Elementary characteristic laws and fixed rational composition

Define $T_0(r,h)=m_0(r,h)+N(r,\mathrm{\infty };h)$. For every finite $w$, ${\log }^{+}\mathrm{\mid }w\mathrm{\mid }\le \frac{1}{2}\log (1+\mathrm{\mid }w{\mathrm{\mid }}^2)\le {\log }^{+}\mathrm{\mid }w\mathrm{\mid }+\frac{1}{2}\log 2$. By [F1], $m(r,\mathrm{\infty };h)$ is the mean of $\frac{1}{2}\log (1+\mathrm{\mid }h{\mathrm{\mid }}^2)$, so averaging gives $0\le T(r,h)-T_0(r,h)\le \frac{1}{2}\log 2$ for every meromorphic $h$ and $r>0$.

For complex $x,y$, ${\log }^{+}\mathrm{\mid }xy\mathrm{\mid }\le {\log }^{+}\mathrm{\mid }x\mathrm{\mid }+{\log }^{+}\mathrm{\mid }y\mathrm{\mid }$ and ${\log }^{+}\mathrm{\mid }x+y\mathrm{\mid }\le {\log }^{+}\mathrm{\mid }x\mathrm{\mid }+{\log }^{+}\mathrm{\mid }y\mathrm{\mid }+\log 2$. At each point the pole order of either $fg$ or $f+g$ is at most the sum of the pole orders of $f$ and $g$; this includes cancellation and the identically zero sum, whose pole order is zero. For $r\ge 1$, every pole-count weight in [F2] is nonnegative, including the centre weight $\log r$. Integrating the logarithmic bounds and the divisor bounds gives both upper laws for $T_0$; step 1.1 transfers them to $T$.

Suppose first that $f$ is nonconstant and not identically zero. Away from its zeros and poles, [F1] gives $\log (1\mathrm{/}\delta (f,0))=\frac{1}{2}\log (1+\mathrm{\mid }f{\mathrm{\mid }}^2)-\log \mathrm{\mid }f\mathrm{\mid }=\frac{1}{2}\log (1+\mathrm{\mid }1\mathrm{/}f{\mathrm{\mid }}^2)$. The poles of $1\mathrm{/}f$ are precisely the zeros of $f$ with the same multiplicities. Therefore $T(r,1\mathrm{/}f)=m(r,0;f)+N(r,0;f)=T(r,f)+C(f,0)$ by [F1] and [F3]. If $f$ is a nonzero constant, both characteristics are constant in $r$. This proves the reciprocal law for $T$; step 1.1 gives the same law for $T_0$ with a bounded error.

Let $P(w)=a_p{w}^p+\cdots +a_0$ with $p\ge 1$ and $a_p\mathrm{\ne }0$. For sufficiently large $\mathrm{\mid }w\mathrm{\mid }$, the leading term bounds $\mathrm{\mid }P(w)\mathrm{\mid }$ above and below by positive constant multiples of $\mathrm{\mid }w{\mathrm{\mid }}^p$; on the remaining compact $w$-disc both ${\log }^{+}\mathrm{\mid }P(w)\mathrm{\mid }$ and $p{\log }^{+}\mathrm{\mid }w\mathrm{\mid }$ are bounded. Hence ${\log }^{+}\mathrm{\mid }P(w)\mathrm{\mid }=p{\log }^{+}\mathrm{\mid }w\mathrm{\mid }+O_P(1)$ uniformly in $w$. Averaging gives $m_0(r,P(f))=p{m}_0(r,f)+O_P(1)$. At every pole of $f$ of order $\lambda$, the leading term of $P$ makes $P(f)$ have pole order exactly $p\lambda$, and $P(f)$ has no other poles. Thus [F2] gives $N(r,\mathrm{\infty };P(f))=pN(r,\mathrm{\infty };f)$, and step 1.1 yields $T(r,P(f))=pT(r,f)+O_P(1)$. If $P$ is constant, $P(f)$ is constant and has bounded characteristic.

