Nevanlinna Second Main Theorem with ramification and truncation

(Reduction to finite targets) Among the $q+1$ distinct sphere points $0,1,\dots ,q$ at most $q$ belong to $\{a_1,\dots ,a_q\}$; let $m$ be the least one that does not, so $m\mathrm{\ne }\mathrm{\infty }$, and put $g:=1\mathrm{/}(f-m)$. Then $g$ is nonconstant meromorphic. For finite $a_j$ set $a_j^{\mathrm{\prime }}:=1\mathrm{/}(a_j-m)$, and for $a_j=\mathrm{\infty }$ set $a_j^{\mathrm{\prime }}:=0$; these are distinct finite values.

(Local computation for the substitution) The Möbius map $w\mapsto 1\mathrm{/}(w-m)$ has local degree one on the sphere, so composition preserves every local degree and ramification multiplicity of $f$. For each finite target $a_j$, the identity $g-a_j^{\mathrm{\prime }}=\frac{a_j-f}{(f-m)(a_j-m)}$ shows that a zero of $g-a_j^{\mathrm{\prime }}$ occurs exactly at $f=a_j$ with the same order. If $a_j=\mathrm{\infty }$, then $a_j^{\mathrm{\prime }}=0$ and $g=1\mathrm{/}(f-m)$ has a zero of order $t$ exactly where $f$ has a pole of order $t$. Thus the target counting functions agree, $N(r,a_j^{\mathrm{\prime }};g)=N(r,a_j;f)$ for every sphere target, and the preserved ramification multiplicities give $N_1(r,g)=N_1(r,f)$.

(Finite-target setup) Henceforth $a_1,\dots ,a_q$ are finite and distinct; put $\delta :={\min }_{i<j}\mathrm{\mid }{a}_i-a_j\mathrm{\mid }>0$ and $H:={\sum }_{j=1}^{q}1\mathrm{/}(f-a_j)$. For each fixed finite $a$, the chordal proximity $m(r,a;f)$ differs from $m_0(r,1\mathrm{/}(f-a))$ by at most a constant depending on $a$: writing $w=f-a$, the ratio $\sqrt{1+\mathrm{\mid }w+a{\mathrm{\mid }}^2}\mathrm{/}\max (1,\mathrm{\mid }w\mathrm{\mid })$ is bounded above and below by positive constants depending only on $a$. Thus the finite number of conversions below contributes only $O_{a_1,\dots ,a_q}(1)$.

(Target separation) At every point $z$ at most one index $j$ satisfies $\mathrm{\mid }f(z)-a_j\mathrm{\mid }<\delta \mathrm{/}2$, since two such indices would give $\mathrm{\mid }{a}_i-a_j\mathrm{\mid }<\delta$; write $u_j:=1\mathrm{/}(f(z)-a_j)$ and $M:={\max }_j\mathrm{\mid }{u}_j\mathrm{\mid }=\mathrm{\mid }{u}_{j_{\ast }}\mathrm{\mid }$. If $M\le \max \{1,4(q-1)\mathrm{/}\delta \}$, then ${\sum }_j{\log }^{+}\mathrm{\mid }{u}_j\mathrm{\mid }\le q{\log }^{+}\max \{1,4(q-1)\mathrm{/}\delta \}\le {\log }^{+}\mathrm{\mid }H(z)\mathrm{\mid }+q{\log }^{+}(4(q-1)\mathrm{/}\delta )$. If $M>\max \{1,4(q-1)\mathrm{/}\delta \}$, then $\mathrm{\mid }f(z)-a_{j_{\ast }}\mathrm{\mid }<\delta \mathrm{/}2$, so $\mathrm{\mid }{u}_j\mathrm{\mid }\le 2\mathrm{/}\delta <M\mathrm{/}2$ for every $j\mathrm{\ne }{j}_{\ast }$, whence $\mathrm{\mid }H(z)\mathrm{\mid }\ge M-{\sum }_{j\mathrm{\ne }{j}_{\ast }}\mathrm{\mid }{u}_j\mathrm{\mid }>M\mathrm{/}2$ and ${\sum }_j{\log }^{+}\mathrm{\mid }{u}_j\mathrm{\mid }\le \log M+(q-1){\log }^{+}(2\mathrm{/}\delta )\le {\log }^{+}\mathrm{\mid }H(z)\mathrm{\mid }+\log 2+(q-1){\log }^{+}(2\mathrm{/}\delta )$. In both cases ${\sum }_{j=1}^q{\log }^{+}\mid \frac{1}{f(z)-a_j}\mid \le {\log }^{+}\mathrm{\mid }H(z)\mathrm{\mid }+C_{q,\delta }$ with $C_{q,\delta }:=\log 2+\max \left\{q{\log }^{+}\right(4(q-1)\mathrm{/}\delta ),(q-1){\log }^{+}(2\mathrm{/}\delta )\}$.

