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| In the mathematical field of [[complex analysis]], '''Nevanlinna theory''' is part of the
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| theory of [[meromorphic function]]s. It was devised in 1925, by [[Rolf Nevanlinna]]. [[Hermann Weyl]] has called it "one of the few great mathematical events of (the twentieth) century."<ref name="weyl">{{cite book|author=H. Weyl| authorlink=Hermann Weyl|title=Meromorphic functions and analytic curves
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| |publisher=[[Princeton University Press]]|year=1943|page=8}}</ref> The theory describes
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| the asymptotic distribution of folutions of the equation $f(z)=a$, as $a$ varies.
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| A fundamental tool is the Nevanlinna characteristic $T(r,f)$ which measures
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| the rate of growth of a meromorphic function.
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| Other main contributors in the first half of the 20th century were [[Lars Ahlfors]], [[André Bloch (mathematician)|André Bloch]], [[Henri Cartan]], [[Edward Collingwood]], [[Otto Frostman]], [[Frithiof Nevanlinna]], Henrik Selberg, Tatsujiro Shimizu, [[Oswald Teichmüller]],
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| and [[Georges Valiron]]. In its original form, Nevanlinna theory deals with [[meromorphic function]]s of one complex variable defined in a disc |''z''| < ''R'' or in the whole complex plane (''R'' = ∞). Subsequent generalizations extended Nevanlinna theory to algebroid functions, [[holomorphic curve]]s, holomorphic maps between [[complex manifold]]s of arbitrary dimension, [[quasiregular map]]s and [[minimal surface]]s.
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| This article describes mainly the classical version for meromorphic functions of one variable, with emphasis on functions meromorphic in the complex plane. General references for this theory are Goldberg & Ostrovskii,<ref name="GO">{{cite book | first=A.|last= Goldberg |first2=I.|last2= Ostrovskii | authorlink=Anatolii Goldberg | title=Distribution of values of meromorphic functions | publisher=[[American Mathematical Society]] | year=2008}}</ref> Hayman<ref name="hayman">{{cite book | first=W.|last=Hayman | authorlink=Walter Hayman | title=Meromorphic functions | publisher=[[Oxford University press]] | year=1964}}</ref> and Lang (1987).
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| == Nevanlinna characteristic ==
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| === Nevanlinna's original definition ===
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| Let ''f'' be a meromorphic function. For every ''r'' ≥ 0, let ''n''(''r'',''f'') be the number of poles, counting multiplicity, of the meromorphic function ''f'' in the disc |''z''| ≤ ''r''. Then define the '''Nevanlinna counting function''' by
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| :<math> N(r,f) = \int\limits_0^r\left( n(t,f) - n(0,f) \right)\dfrac{dt}{t} + n(0,f)\log r.\,</math> | |
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| This quantity measures the growth of the number of poles in the discs |''z''| ≤ ''r'', as
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| ''r'' increases.
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| Let log<sup>+</sup>''x'' = max(log ''x'', 0). Then the '''proximity function''' is defined by
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| :<math> m(r,f)=\frac{1}{2\pi}\int_{0}^{2\pi}\log^+ \left| f(re^{i\theta})\right| d\theta. \,</math>
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| Finally, define the '''Nevanlinna characteristic''' by
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| : <math>T(r,f) = m(r,f) + N(r,f).\,</math>
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| === Ahlfors–Shimizu version ===
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| A second method of defining the Nevanlinna characteristic is based on the formula
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| :<math> \int_0^r\frac{dt}{t}\left(\frac{1}{\pi}\int_{|z|\leq t}\frac{|f'|^2}{(1+|f|^2)^2}dm\right)=T(r,f)+O(1), \,</math>
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| where ''dm'' is the area element in the plane. The expression in the left hand side is called the
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| Ahlfors–Shimizu characteristic. The bounded term ''O(1)'' is not important in most questions.
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| The geometric meaning of the Ahlfors—Shimizu characteristic is the following. The inner integral ''dm'' is the spherical area of the image of the disc |''z''| ≤ ''t'', counting multiplicity (that is, the parts of the [[Riemann sphere]] covered ''k'' times are counted ''k'' times). This area is divided by ''π'' which is the area of the whole Riemann sphere. The result can be interpreted as the average number of sheets in the covering of the Riemann sphere by the disc |''z''| < ''t''. Then this average covering number is integrated with respect to ''t'' with weight 1/''t''.
