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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.

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.

Use the Nevanlinna proximity function $m(r,f)=(2\pi {)}^{-1}\int {\log }^{+}|f(r{e}^{i\theta })|d\theta$ and the Nevanlinna integrated counting function $N(r,f)$ for poles, counting multiplicity. If the pole order at zero is ${\nu }_0$, thenBoundedness of the Nevanlinna characteristic refers to $r\uparrow 1$, equivalently to a uniform bound on $r_0\le r<1$ for any fixed $r_0>0$. This avoids the harmless origin normalization when ${\nu }_0>0$.

The Nevanlinna first main theorem states, for every fixed finite value $a$ and every nonconstant meromorphic function on the disk,For $a=\infty$, this is the definition of $T$. The error is bounded independently of $r$. Translation changes $T$ by $O(1)$, while Jensen's formula gives $T(r,h)-T(r,1/h)=\log |c|$ when $h(z)=c{z}^{\nu }(1+O(z))$ near zero; these observations explain the first theorem.

The Nevanlinna second main theorem in the unit disc states that for $q\ge 3$ distinct values $a_j$ on the Riemann sphere,as $r\uparrow 1$ outside an exceptional set $E$ satisfying ${\int }_{E}dr/(1-r)<\infty$. The truncated Nevanlinna counting function $\overline{N}$ counts each distinct preimage once. The boundary-distance error cannot simply be suppressed: on the disk it need not be $o(T)$. The disk error and exceptional-set estimates follow from the logarithmic-derivative estimates in mathweb.tifr.res.in/Documents/Publications/Lectures/tifr17.pdf .

Suppose first that $T(r,f)$ is bounded. Since $m(r,f)\ge 0$, its poles satisfy the Blaschke condition, counting their orders: let $r\uparrow 1$ in their integrated count and use $1-|b|\le -\log |b|$. Let $B$ be their Blaschke product, including the origin factor if necessary. Then $g=Bf$ extends to a holomorphic function on the whole disk, and $|B|\le 1$ givesHere the last bound follows from bounded $T$ and the uniform lower bound $N(r,f)\ge {\nu }_0\log r_0$ on $r\ge r_0$.

The nonnegative subharmonic function ${\log }^{+}|g|$ has a finite harmonic majorant. Indeed its Poisson integrals $H_R$ on expanding disks dominate it by the maximum principle for subharmonic functions, increase with $R$, and have $H_R(0)\le C$. The Harnack inequality for harmonic functions bounds them on every compact subdisk, so their increasing limit is a finite harmonic function $H\ge {\log }^{+}|g|$. Since the disk is simply connected, $H$ has a harmonic conjugate, giving a holomorphic function $A$ with $\mathrm{Re}A=H$. ConsequentlyThis proves the constructive half of the quotient characterization of bounded characteristic.

Conversely, if $f=u/v$ with $u,v$ bounded holomorphic functions and $v\not\equiv 0$, each pole of $f$ is a zero of $v$ of at least the same order. The elementary inequality ${\log }^{+}|u/v|\le {\log }^{+}|u|+{\log }^{+}|1/v|$ and the Nevanlinna first main theorem giveAny cancellation at an origin zero contributes only a bounded multiple of $\log r$ on $r\ge r_0$, which is included in this $O(1)$. The zero function is immediate; constant nonzero $v$ is handled directly. Thus bounded characteristic is exactly a quotient of two bounded analytic functions.

An unbounded example with bounded characteristic isIt is a ratio of the bounded holomorphic functions $1$ and $1-z$. More explicitly, Jensen's formula gives the mean of $\log |1-r{e}^{i\theta }|$ as zero, so $m(r,1/(1-z))=m(r,1-z)\le \log 2$, although $g(r)\to \infty$.

Finally, set $Y=(1+|g{|}^2{)}^{p/2}\ge 1$. Pointwise ${\log }^{+}|g|\le p^{-1}\log Y$. Jensen's inequality for the concave logarithm givesThere is no pole term because $g$ is holomorphic. Hence the stated integral bound implies bounded characteristic. This argument in fact works for every $p>0$.

Write ${\log }^{+}x=\max (0,\log x)$. The Nevanlinna proximity function and Nevanlinna integrated counting function areThe pole sum counts multiplicity; it is finite for each $r$ because $f(0)$ is finite and nonzero and poles of a meromorphic function are isolated. Equivalently $N(r,f)={\int }_0^{r}n(t,f)dt/t$, where $n(t,f)$ counts poles in the disc. The Nevanlinna characteristic isFor a general nonzero meromorphic function $g$ with signed order $\nu$ at zero, use $g(z)=c{z}^{\nu }(1+O(z))$, $c\ne 0$. The general counting convention includes a pole at zero:where $n(0,g)=\max (0,-\nu )$. This convention will also cover $g=f-a$ when the target $a=f(0)$.

Here is a direct proof of the exact reciprocal identity underlying the Nevanlinna first main theorem. Choose a circle with no zero or pole of $g$ on it, and a slightly larger disc with the same interior zeros and poles. Divide $g$ by $z^{\nu }$ and its nonzero zero factors and multiply by its nonzero pole factors, obtaining a holomorphic, nowhere-zero function $H$ on that disc. For any factor $z-a$ with $|a|<r$,Indeed, factor out $r{e}^{i\theta }$; the remaining logarithm has real part equal to that of $-{\sum }_{k\ge 1}(a/r{)}^k{e}^{-ik\theta }/k$, whose angular mean is zero. Also, $\log |H|$ is harmonic, so the mean value property for harmonic functions equates its circle mean to $\log |H(0)|$. Putting back the zero and pole factors provesThe identity ${\log }^{+}x-{\log }^{+}(1/x)=\log x$ now yieldsZeros or poles on the circle cause only integrable logarithmic singularities; the same identities hold at those radii by limits. A boundary zero or pole contributes $\log (r/r)=0$ to its counting term.

For a finite target $a$, set $g=f-a$. Its finite poles and their orders equal those of $f$. The elementary estimatesgive $|T(r,f-a)-T(r,f)|\le \log (1+|a|)$. Combining this with the reciprocal identity proves the first main theorem:The error is bounded independently of $r$, with an explicit bound $\log (1+|a|)+|\log |c_a||$, where $c_a$ is the first nonzero local coefficient of $f-a$ at zero. The count on the right records the occurrences of the value $a$ with multiplicity. The target infinity is the defining identity $T=m+N$. If $f$ is a constant equal to $a$, the reciprocal expression is undefined; the value-distribution assertion assumes $f-a\not\equiv 0$, as it automatically does for nonconstant $f$. The exact reciprocal identity remains valid for every nonzero constant $f$.

Taking $g=f$, whose signed order is zero and whose local coefficient is $f(0)$, rearranges that exact identity into Jensen's formula:Thus the zero and pole terms, their signs and the initial-value constant follow from the same proved identity, rather than from a theorem citation without proof.

Finally let $f$ be a nonzero bounded holomorphic function on the unit disc, with $|f|\le M$. If its order at zero is $m$, write $f(z)=z^{m}h(z)$, $h(0)\ne 0$. The factorization argument above works on every compact subdisc and gives, for its nonzero zeros,For $1/2\le r<1$ this is a uniform finite upper bound. As $r\uparrow 1$, monotone convergence givesSince $1-u\le -\log u$ for $0<u<1$, and there are only $m$ zeros at zero, we obtainThis is precisely the Blaschke condition, so the zeros, counted with multiplicity, form a Blaschke sequence.

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.