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

(The term $m(1\mathrm{/}{f}^{\mathrm{\prime }})+N_1$) Since $f$ is nonconstant, $f^{\mathrm{\prime }}\equiv ̸0$. If $f^{\mathrm{\prime }}$ is nonconstant, [F2] gives $m(r,0;f^{\mathrm{\prime }})=T(r,f^{\mathrm{\prime }})-N(r,0;f^{\mathrm{\prime }})+O_f(1)$; if $f^{\mathrm{\prime }}=c\mathrm{\ne }0$ is constant, the same relation follows directly from the definitions, since $N(r,0;f^{\mathrm{\prime }})=0$ and $m(r,0;c)=T(r,c)-\log \mathrm{\mid }c\mathrm{\mid }$. In either case [F5] and $T(r,f^{\mathrm{\prime }})=m(r,\mathrm{\infty };f^{\mathrm{\prime }})+N(r,\mathrm{\infty };f^{\mathrm{\prime }})$ give $m(r,0;f^{\mathrm{\prime }})+N_1(r,f)=m(r,\mathrm{\infty };f^{\mathrm{\prime }})+2N(r,\mathrm{\infty };f)+O_f(1),$ using $N_1(r,f)=N(r,0;f^{\mathrm{\prime }})+2N(r,\mathrm{\infty };f)-N(r,\mathrm{\infty };f^{\mathrm{\prime }})$.

(Characteristic transfer) $T(r,g)=T(r,1\mathrm{/}(f-m))=T(r,f-m)+O_f(1)=T(r,f)+O_f(1)$ by [F3]; consequently by [F2], $m(r,a_j^{\mathrm{\prime }};g)=T(r,g)-N(r,a_j^{\mathrm{\prime }};g)+C(g,a_j^{\mathrm{\prime }})=T(r,f)-N(r,a_j;f)+O_f(1)=m(r,a_j;f)+O_f(1)$, and the finite sum of $O_f(1)$ errors is absorbed into $S(r,f)$ for all large $r$; hence it suffices to prove the inequalities for finite distinct targets.

(Means) Taking angular means of step 1.4 and using the integrability of the logarithmic singularities and the finite-target comparison of step 1.3 gives ${\sum }_{j=1}^{q}m(r,a_j;f)\le m(r,\mathrm{\infty };H)+C_{a_1,\dots ,a_q}$ for every $r>0$.

(Logarithmic derivative of the shifted functions) For each $j$ the function $f-a_j$ is nonconstant meromorphic and $(f-a_j{)}^{\mathrm{\prime }}\mathrm{/}(f-a_j)=f^{\mathrm{\prime }}\mathrm{/}(f-a_j)$, so [F4] gives $m(r,\mathrm{\infty };f^{\mathrm{\prime }}\mathrm{/}(f-a_j))=S(r,f-a_j)$; since $T(r,f-a_j)\le T(r,f)+O_f(1)$ by [F3], we have $S(r,f-a_j)\le S(r,f)+O_f(1)$, and likewise $m(r,f^{\mathrm{\prime }}\mathrm{/}f)=S(r,f)$; taking the union of the exceptional sets of these $q+1$ occurrences gives a set $E$ of finite linear measure such that ${\sum }_{j=1}^{q}m(r,a_j;f)\le m(r,\mathrm{\infty };1\mathrm{/}{f}^{\mathrm{\prime }})+qS(r,f)+O_f(1)$ for every $r\mathrm{\notin }E$.