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| === Properties ===
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| The role of the characteristic function in the theory of meromorphic functions in the plane is similar to that of
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| :<math>\log M(r, f) = \log \max_{|z|\leq r} |f(z)| \,</math>
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| in the theory of [[entire function]]s. In fact, it is possible to directly compare ''T''(''r'',''f'') and ''M''(''r'',''f'') for an entire function:
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| :<math>T(r,f) \leq \log^+ M(r,f) \,</math>
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| and
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| :<math>\log M(r,f) \leq \left(\dfrac{R+r}{R-r}\right)T(R,f),\,</math>
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| for any ''R'' > ''r''.
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| If ''f'' is a [[rational function]] of degree ''d'', then ''T''(''r'',''f'') ~ ''d'' log ''r''; in fact, ''T''(''r'',''f'') = ''O''(log ''r'') if and only if ''f'' is a rational function.
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| The '''order''' of a meromorphic function is defined by
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| :<math>\rho(f) = \limsup_{r \rightarrow \infty} \dfrac{\log^+ T(r,f)}{\log r}.</math>
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| Functions of finite order constitute an important subclass which was much studied.
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| When ''R'' < ∞, characteristic can be bounded. Functions in a disc with bounded characteristic, also known as functions of [[bounded type (mathematics)|bounded type]], are exactly the ratios of bounded analytic functions.
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| == First fundamental theorem ==
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| Let ''a'' ∈ '''C''', and define
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| :<math>
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| \quad N(r,a,f) = N\left(r,\dfrac{1}{f-a}\right),
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| \quad m(r,a,f) = m\left(r,\dfrac{1}{f-a}\right).\,</math>
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| For ''a'' = ∞, we set ''N''(''r'',∞,''f'') = ''N''(''r'',''f''), ''m''(''r'',∞,''f'') = ''m''(''r'',''f'').
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| The '''First Fundamental Theorem''' of Nevanlinna theory states that for every ''a'' in the [[Riemann sphere]],
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| :<math> T(r,f) = N(r,a,f)+m(r,a,f) + O(1),\,</math>
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| where the bounded term ''O''(1) may depend on ''f'' and ''a''.<ref>Ru (2001) p.5</ref> For non-constant meromorphic functions in the plane, ''T''(''r'', ''f'') tends to infinity as ''r'' tends to infinity,
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| so the First Fundamental Theorem says that the sum ''N''(''r'',''a'',''f'') + ''m''(''r'',''a'',''f''), tends to infinity at the rate which is independent of ''a''. The first Fundamental theorem is a simple consequence
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| of [[Jensen's formula]].
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| The characteristic function has the following properties of the degree:
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| :<math>\begin{array}{lcl}
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| T(r,fg)&\leq&T(r,f)+T(r,g)+O(1),\\
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| T(r,f+g)&\leq& T(r,f)+T(r,g)+O(1),\\
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| T(r,1/f)&=&T(r,f)+O(1),\\
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| T(r,f^m)&=&mT(r,f)+O(1), \,
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| \end{array}</math>
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| where ''m'' is a natural number. The bounded term ''O''(1) is negligible when ''T''(''r'',''f'') tends to infinity. These algebraic properties are easily obtained from Nevanlinna's definition and Jensen's formula.
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| ==Second fundamental theorem==
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| We define {{overline|''N''}}(''r'', ''f'') in the same way as ''N''(''r'',''f'') but without taking multiplicity into account (i.e. we only count the number of distinct poles). Then ''N''<sub>1</sub>(''r'',''f'')
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| is defined as the Nevanlinna counting function of critical points of ''f'', that is | |
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| :<math> N_1(r,f) = 2N(r,f) - N(r,f') + N\left(r,\dfrac{1}{f'}\right) = N(r,f) + \overline{N}(r,f) + N\left(r,\dfrac{1}{f'}\right).\,</math>
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| The Second Fundamental theorem says that for every ''k'' distinct values ''a<sub>j</sub>'' on the Riemann sphere, we have
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| :<math> \sum_{j=1}^k m(r,a_j,f) \leq 2 T(r,f) - N_1(r,f) + S(r,f). \,</math>
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| This implies
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| :<math> (k-2)T(r,f) \leq \sum_{j=1}^k \overline{N}(r,a_j,f) + S(r,f),\,</math>
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| where ''S''(''r'',''f'') is a "small error term".