Let $R=P\mathrm{/}Q$ be nonconstant of degree $d\ge 1$, with $P,Q$ coprime. The composition $R(f)$ is nonconstant: otherwise the connected image of the nonconstant meromorphic map $f:\mathbb{C}\to \widehat{\mathbb{C}}$ would lie in a finite fiber of $R$. By step 2.2, replacing $R$ by $1\mathrm{/}R$ changes the characteristic of its composition by only $O_{R,f}(1);$ this handles $\mathrm{deg}P=d>\mathrm{deg}Q$. If $\mathrm{deg}P=\mathrm{deg}Q=d$, subtract $c=R(\mathrm{\infty })$: translating a meromorphic function by a constant changes $m_0$ by a bounded amount and leaves its pole orders unchanged, while $P-cQ$ has degree less than $d$. Thus it suffices to prove the result when $\mathrm{deg}Q=d$ and $\mathrm{deg}P<d$.

In this normalized case, the nonempty finite zero set of $Q$ is disjoint from that of $P$. If $P$ has zeros, let $ϵ$ be one third of the minimum distance between the two finite zero sets; if $P$ has none, take any $ϵ>0$. Let $\mathrm{\Gamma }$ be the union of the open $ϵ$-discs around the zeros of $Q$ and put $\mathrm{\Omega }=\widehat{\mathbb{C}}\setminus \mathrm{\Gamma }$. Its closure avoids the zeros of $P$. On $\bar{\mathrm{\Gamma }}$, $\mathrm{\mid }P\mathrm{\mid }$ has a positive lower bound and a finite upper bound, so $\mathrm{\mid }R\mathrm{\mid }=\mathrm{\mid }P\mathrm{/}Q\mathrm{\mid }$ is bounded above and below by positive constant multiples of $\mathrm{\mid }1\mathrm{/}Q\mathrm{\mid }$. On $\mathrm{\Omega }$, both $R$ and $1\mathrm{/}Q$ are bounded, including at infinity because $\mathrm{deg}P<\mathrm{deg}Q$. Splitting each circle into the sets where $f$ lies in $\mathrm{\Gamma }$ and $\mathrm{\Omega }$, these comparisons and the boundedness of ${\log }^{+}$ on bounded values give $m_0(r,R(f))=m_0(r,1\mathrm{/}Q(f))+O_R(1)$. Values at isolated poles are interpreted through their integrable logarithmic singularities.

The pole divisors of $R(f)$ and $1\mathrm{/}Q(f)$ agree. At a point where $f$ is finite, a pole occurs exactly when $Q(f)=0$; coprimality makes $P(f)\mathrm{\ne }0$ there, so its order is the zero order of $Q(f)$. At a pole of $f$, the inequalities $\mathrm{deg}P<\mathrm{deg}Q=d$ imply both $R(f)\to 0$ and $1\mathrm{/}Q(f)\to 0$, so neither has a pole. Hence [F2] gives $N(r,\mathrm{\infty };R(f))=N(r,\mathrm{\infty };1\mathrm{/}Q(f))$. Combining this with step 4.1 yields $T_0(r,R(f))=T_0(r,1\mathrm{/}Q(f))+O_R(1)$.

Since $f$ is nonconstant, $Q(f)$ is not identically zero: otherwise the connected image of $f$ would lie in the finite zero set of $Q$, forcing $f$ to be constant. Applying the reciprocal law of step 2.2 and the polynomial law of step 2.3 gives $T_0(r,1\mathrm{/}Q(f))=T_0(r,Q(f))+O_{Q,f}(1)=d{T}_0(r,f)+O_{Q,f}(1)$. Step 1.1 transfers this estimate to $T$, while steps 3.1 and 5.1 reduce the original composition to this normalized estimate.

If $d=0$, $R$ is a constant $c$ and $R(f)$ has no poles; by [F1], $T(r,R(f))=\frac{1}{2}\log (1+\mathrm{\mid }c{\mathrm{\mid }}^2)$ for every $r$, which is bounded.