Nevanlinna characteristic

Remarks

Example: for $f(z) = e^z$, $\log^+ \abs{f(re^{i\theta})} = \max(r\cos\theta, 0)$ and there are no poles, so $T(r, f) = \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2} r \cos\theta \, d\theta = r/\pi$. Non-example: the maximum modulus $\max_{\abs{z} = r} \abs{f(z)}$ is no growth measure for $f(z) = \tan z$. It is infinite whenever the circle $\abs{z} = r$ passes through a pole $\pi/2 + k\pi$. The characteristic avoids this by counting the poles in $N(r, f)$. The function $T(r, f)$ is increasing and a convex function of $\log r$. For an entire function $f$ and $M(r) = \max_{\abs{z} = r} \abs{f(z)}$, one has $T(r, f) \le \log^+ M(r) \le 3\, T(2r, f)$, so $T$ and $\log M$ define the same order. A meromorphic function is rational exactly when $T(r, f) = O(\log r)$, and a rational function of degree $d$ has $T(r, f) = d \log r + O(1)$. The Ahlfors–Shimizu characteristic, defined from the spherical area of the image of the disc $\abs{z} < r$, differs from $T(r, f)$ by a bounded function.