Quotient characterization of bounded characteristic ($f=u/v,u,v\in H^{\infty }(\mathbb{D}),v\not\equiv 0$)

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.