The local Second Main Theorem on a punctured disc

(The regular radius exists) On $D^{\ast }$ the pole set of $f$ is closed and discrete, and for each finite $a_j$ the $a_j$-points are isolated: near such a point $f$ is holomorphic, and the zero is isolated unless $f-a_j$ vanishes on a neighbourhood, which by [F6] would force $f\equiv a_j$ on the connected domain $D^{\ast }$, contrary to nonconstancy. On each compact annulus $\frac{1}{k+1}\le \mathrm{\mid }z-z_0\mathrm{\mid }\le r^{\ast }(1-\frac{1}{k+1})$ these sets have only finitely many points. Hence only countably many radii meet a pole or a preimage of one of the finitely many targets; using the countable-choice interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) to run through the compact annuli, some $0<\rho <r^{\ast }$ avoids them.

(Exterior setup) The inversion $w\mapsto z_0+\rho \mathrm{/}w$ is a biholomorphism of $\{\mathrm{\mid }w\mathrm{\mid }>1\}$ onto $\{0<\mathrm{\mid }z-z_0\mathrm{\mid }<\rho \}$, so $F$ is nonconstant and meromorphic on the neighbourhood $\{\mathrm{\mid }w\mathrm{\mid }>\rho \mathrm{/}{r}^{\ast }\}\supseteq \{\mathrm{\mid }w\mathrm{\mid }\ge 1\}$ of the closed exterior, and the circle $\mathrm{\mid }w\mathrm{\mid }=1$ carries no pole and no $a_j$-preimage of $F$.

(Annular Jensen identity) Fix a finite value $a$ with $F-a\mathrm{\ne }0$ on $\mathrm{\mid }w\mathrm{\mid }=1$, put $M_a(t)=\frac{1}{2\pi }{\int }_0^{2\pi }\log \mathrm{\mid }F(t{e}^{i\theta })-a\mathrm{\mid }d\theta$ for $t>1$, and $k_a:=\frac{1}{2\pi i}{\int }_{\mathrm{\mid }w\mathrm{\mid }=1}\frac{F^{\mathrm{\prime }}(w)}{F(w)-a}dw\in \mathbb{Z}$; then for every $R>1$, $M_a(R)-M_a(1)=k_a\log R+N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;F)-N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };F)$. Differentiating under the integral gives $\frac{d}{dt}{M}_a(t)=\frac{1}{t}\cdot \frac{1}{2\pi i}{\int }_{\mathrm{\mid }w\mathrm{\mid }=t}\frac{F^{\mathrm{\prime }}(w)}{F(w)-a}dw$, an integer by [F3]; at a zero of $F-a$ of order $e$ the local factorisation $F-a=(w-p{)}^{e}u$ raises that winding number by $e$ as $t$ crosses $\mathrm{\mid }p\mathrm{\mid }$, and at a pole of $F$ of order $e$ the factorisation $F-a=(w-p{)}^{-e}u$ lowers it by $e$, so the integral equals $k_a+n_{\mathrm{e}\mathrm{x}\mathrm{t}}(t,a;F)-n_{\mathrm{e}\mathrm{x}\mathrm{t}}(t,\mathrm{\infty };F)$ for almost every $t$ and integration against $\frac{dt}{t}$ yields the identity, which extends to all $R>1$ by continuity.

(Two-sided exterior First Main Theorem) Let $g$ be meromorphic on a neighbourhood of $\{\mathrm{\mid }w\mathrm{\mid }\ge 1\}$, and suppose its inner circle contains no pole of $g$ and no point with $g=a$, where $a\in \mathbb{C}$. Put $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;g)=\frac{1}{2\pi }{\int }_0^{2\pi }{\log }^{+}\frac{1}{\mathrm{\mid }g(R{e}^{i\theta })-a\mathrm{\mid }}d\theta$. Then $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;g)+N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;g)=T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,g)+O_a(\log R)$ two-sidedly. Indeed step 3.1 applied to $g$ and the identity $\log \mathrm{\mid }g-a\mathrm{\mid }={\log }^{+}\mathrm{\mid }g-a\mathrm{\mid }-{\log }^{+}\frac{1}{\mathrm{\mid }g-a\mathrm{\mid }}$ give $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;g)=m_{\mathrm{e}\mathrm{x}\mathrm{t}}^{+}(R,a;g)-M_a(1)-k_a\log R-N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;g)+N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };g)$, and by [F1] the comparison $\mid {\log }^{+}\mathrm{\mid }g-a\mathrm{\mid }-{\log }^{+}\mathrm{\mid }g\mathrm{\mid }\mid \le {\log }^{+}\mathrm{\mid }a\mathrm{\mid }+\log 2$ is two-sided, so substituting $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };g)-O_a(1)\le m_{\mathrm{e}\mathrm{x}\mathrm{t}}^{+}(R,a;g)\le m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };g)+O_a(1)$ proves the claim. The same Jensen calculation may be anchored at any regular circle $\mathrm{\mid }w\mathrm{\mid }=\sigma >1$; its integrated counts differ from those anchored at $1$ by $O(\log R)$ because only finitely many divisor points lie in $1<\mathrm{\mid }w\mathrm{\mid }\le \sigma$.