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| For functions meromorphic in the plane,
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| ''S''(''r'',''f'') = o(''T''(''r'',''f'')), outside a set of finite length i.e. the error term is small in comparison with the characteristic for "most" values of ''r''. Much better estimates of
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| the error term are known, but Andre Bloch conjectured and Hayman proved that one cannot dispose of an | |
| exceptional set.
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| The Second Fundamental Theorem allows to give an upper bound for the characteristic function in terms of ''N(r,a)''. For example, if ''f'' is a transcendental entire function, using the Second Fundamental theorem with ''k'' = 3 and ''a''<sub>3</sub> = ∞, we obtain that ''f'' takes every value infinitely often, with at most two exceptions,
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| proving [[Picard's Theorem]].
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| As many other important theorems, the Second Main Theorem has several different proofs.
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| The original proof of Nevanlinna was based on the so-called Lemma on the [[logarithmic derivative]], which says that ''m''(''r'',''f'''/''f'') = ''S''(''r'',''f''). Similar proof also applies to many multi-dimensional generalizations. There are also differential-geometric proofs which relate it to the [[Gauss–Bonnet theorem]]. The Second Fundamental Theorem can also be derived from the metric-topological [[Ahlfors theory|theory of Ahlfors]], which can be considered as an extension of the [[Riemann–Hurwitz formula]] to the coverings of infinite degree.
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| The proofs of Nevanlinna and Ahlfors indicate that the constant 2 in the Second Fundamental Theorem is related to the [[Euler characteristic]] of the Riemann sphere. However, there is a very different explanations of this 2, based on a deep analogy with number theory discovered by Charles Osgood and [[Paul Vojta]]. According to this analogy, 2 is the exponent in the [[Thue–Siegel–Roth theorem]]. On this analogy with number theory we refer to the survey of Lang (1987) and the book by Min Ru (2001).
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| == Defect relation ==
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| This is one of the main corollaries from the Second Fundamental Theorem. The ''defect'' of a meromorphic function at the point ''a'' is defined by the formula
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| :<math> \delta(a,f)=\liminf_{r \rightarrow \infty}\frac{m(r,a,f)}{T(r,f)} = 1 - \limsup_{r \rightarrow \infty} \dfrac{N(r,a,f)}{T(r,f)}. \, </math>
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| By the First Fundamental Theorem, 0 ≤ ''δ''(''a'',''f'') ≤ 1, if ''T''(''r'',''f'') tends to infinity (which is always the case for non-constant functions meromorphic in the plane). The points ''a'' for which ''δ''(''a'',''f'') > 0 are called '''deficient values'''. The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most [[countable]] and the following relation holds:
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| : <math> \sum_{a}\delta(a,f)\leq 2, \,</math>
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| where the summation is over all deficient values.<ref>Ru (2001) p.61</ref> This can be considered as a generalization of [[Picard's theorem]]. Many other Picard-type theorems can be derived from the Second Fundamental Theorem.
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| As another corollary from the Second Fundamental Theorem, one can obtain that
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| : <math> T(r,f')\leq 2 T(r,f)+S(r,f),\,</math>
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| which generalizes the fact that a rational function of degree ''d'' has 2''d'' − 2 < 2''d'' critical points. | |
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| ==Applications==
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| Nevanlinna theory is useful in all questions where transcendental meromorphic functions arise,
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| like analytic theory of [[differential equation|differential]] and [[functional equation|functional]] equations<ref name="laine">{{cite book|author=Ilpo Laine|title=Nevanlinna theory and complex differential equations|publisher=[[Walter de Gruyter]]|location=Berlin|year=1993}}
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| </ref><ref name="erem">{{cite journal|first=A.|last=[[Alexandre Eremenko|Eremenko]]|title=Meromorphic solutons of algebraic differential equations|doi=10.1070/RM1982v037n04ABEH003967|journal=Russian Math. Surv.|volume=37|issue=4|year=1982|pages=61–95}}</ref> [[holomorphic dynamics]], [[minimal surface]]s, and
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| complex hyperbolic geometry, which deals with generalizations of Picard's theorem to higher
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| dimensions.<ref>Lang (1987) p.39</ref>
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| == Further development ==
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| A substantial part of the research in functions of one complex variable in the 20th century was focused on
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| Nevanlinna theory. One direction of this research was to find out whether the main conclusions of Nevanlinna
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| theory are best possible. For example, the ''Inverse Problem'' of Nevanlinna theory consists in
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| constructing meromorphic functions with pre-assigned deficiencies at given points. This was solved
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| by David Drasin in 1975. Another direction was concentrated on the study of various subclasses of the class
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| of all meromorphic functions in the plane. The most important subclass consists of functions of finite order.
