Nevanlinna class ($N(\mathbb{D})=\{f:{\sup }_{r<1}\int {\log }^{+}|f(r\xi )|dm(\xi )<\infty \}$)

The analytic Nevanlinna class consists of holomorphic functions on the unit disc with uniformly bounded means of their positive logarithmic modulus. Here $m$ is normalized angular measure. A bounded holomorphic function belongs to this class, but the class also permits unbounded functions. It is the analytic version of bounded characteristic.

A holomorphic function on the unit disc has bounded characteristic when ${\sup }_{r<1}\int {\log }^{+}|f(r\xi )|dm(\xi )<\infty$, where ${\log }^{+}t=\max (0,\log t)$. Thus it belongs to the Nevanlinna class. A general meromorphic function requires the additional pole-counting term in its characteristic; the displayed analytic definition has no such term.

A meromorphic function on the unit disc has bounded characteristic if and only if it has this representation. Its pole Blaschke product $B$ makes $g=Bf$ analytic with bounded positive logarithmic means. The least harmonic majorant by expanding disk lifts gives $H\ge {\log }^{+}|g|$. A harmonic conjugate gives an analytic $A$ with $\mathrm{Re}A=H$, so $u=g{e}^{-A}$ and $v=B{e}^{-A}$ are bounded. Conversely the Nevanlinna first main theorem bounds the characteristic of $u/v$ by that of $u$ and $1/v$, hence by a constant.