= Quotient characterization of bounded characteristic
{title2=$f=u/v,\qquad u,v\in H^\infty(\mathbb D),\quad v\not\equiv0$}
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 $\operatorname{Re}A=H$, so $u=ge^{-A}$ and $v=Be^{-A}$ 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.
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