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| It turns out that for this class, deficiencies are subject to several restrictions, in addition
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| to the defect relation (Norair Arakelyan, David Drasin, Albert Edrei, [[Alexandre Eremenko]],
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| [[Wolfgang Heinrich Johannes Fuchs|Wolfgang Fuchs]],
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| [[Anatolii Goldberg]], [[Walter Hayman]], Joseph Miles, Daniel Shea,
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| [[Oswald Teichmüller]], Alan Weitsman and others).
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| [[Henri Cartan]], Joachim and [[Hermann Weyl]]<ref name="weyl" /> and [[Lars Ahlfors]] extended Nevanlinna theory to [[holomorphic curve]]s. This extension is the main tool of Complex Hyperbolic Geometry.<ref>Lang (1987) ch.VII</ref> Intensive research in the classical one-dimensional theory still continues.<ref name="el">[[Alexandre Eremenko|A. Eremenko]] and J. Langley
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| (2008).[http://arxiv.org/abs/0801.0692 Meromorphic functions of one complex variable. A survey], appeared as appendix to {{cite book | first=A.|last= Goldberg |first2=I.|last2= Ostrovskii | authorlink=Anatolii Goldberg | title=Distribution of values of meromorphic functions | publisher=[[American Mathematical Society]] | year=2008}}</ref>
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| ==See also==
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| * [[Nevanlinna function]]
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| * [[Vojta's conjecture]]
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| ==References==
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| {{Reflist}}
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| * {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Introduction to complex hyperbolic spaces | publisher=[[Springer-Verlag]] | place=New York | year=1987 | isbn=0-387-96447-9 | zbl=0628.32001 }}
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| * {{cite book | first=Serge |last= Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=192–204 | zbl=0869.11051 }}
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| *{{Citation | last1=Nevanlinna | first1=Rolf | title=Zur Theorie der Meromorphen Funktionen | publisher=Springer Netherlands | year=1925 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=46 | pages=1–99 | doi=10.1007/BF02543858}}
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| *{{Citation | last1=Nevanlinna | first1=Rolf | title=Analytic functions | origyear=1936 | url=http://books.google.com/books?id=LDLvAAAAMAAJ | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Die Grundlehren der mathematischen Wissenschaften | id={{MR|0279280}} | year=1970 | volume=162}}
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| * {{cite book | title=Nevanlinna Theory and Its Relation to Diophantine Approximation | first=Min |last=Ru | publisher=World Scientific Publishing | year=2001 | isbn=981-02-4402-9 }}
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| ==Further reading==
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| * {{cite book | first1=Enrico | last1=Bombieri | authorlink1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | chapter=13. Nevanlinna Theory | pages=444–478 | zbl=1115.11034 }}
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| * {{cite book | first=Paul | last=Vojta | authorlink=Paul Vojta | title=Diophantine Approximations and Value Distribution Theory | series=Lecture Notes in Mathematics | volume=1239 | publisher=[[Springer-Verlag]] | year=1987 | isbn=978-3-540-17551-3 | zbl=0609.14011 }}
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| * {{cite book | first=Paul | last=Vojta | authorlink=Paul Vojta | chapter=Diophantine approximation and Nevanlinna theory | editor1-last=Corvaja | editor1-first=Pietro | editor2-last=Gasbarri | editor2-first=Carlo | title= Arithmetic geometry. Lectures given at the C.I.M.E summer school, Cetraro, Italy, September 10--15, 2007 | location=Berlin | publisher=[[Springer-Verlag]] | series=Lecture Notes in Mathematics | volume=2009 | pages=111-224 | year=2011 | isbn=978-3-642-15944-2 | zbl=1258.11076 }}
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| ==External links==
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| * {{SpringerEOM|title=Value-distribution theory|id=Value-distribution_theory|first=V.P.|last=Petrenko}}
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| * {{SpringerEOM|title=Nevanlinna theorems|id=Nevanlinna_theorems|first=V.P.|last=Petrenko}}
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| [[Category:Meromorphic functions]]
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