First Main Theorem
For any function \(w(z)\) meromorphic in the disk \(|z| < R < \infty\), a function \(T(r, w)\) can be defined for \(0 < r < R\) so that
\(T(r)\) is an increasing function of \(r\) and a convex function of \(\log r\).
If \(a\) denotes any complex number that is independent of \(z\), finite or infinite, then is \[m(r, a) + N(r, a) = T (r) + O(1). \tag{I}\] where the two terms on the left of relation (I) are nonnegative.
Lemma 1
If \(\alpha\) is a nonnegative number, let \[\log^{+} \alpha = \begin{cases}\log \alpha, & \alpha > 1\\ 0, & 0 < \alpha \leq 1\end{cases}\] be the larger of the numbers \(\log \alpha\) and \(0\); it follows that \[ \begin{aligned} \log \alpha = \log^{+} \alpha - \log^{+} \frac{1}{\alpha}\\ |\log \alpha| = \log^{+} \alpha + \log^{+} \frac{1}{\alpha} \end{aligned} \]
Lemma 2
Notation as above lemma 1, and further, as is easily confirmed, \[ \begin{equation} \begin{aligned} \log^{+} \alpha_{1}\cdot \alpha_{2}\cdots \alpha_{p} \leq \sum_{i=1}^{p} \log^{+} {\alpha_{i}}\\ \log^{+} \sum_{i=1}^{p} {\alpha_{i}} \leq \sum_{i=1}^{p} \log^{+} \alpha_{i} + \log p \end{aligned} \end{equation} \]
Proof of Theorem 1
In Jensen's formula, we now set \(\log |w| = \log^{+} |w| - \log |\frac{1}{w}|\) and for short write
\[\begin{aligned}N(r,a,w) =& \int_{0}^{r} \frac{n(t,a) - n(0,a)}{t} dt + n(0,a) \log r\\ m(r, a, w) =& \frac{1}{2\pi} \int_{0}^{2\pi} \log^{+} \left|\frac{1}{w(re^{i\varphi}) -a }\right|d\varphi \end{aligned}\]
\[\begin{aligned}N(r,\infty,w) =& \int_{0}^{r} \frac{n(t,\infty) - n(0,\infty)}{t} dt + n(0,\infty) \log r\\ m(r, \infty, w) =& \frac{1}{2\pi} \int_{0}^{2\pi} \log^{+} \left|w(re^{i\varphi})\right|d\varphi \end{aligned}\]
where \(n(r, a)\) is the number of roots of the equation \(w- a = 0\) in the disk \(|z|\leq r\)
Jensen's formula then becomes
\[m(r, w) + N(r, w) = m (r, \frac{1}{w}) + N (r,\frac{1}{w}) +\log|c_{\lambda}|\]
where \(c_{\lambda}\) is the first nonvanishing coefficient in the Laurent expansion of \(w-a\) at the origin \(z = 0\).
The result is then
\[m(r, w - a) + N(r, w - a) = m (r, \frac{1}{w-a}) + N (r, \frac{1}{w-a}) + \text{const.}\]
But \(N(r, w - a) = N(r, w)\)
and by Lemma 2
\[\left| m(r, w- a)- m(r, w) \right| < \log^{+} |a|+ \log 2\]
so we conclude from (2.3) that
\[m(r, a) + N(r, a) = m(r, \infty) + N(r,\infty) + \varphi(r, a),\]
In addition we set \(T(r, w) = T(r) = m(r, \infty) + N(r, \infty)\), This is the desired results. \(\square\)
Second Main Theorem
- For \(|z| < R < \infty\) let \(w(z)\) be a nonconstant meromorphic function. If \(w_{1}, \cdots , w_{q} (q \geq 1)\) are mutually distinct finite or infinite complex numbers, then for \(0 \leq r < R\)
\[\sum m(r, \omega_{\nu}) < 2 T(r) - N_{1}(r) + S(r)\]
- the remainder term satisfies the following conditions:
If \(R = \infty\), then \[S(r) = O \left\{\log \left[rT(r)\right]\right\}\] with at most the exception of a set of values \(\{r\}\) of finite total measure
If \(R = 1\), then
\[S(r) = O \left\{\frac{1}{1-r}T(r)\right\}\]
with at most the exception of a set of values \(\{r\}\) for which the variation of \(\frac{1}{1-r}\) is fintte
Proof
Suppose, therefore, that \(w(z) = c_{0} + c_{k}z^{k} + \cdots (c_{0}\neq 0, c_{k} \neq 0)\) is a function that is meromorphic for \(|z| < R \leq \infty\), and let \(a_{1}, \cdots, a_{p}\) be a system of \(p \geq 2\) different finite complex numbers.
