Nevanlinna theory

Nevanlinna theory studies the growth and value distribution of meromorphic functions through their integrated pole counts and logarithmic proximity to target values. The Nevanlinna characteristic combines both quantities, while the Nevanlinna first main theorem compares it with the corresponding counts and proximity for any fixed target.

This theory compares the area, boundary length and Euler characteristic of a surface mapped holomorphically to another surface, using the pulled-back metric. It extends covering-degree and Riemann-Hurwitz formula estimates when some boundary maps into the interior of the target. The Ahlfors second fundamental theorem and Ahlfors five islands theorem are important consequences.

For a nonconstant meromorphic function on the plane, let $S(R)$ be its normalized pulled-back spherical area on $|z|<R$, and $L(R)$ the spherical length of its boundary image counted with multiplicity. The Cauchy-Schwarz inequality gives $L(R{)}^2\le 8{\pi }^{2}R{S}^{\prime }(R)$. If $L/S$ stayed above a positive constant for all large $R$, integration of $(1/S{)}^{\prime }\le -c/R$ would make $1/S$ negative. Thus there are arbitrarily large radii with $L/S$ arbitrarily small, even when the total area is finite.

For a finite simply connected covering surface over the Riemann sphere and $q\ge 3$ fixed separated Jordan domains, let $n_j$ count its simply connected islands over the $j$th domain once each, regardless of mapping degree. Let $S$ be its average sheet number and $L$ its spherical boundary length. The displayed inequality holds with $h$ depending only on the target domains. For regular bordered surfaces $f:X\to Y$, a general formulation is $\max \{-\chi (X),0\}\ge -S\chi (Y)-hL$, where $L$ counts only the boundary mapped into the interior of $Y$ and $\chi$ is the modern Euler characteristic. The boundary error disappears for an actual branched covering. The covering-surface formulation is explained in www.math.purdue.edu/~eremenko/dvi/ahl.pdf ; the simply connected island estimate is also recorded in Lemma 2 of pdfs.semanticscholar.org/d247/562607e237af7c8fc6e683b77be6b94829fa.pdf .

Every nonconstant meromorphic function on the complex plane has a simple island over at least one of any five Jordan domains on the Riemann sphere with disjoint closures. The Ahlfors second fundamental theorem, area comparison over target domains, and a length-area exhaustion of the complex plane prove this: without a simple island every counted island has degree at least two, forcing $3S\le (5/2)S+O(L)$.

An island over a Jordan domain $D$ is a relatively compact component $U$ of its inverse image on which the meromorphic function is a proper map onto $D$. A component reaching the boundary of the source is instead a peninsula. The island has an integer mapping degree; degree one means that the map is a biholomorphism. A simply connected island can have larger degree and need not be a simple island.

A simple island is an island of a meromorphic function of mapping degree one. It is mapped biholomorphically onto the target Jordan domain. This is stronger than the source island being simply connected.

For a holomorphic map between metric Riemann surfaces, the average sheet number is the pulled-back source area divided by the target area. The area formula identifies it with the average number of preimages counted with multiplicity. The same definition over a target region $D$ gives $S_D=A_f(f^{-1}D)/A(D)$.

For distinct fixed target values and a nonconstant meromorphic function, the truncated second main theorem has the displayed form, with $S_f=O({\log }^{+}{T}_f+\log r)$ outside a set of finite linear measure. The truncated Nevanlinna counting function retains only one unit at each preimage. For finite-order functions the error is $O(\log r)$; for transcendental functions it is $o(T_f)$ along the nonexceptional radii.

For a nonconstant meromorphic function on the unit disc and $q\ge 3$ distinct spherical values, this version of the Nevanlinna second main theorem holds outside a set $E$ with ${\int }_{E}dr/(1-r)<\infty$, as $r\uparrow 1$. The truncated Nevanlinna counting function counts distinct preimages rather than multiplicity. The boundary-distance error is essential and need not be small relative to the characteristic. It follows from the disk Nevanlinna logarithmic derivative lemma and the growth lemma in the variable $\log (1/(1-r))$. The disk error estimate and weighted exceptional-set argument appear in mathweb.tifr.res.in/Documents/Publications/Lectures/tifr17.pdf .

If a transcendental meromorphic function had only finitely many simple preimages over each of five distinct finite values, their simple counting functions would be $O(\log r)$. The displayed multiplicity inequality and the Nevanlinna first main theorem would bound the truncated sum by $5T_f/2+O(\log r)$. The Nevanlinna second main theorem requires at least $3T_f$ up to its $o(T_f)$ error, a contradiction. Thus at least one value has infinitely many simple preimages; four other values can remain totally ramified.