(Möbius normalisation and characteristic comparison) If $q\le 2$ the asserted inequality is trivial with $C=0$; assume $q\ge 3$. Choose any finite $b\mathrm{\notin }\{a_1,\dots ,a_q\}$ and let $M$ be the Möbius transformation with $M(b)=\mathrm{\infty }$, $M(a_1)=0$, $M(a_2)=1$ [F4]; put $c_j:=M(a_j)$ and $G:=M\circ F$. Then each $c_j$ is finite and the $c_j$ are distinct; $G$ is nonconstant and meromorphic on a neighbourhood of $\{\mathrm{\mid }w\mathrm{\mid }\ge 1\}$, and the circle $\mathrm{\mid }w\mathrm{\mid }=1$ carries no $c_j$-point of $G$. By [F4] the $c_j$-divisor of $G$ equals the $a_j$-divisor of $F$ with multiplicities and the poles of $G$ are exactly the $b$-points of $F$ in $\mathrm{\mid }w\mathrm{\mid }>1$ with equal orders, so ${\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)={\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a_j;F)$ and $N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G)=N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,b;F)$. Since $M(u)=A\mathrm{/}(u-b)+D$ for constants $A\mathrm{\ne }0,D$, $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G)=m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,b;F)+O(1)$. Choose one $r_1\in (1,2)$ whose circle avoids the poles and $b$-points of $F$, the poles and $c_j$-points of $G$, and the zeros and poles of $G^{\mathrm{\prime }}$. These divisors are locally finite in the compact annulus $1\le \mathrm{\mid }w\mathrm{\mid }\le 2$, so only finitely many radii are excluded. Applying step 4.1 at $r_1$ and using its base-radius observation gives $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,b;F)+N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,b;F)=T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,F)+O(\log R)$. Therefore $T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)=T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,F)+O(\log R)$ two-sidedly.

(Target separation) Put $\delta :={\min }_{i<j}\mathrm{\mid }{c}_i-c_j\mathrm{\mid }>0$ and $H:={\sum }_{j=1}^q\frac{1}{G-c_j}$; the pointwise separation estimate of the plane Second Main Theorem Nevanlinna Second Main Theorem with ramification and truncation, valid for an arbitrary meromorphic function and reproduced here in the exterior normalisation, gives ${\sum }_{j=1}^q{\log }^{+}\frac{1}{\mathrm{\mid }G(w)-c_j\mathrm{\mid }}\le {\log }^{+}\mathrm{\mid }H(w)\mathrm{\mid }+C_{q,\delta }$ on every outer circle, with $C_{q,\delta }=\log 2+\max \{q{\log }^{+}(4(q-1)\mathrm{/}\delta ),(q-1){\log }^{+}(2\mathrm{/}\delta )\}$; points with $G(w)=c_j$ are covered by the convention $+\mathrm{\infty }\le +\mathrm{\infty }$.

(Exterior ramification bookkeeping) Define $N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G):=N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)-{\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)$ and $N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G):=N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,0;G^{\mathrm{\prime }})+2N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G)-N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }})$. The pointwise computation of [F2] applied at each point $w$ of the annulus $1<\mathrm{\mid }w\mathrm{\mid }\le R$ with local degree $e_w$ of $G$ gives $N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)={\sum }_{G(w)=a}(e_w-1)\log \frac{R}{\mathrm{\mid }w\mathrm{\mid }}$, $N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)={\sum }_{e_w\ge 2}(e_w-1)\log \frac{R}{\mathrm{\mid }w\mathrm{\mid }}$, hence $N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)={\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)+N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)$ with $0\le N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)\le N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a;G)$, and ${\sum }_{j=1}^q{N}_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)\le N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)$ because each ramified point contributes to at most one of the disjoint target classes.

(Exterior logarithmic-derivative bounds) Apply [F5] to $G$ and each $G-c_j$ with inner radius $\sigma =r_1$ from step 5.1. All are nonconstant and meromorphic on a neighbourhood of that closed exterior; its inner circle has no pole or zero of these functions. Thus the source characteristics obey $T_1(R,G)\le 2\pi T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)$ and $T_1(R,G-c_j)\le 2\pi T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G-c_j)$, with the right sides based at $1$. Their pole divisors agree and $\mathrm{\mid }{\log }^{+}\mathrm{\mid }G-c_j\mathrm{\mid }-{\log }^{+}\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }\le \log 2+{\log }^{+}\mathrm{\mid }{c}_j\mathrm{\mid }$, so $T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G-c_j)=T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)+O(1)$. Taking the finite union of the $q+1$ source exceptional sets, we obtain a measurable $E$ of finite linear measure such that $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }}\mathrm{/}G)$ and every $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }}\mathrm{/}(G-c_j))$ are bounded by $C_j({\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)+\log R)$ at all sufficiently large $R\mathrm{\notin }E$.