We first compare the proximity functions \(m(r, w)\) and \(m(r, w^{\prime})\) with one another. By means of the elementary inequalities (Lemma 2 in (I)), one finds immediately that
\[\begin{equation} m(r,w) = m (r,w^{\prime}\frac{w}{w^{\prime}}) \leq m (r,w^{\prime}) + m (r,\frac{w}{w^{\prime}}) \end{equation}\]
To estimate the mean value \(m ( r, \frac{1}{w^{\prime}})\) , consider the sum
\[f(z) = \sum_{\nu=1}^{p} \frac{1}{w(z) - a_{\nu}}\]
We have
\[\begin{equation} m(r,f) = m (r,fw^{\prime}\frac{1}{w^{\prime}}) \leq m (r,\frac{1}{w^{\prime}}) + m (r,\sum\frac{w^{\prime}}{w- a_{\nu}}) \end{equation}\]
On the other hand, for a given \(\mu\) (\(\mu = 1, \cdots , p\))
\[f = \frac{1}{w - a_{\mu}}\left(1 + \sum_{\nu\neq\mu} \frac{w - a_{\mu}}{w - a_{\nu}}\right)\]
If \(\delta = \min (|a_{h} - a_{k}|, 1) (h \neq k)\), then at every point \(z\) where
\[\begin{equation} \left|w(z) - a_{\mu}\right| < \frac{\delta}{2p} \left(\leq \frac{1}{2p}\right) \end{equation}\]
for \(\nu \neq \mu\) \[\left|w - a_{\nu}\right| \geq \left|a_{\mu} - a_{\nu}\right| - \left|w - a_{\mu}\right| > \delta - \frac{\delta}{2p} \geq \frac{3\delta}{4}\]
and hence \[\sum_{\nu\neq\mu} \left|\frac{w - a_{\mu}}{w - a_{\nu}}\right| < (p-1) \frac{2}{3p} < \frac{2}{3}\]
so that \[\left|1 + \sum_{\nu\neq\mu} \frac{w - a_{\mu}}{w - a_{\nu}}\right| > \frac{1}{3}\]
From this it follows that \[\log^{+}\left|f(z)\right| > \log^{+}\left|\frac{1}{w - a_{\mu}}\right| - \log 3\]
at every point z where condition (4) is satisfied.