For a nonconstant meromorphic function on the plane, the displayed estimate holds outside an exceptional set of radii of finite linear measure, for large $r$. The logarithmic derivative has only simple principal parts at zeros and poles; Poisson-Jensen formula estimates and control of increasing auxiliary growth functions produce the logarithmic error. It supplies the error term in the Nevanlinna second main theorem.

For a nonnegative increasing continuously differentiable function not identically zero, choose $x_0$ with $\varphi (x_0)>0$. On $I=\{x\ge x_0:{\varphi }^{\prime }>{\varphi }^2\}$, $1<{\varphi }^{\prime }/{\varphi }^2$. Thus $|I|\le {\int }_{x_0}^{\infty }{\varphi }^{\prime }/{\varphi }^2\le 1/\varphi (x_0)$. Adding the finite initial interval controls all exceptions. For the zero function there are none. Taking logarithms turns derivative growth into logarithmic auxiliary-function growth in value-distribution estimates.

The Nevanlinna ramification index is the normalized lower limiting integrated excess multiplicity over a target value, with pole orders used for infinity. It is an asymptotic value-distribution quantity, distinct from the integer ramification index of a holomorphic map at one source point. It can be positive even when the target has positive Nevanlinna deficiency.

For a nonconstant meromorphic function, its deficiency at a point of the Riemann sphere is the displayed asymptotic proportion missing from the full Nevanlinna integrated counting function. The Nevanlinna first main theorem bounds $N(r;a)$ by $T_f(r)+O(1)$ and identifies deficiency with the lower limiting normalized proximity. An omitted value has deficiency one.

The rational composition law for the Nevanlinna characteristic gives $T_f(r)=3r/\pi +O(1)$. Its zeros are precisely $2\pi i\mathbb{Z}$, all of multiplicity two, so $N(r;0)=2r/\pi +O(\log r)$ and $\overline{N}(r;0)=r/\pi +O(\log r)$. Hence both its Nevanlinna deficiency and Nevanlinna ramification index at zero equal one third.

For a fixed finite target $a$, the displayed identity compares growth with proximity to and occurrences of $a$. If $g(z)=c{z}^{\nu }(1+O(z))$ near zero, where $\nu$ is its signed zero order, the exact reciprocal identity is $T(r,g)-T(r,1/g)=\log |c|$. It follows by factoring out zeros and poles on a disc, applying the mean value property for harmonic functions to the remaining nonvanishing factor, and using ${\log }^{+}u-{\log }^{+}(1/u)=\log u$. Translation changes $T$ by a bounded amount, proving the general assertion.

The characteristic of a meromorphic function is the sum of its Nevanlinna proximity function and Nevanlinna integrated counting function. It treats large values and actual poles together. Replacing $f$ by $f-a$ changes the characteristic by at most $\log (1+|a|)$.

The growth order of a meromorphic function is the upper power exponent of its Nevanlinna characteristic, with constants assigned order zero. It is different from the local signed order at a zero or pole. A Nevanlinna integrated counting function has an analogous growth order. The Nevanlinna first main theorem bounds every counting order by the function order.

At most two values can have counting order smaller than a positive finite function order. Otherwise the Nevanlinna second main theorem with three such values bounds the characteristic by a smaller power outside a finite-length exceptional set. Monotonicity extends this bound to all large radii by choosing a good radius in $[r,r+1]$, contradicting the original order. Order zero is immediate from nonnegativity and the first theorem.

For a degree-$d$ rational function $U$ of the Riemann sphere, choose relatively prime homogeneous polynomials $(P,Q)$ of degree $d$. Their joint norm on the unit three-sphere has positive lower and finite upper bounds. Thus $\log \Vert (P,Q)(v)\Vert -d\log \Vert v\Vert$ is bounded. Averaging over a circle using a nonvanishing entire homogeneous lift of a meromorphic function proves the displayed law for the spherical characteristic. Jensen's formula and $0\le \frac{1}{2}\log (1+x^2)-{\log }^{+}x\le \frac{1}{2}\log 2$ show that it also holds for the ordinary Nevanlinna characteristic.

The number $n(t,f)$ counts poles in $|z|<t$ with multiplicity, and $n(0,f)$ is the pole order at zero. Thus $N(r,f)=n(0,f)\log r+{\sum }_{0<|p|<r}\log (r/|p|)$. Applying this to $1/(f-a)$ counts occurrences of the value $a$.

The truncated counting function counts each distinct $a$-point of a meromorphic function once, rather than with its local multiplicity. It uses the same logarithmic radial weights as the Nevanlinna integrated counting function, including an origin contribution $\log r$ when appropriate. The difference $N-\overline{N}$ measures integrated excess multiplicity.

Here ${\log }^{+}x=\max (0,\log x)$. The function $m(r,f)$ measures logarithmic proximity of $f$ to infinity. For a finite target $a$, proximity is $m(r,1/(f-a))$. Logarithmic singularities on a circle are integrable.