(Reduction to logarithmic derivatives) Since $H=(H{G}^{\mathrm{\prime }})\cdot \frac{1}{G^{\mathrm{\prime }}}$ and $H{G}^{\mathrm{\prime }}={\sum }_{j=1}^q\frac{G^{\mathrm{\prime }}}{G-c_j}$, the pointwise inequalities ${\log }^{+}\mathrm{\mid }uv\mathrm{\mid }\le {\log }^{+}\mathrm{\mid }u\mathrm{\mid }+{\log }^{+}\mathrm{\mid }v\mathrm{\mid }$ and ${\log }^{+}\mathrm{\mid }{\sum }_{j\le q}{u}_j\mathrm{\mid }\le {\sum }_{j\le q}{\log }^{+}\mathrm{\mid }{u}_j\mathrm{\mid }+\log q$ give $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };H)\le m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,0;G^{\mathrm{\prime }})+{\sum }_{j=1}^q{m}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }}\mathrm{/}(G-c_j))+\log q$ for every $R>0$.

(The term $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(0;G^{\mathrm{\prime }})+N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}$) By the choice in step 5.1, $\mathrm{\mid }w\mathrm{\mid }=r_1$ contains no zero or pole of $G^{\mathrm{\prime }}$. Apply the annular Jensen identity of step 4.1 to $G^{\mathrm{\prime }}$ with inner circle $\mathrm{\mid }w\mathrm{\mid }=r_1$; its counts anchored at $r_1$ differ from $N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\cdot ;G^{\mathrm{\prime }})$, anchored at $1$, by $O(\log R)$, since only finitely many zeros and poles lie in $1<\mathrm{\mid }w\mathrm{\mid }\le r_1$. Thus $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,0;G^{\mathrm{\prime }})=m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }})+N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }})-N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,0;G^{\mathrm{\prime }})+O(\log R)\mathrm{.}$ Adding the definition of $N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)$ from step 6.2 yields $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,0;G^{\mathrm{\prime }})+N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)=m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }})+2N_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G)+O(\log R)$.

(Bounding $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(\mathrm{\infty };G^{\mathrm{\prime }})$) The pointwise bound ${\log }^{+}\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }\le {\log }^{+}\mathrm{\mid }G\mathrm{\mid }+{\log }^{+}\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{/}G\mathrm{\mid }$ gives $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }})\le m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G)+m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G^{\mathrm{\prime }}\mathrm{/}G)\le m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,\mathrm{\infty };G)+C_0({\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)+\log R)$ for $R$ large outside the exceptional set of step 6.3.

(Ramified exterior Second Main Theorem) Combining steps 6.1, 7.1, 6.3, 7.2 and 7.3, for all large $R$ outside the finite-measure exceptional set $E$ one has ${\sum }_{j=1}^q{m}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)+N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)\le 2T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)+C_1({\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)+\log R)$: the separation and logarithmic-derivative steps bound the proximity sum by $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,0;G^{\mathrm{\prime }})+C$, and steps 7.2 and 7.3 bound $m_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,0;G^{\mathrm{\prime }})$ by $2T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)-N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)+C_1({\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}+\log R)$.

(Truncated exterior Second Main Theorem) Step 4.1 at the regular radius $r_1$ of step 5.1 gives ${\sum }_{j=1}^q{m}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)=q{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)-{\sum }_{j=1}^q{N}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)+O(\log R)$; the base-radius observation in step 4.1 accounts for the divisor terms between radii $1$ and $r_1$. Substituting into step 8.1 and using $N_{\mathrm{e}\mathrm{x}\mathrm{t}}(c_j)={\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(c_j)+N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(c_j)$ together with ${\sum }_j{N}_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)\le N_{1,\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)$ from step 6.2 gives $(q-2)T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)\le {\sum }_{j=1}^q{\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)+C_2({\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)+\log R)$ for all large $R\mathrm{\notin }E$.

(Transfer back to $F$) Using ${\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,c_j;G)={\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a_j;F)$, the two-sided comparison $T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)=T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,F)+O(\log R)$ of step 5.1, and ${\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,G)\le {\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,F)+O(\log R)$ for large $R$, the inequality of step 9.1 becomes $(q-2)T_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,F)\le {\sum }_{j=1}^q{\bar{N}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,a_j;F)+C({\log }^{+}{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}(R,F)+\log R)$ for all large $R$ outside $E$, with a suitably enlarged constant $C$.

(The exceptional set in the puncture radius) The substitution $s=\rho \mathrm{/}R$ maps $[R_0,\mathrm{\infty })$ bijectively onto $(0,\rho \mathrm{/}{R}_0]$ with $\mid \frac{ds}{dR}\mid =\rho R^{-2}\le \rho R_0^{-2}$, so by the change-of-variables formula for a measurable set the image $\{\rho \mathrm{/}R:R\in E\}$ of the exceptional set $E$ of step 6.3 has linear measure at most $\rho R_0^{-2}\mathrm{\mid }E\mathrm{\mid }<\mathrm{\infty }$.