The arcs determined on the circle by (4) are disjoint for different values of \(p\), and therefore one concludes that
\[\begin{aligned} m(r,f) \geq & \frac{1}{2\pi} \sum_{\mu =1}^{p} \int_{|w - a_{\mu}| < \frac{\delta}{2p}} \log^{+} |f(re^{i\varphi})| d \varphi\\ < & \frac{1}{2\pi} \sum_{1}^{p} \int_{|w - a_{\mu}| < \frac{\delta}{2p}} \left(\log^{+} \left|\frac{1}{w(re^{i\varphi})- a_{\mu}}\right| - \log 3\right) d \varphi \end{aligned}\]
Further, \[\begin{aligned} \frac{1}{2\pi} \sum_{1}^{p} \int_{|w - a_{\mu}| < \frac{\delta}{2p}} \log^{+} \left|\frac{1}{w- a_{\mu}}\right| = & m(r,a_{\mu}) - \frac{1}{2\pi} \sum_{1}^{p} \int_{|w - a_{\mu}| \geq \frac{\delta}{2p}}\log^{+} \left|\frac{1}{w- a_{\mu}}\right|\\ \geq & m(r,a_{\mu}) - \log \frac{2p}{\delta} \end{aligned}\]
and finally
\[m(r,f) > \sum_{1}^{p} m(r,a_{\mu}) - p \log\frac{2p}{\delta} - \log 3\]
or in conjunction with (2)
\[\begin{equation} m (r,\frac{1}{w^{\prime}}) > \sum_{1}^{p} m(r,a_{\mu}) - m (r,\sum\frac{w^{\prime}}{w- a_{\nu}}) - p \log\frac{2p}{\delta} - \log 3 \end{equation}\]
If the quantities \(N(r, w^{\prime})\) and \(N(r, \frac{1}{w^{\prime}})\) are now added to both sides of inequalities (2) and (5), respectively, then using the first main theorem
\[T(r, w^{\prime}) = T (r, \frac{1}{w^{\prime}}) + \log |kc_{k}|\]
one obtains the following result, which is to be stated as a special lemma 1 :
The characteristic \(T(r, w^{\prime})\) for the derivative of the meromorphic function w(z) lies between the bounds \[m(r, w) + N(r, w^{\prime}) + m (r,\frac{w^{\prime}}{w})\]
and
\[ \sum_{1}^{p} m(r,\frac{1}{w - a_{\mu}}) + N (r,\frac{1}{w^{\prime}}) - m (r,\sum\frac{w^{\prime}}{w- a_{\nu}}) - p \log\frac{2p}{\delta} - \log 3\]
If we leave out T(r, w') and introduce
\[N_{1}(r) = N(r,\frac{1}{w^{\prime}}) + \left(2 N(r,w) - N (r,w^{\prime})\right)\]
it becomes a version of this theorem. \(\square\)
The estimation is rather lengthy, readers can refer to Hayman, W.[1, Chapter III] and Nevanlinna, Rolf [3, Chapter IX].
Remark: The quantity \(N_{1}(r)\) measures the number of multiple points of \(w(z)\). It can be written in the form \[N_{1}(r) = \int_{0}^{r} \frac{n_{1}(t) - n_{1}(0)}{t} dt + n_{1}(0) \log r, \] where \(n_{1}(r)\) indicates the number of multiple points of \(w(z)\) in the disk \(|z| < r\), each \(k\)-fold point being counted \(k - 1\) times.
Vojta's conjecture: Let \(F\) be a number field, let \(X/F\) be a non-singular algebraic variety, let \(D\) be an effective divisor on \(X\) with at worst normal crossings, let \(H\) be an ample divisor on \(X\), and let \(K_{X}\) be a canonical divisor on \(X\). Choose Weil height functions \(h_{H}\) and \(h_{K_{X}}\) and, for each absolute value \(v\) on \(F\), a local height function \(\lambda_{D,v}\). Fix a finite set of absolute values \(S\) of \(F\), and let \(\epsilon>0\). Then there is a constant \(C\) and a non-empty Zariski open set \(U\subseteq X\), depending on all of the above choices, such that \[\sum_{v\in S}\lambda_{D,v}(P)+h_{K_{X}}(P)\leq \epsilon h_{H}(P)+C \quad {\hbox{for all }}P\in U(F).\]
Reference
[1] Hayman, W. (1964). Meromorphic functions. Oxford University press.
[2] Nevanlinna, Rolf (1925), "Zur Theorie der Meromorphen Funktionen", Acta Mathematica, 46 (1–2): 1–99, doi:10.1007/BF02543858, ISSN 0001-5962
[3] Nevanlinna, Rolf (1970), Analytic functions, Die Grundlehren der mathematischen Wissenschaften, vol. 162, Berlin, New York: Springer-Verlag, MR 0279280