(Bounding $m(f^{\mathrm{\prime }})$) The pointwise bound ${\log }^{+}\mathrm{\mid }{f}^{\mathrm{\prime }}\mathrm{\mid }\le {\log }^{+}\mathrm{\mid }f\mathrm{\mid }+{\log }^{+}\mathrm{\mid }{f}^{\mathrm{\prime }}\mathrm{/}f\mathrm{\mid }$ gives $m(r,\mathrm{\infty };f^{\mathrm{\prime }})\le m(r,\mathrm{\infty };f)+m(r,f^{\mathrm{\prime }}\mathrm{/}f)\le m(r,\mathrm{\infty };f)+S(r,f)$ for $r\mathrm{\notin }E$.

(Conclusion of the finite-target case) Combining steps 3.1, 1.6 and 4.1 for $r\mathrm{\notin }E$ gives ${\sum }_{j}m(r,a_j;f)+N_1(r,f)\le m(r,\mathrm{\infty };f)+2N(r,\mathrm{\infty };f)+qS(r,f)+O_f(1)=T(r,f)+N(r,\mathrm{\infty };f)+qS(r,f)+O_f(1)=2T(r,f)-m(r,\mathrm{\infty };f)+qS(r,f)+O_f(1)\le 2T(r,f)+qS(r,f)+O_f(1)$, and the constant is absorbed into the error term, so ${\sum }_{j=1}^{q}m(r,a_j;f)+N_1(r,f)\le 2T(r,f)+S(r,f)$ outside $E$.

(Equivalent forms) By [F2], ${\sum }_{j=1}^{q}m(r,a_j;f)=qT(r,f)-{\sum }_{j=1}^{q}N(r,a_j;f)+O_f(1)$; substituting into step 5.1 and absorbing $O_f(1)$ yields $(q-2)T(r,f)\le {\sum }_{j=1}^{q}N(r,a_j;f)-N_1(r,f)+S(r,f)$ outside $E$. Conversely, rearranging this displayed bound using the same equality from [F2] recovers step 5.1 up to $O_f(1)$; [F7] absorbs that bounded term into an error of the same class, enlarging the finite-measure exceptional set if needed, so the first two displayed inequalities are equivalent. Moreover $N(r,a_j;f)=\bar{N}(r,a_j;f)+N_1(r,a_j;f)$ and ${\sum }_j{N}_1(r,a_j;f)\le N_1(r,f)$ by [F5] and [F6], so ${\sum }_{j}N(r,a_j;f)-N_1(r,f)\le {\sum }_j\bar{N}(r,a_j;f)$ and $(q-2)T(r,f)\le {\sum }_{j=1}^q\bar{N}(r,a_j;f)+S(r,f)$ outside $E$.

(Refinements) If $f$ has finite order then $g=1\mathrm{/}(f-m)$ and every $f-a_j$ have finite order with characteristics $T(r,f)+O_f(1)$, so the errors in step 3.1 are $O_f(\log r)$ at every large radius by [F4], and all other errors above are $O_f(1)$ by [F2] and [F3]; hence the inequalities hold with $O_f(\log r)$ at every sufficiently large $r$, with no exceptional set in this case. If $f$ is rational the same argument gives $O_f(1)$ at every sufficiently large $r$, again with no exceptional set.

(Unwinding) Applying the finite-target argument of steps 1.3–7.1 to the transform of step 1.1 and transferring back by steps 1.2 and 2.1 proves the three displayed inequalities for the original targets, including the case in which some $a_j$ equals $\mathrm{\infty }$; the finite union $E$ of the exceptional sets of the finitely many logarithmic-derivative applications still has finite linear measure, and all conversion constants are absorbed into $S(r,f)$.