosing a good radius in $[r,r+1]$, contradicting the original order. Order zero is immediate from nonnegativity and the first theorem.