Value-distribution theory

2020 Mathematics Subject Classification: Primary: 30D35 [MSN][ZBL]

Nevanlinna theory

The theory of the distribution of values of meromorphic functions developed in the 1920's by R. Nevanlinna (see [1]). The basic problem is the study of the set $ \{ z _ {n} \} $ of points in a domain $ G $ at which a function $ w ( z) $ takes a prescribed value $ w = a $( so-called $ a $- points), where $ a \in \mathbf C \cup \{ \infty \} $.

Basic concepts.

The fundamental aspects of Nevanlinna theory can be illustrated by taking the case where $ w = f ( z) $ is a transcendental meromorphic function on the open complex plane $ \mathbf C $. Let $ n ( t, a, f ) $ denote the number of $ a $- points of $ f ( z) $( counted with multiplicities) lying in the disc $ \{ | z | \leq t \} $. Further, for any $ a \in \mathbf C $, define

$$ N ( r, a, f ) = \ \int\limits _ { 0 } ^ { r } [ n ( t, a, f ) - n ( 0, a, f )] d \mathop{\rm ln} t + n ( 0, a, f ) \mathop{\rm ln} r, $$

$$ m ( r, a, f ) = m \left ( r, \infty , { \frac{1}{f - a } } \right ) ,\ a \neq \infty , $$

$$ m ( r, \infty , f ) = { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {+} | f ( re ^ {i \theta } ) | d \theta , $$

$$ T ( r, f ) = m ( r, \infty , f ) + N ( r, \infty , f ). $$

$ T ( r, f ) $ is called the Nevanlinna characteristic (or characteristic function) of $ f ( z) $. The function $ m ( r, a, f ) $ describes the average rate of convergence of $ f ( z) $ to $ a $ as $ | z | \rightarrow \infty $, and the function $ N ( r, a, f ) $ describes the average density of the distribution of the $ a $- points of $ f ( z) $. The following theorem yields a geometric interpretation of the Nevanlinna characteristic $ T ( r, f ) $. Let $ F _ {r} $ denote the part of the Riemann surface of $ f ( z) $ corresponding to the disc $ \{ | z | \leq r \} $, and let $ \pi A ( r, f ) $ be the spherical area of the surface $ F _ {r} $. Then

$$ T ( r, f ) = \ \int\limits _ { 0 } ^ { r } A ( s, f ) d \mathop{\rm ln} s + O ( 1) \ \ ( r \rightarrow \infty ). $$

$ T ( r, f ) $ can be used to determine the order of growth $ \rho $ of $ f ( z) $ and its lower order of growth $ \lambda $:

$$ \rho = \ \overline{\lim\limits}\; _ {r \rightarrow \infty } \ \frac{ \mathop{\rm ln} T ( r, f ) }{ \mathop{\rm ln} r } ,\ \ \lambda = \ \lim\limits _ {\overline{ {r \rightarrow \infty }}\; } \ \frac{ \mathop{\rm ln} T ( r, f ) }{ \mathop{\rm ln} r } . $$

Nevanlinna's first main theorem. As $ r \rightarrow \infty $,

$$ m ( r, a, f ) + N ( r, a, f ) = T ( r, f ) + O ( 1), $$

that is, up to a term that is bounded as $ r \rightarrow \infty $, the left-hand side takes the constant value $ T ( r, f ) $( whatever the value of $ a $). In this sense, all values $ w $ of the meromorphic function $ f ( z) $ are equivalent. Of special interest is the behaviour of the function $ N ( r, a, f ) $ as $ r \rightarrow \infty $. In value-distribution theory, use is made of the following quantitative measures of growth of the functions $ N ( r, a, f ) $ and $ m ( r, a, f ) $ relative to the growth of the characteristic $ T ( r, f ) $:

$$ \delta ( a, f ) = 1 - \overline{\lim\limits}\; _ {r \rightarrow \infty } \ \frac{N ( r, a, f ) }{T ( r, f ) } = \ \lim\limits _ {\overline{ {r \rightarrow \infty }}\; } \ \frac{m ( r, a, f ) }{T ( r, f ) } \leq 1, $$

$$ \Delta ( a, f ) = 1 - \lim\limits _ {\overline{ {r \rightarrow \infty }}\; } \frac{N ( r, a, f ) }{T ( r, f ) } = \ \overline{\lim\limits}\; _ {r \rightarrow \infty } \frac{m ( r, a, f ) }{T ( r, f ) } \leq 1. $$

The quantity $ \delta ( a, f ) $ is called the Nevanlinna defect of $ f ( z) $ at $ a $ and $ \Delta ( a, f ) $ is called the Valiron defect of $ f ( z) $ at $ a $. Let

$$ D ( f ) = \{ {a } : {\delta ( a, f ) > 0 } \} ,\ \ V ( f ) = \{ {a } : {\Delta ( a, f ) > 0 } \} . $$

$ D ( f ) $ is called the set of deficient values (cf. Defective value) of $ f ( z) $ in the sense of Nevanlinna, and $ V ( f ) $ is called the set of deficient values of $ f ( z) $ in the sense of Valiron. Nevanlinna's theorem on the magnitudes of the defects and on the set of deficient values of $ f ( z) $ is as follows. For an arbitrary meromorphic function $ f ( z) $: a) the set $ D ( f ) $ is at most countable; and b) the defects of $ f ( z) $ satisfy the relation

$$ \tag{1 } \sum _ { ( } a) \delta ( a, f ) \leq 2 $$

(the defect relation). The constant 2 figuring in (1) is the Euler characteristic of the extended complex plane $ \mathbf C \cup \{ \infty \} $, which is covered by the Riemann surface of $ f ( z) $.

The structure of the set $ D ( f ) $.

Nevanlinna's assertion that the set $ D ( f ) $ is at most countable cannot be strengthened. In fact, given any finite or countable set of points $ E $ in the extended complex plane and any value of $ \rho $, $ 0 < \rho \leq \infty $, there is a meromorphic function $ f _ \rho ( z) $ of order $ \rho $ for which $ E $ coincides with $ D ( f _ \rho ) $. For meromorphic functions whose lower order is zero, $ D ( f ) $ can contain at most one point. Thus, the question on the structure of $ D ( f ) $ is completely solved.

Moreover, it can be shown that for any $ \rho > 0.5 $ there is an entire function $ g _ \rho ( z) $ of order $ \rho $ for which the set $ D ( g _ \rho ) $ is countable. Entire functions of lower order $ \lambda \leq 0.5 $ cannot have finite deficient values.

The structure of the set $ V ( f ) $.

study of the set $ V ( f ) $ of Valiron deficient values is as yet (1992) incomplete. G. Valiron showed that there is an entire function $ g ( z) $ of order one for which the set $ V ( g) $ has the cardinality of the continuum. On the other hand, it can be shown that, for an arbitrary meromorphic function $ f ( z) $, the set $ V ( f ) $ always has zero logarithmic capacity.

For every set $ E $ of class $ F _ \sigma $ of zero logarithmic capacity there is an entire function $ g ( z) $ of infinite order for which $ E \subset V ( g) $.

Properties of defects of meromorphic functions of finite lower order.

For meromorphic functions of infinite lower order, the defects do not, in general, satisfy any relations other than the defect relation (1). However, if one restricts to meromorphic functions of finite lower order, then the picture changes considerably. In fact, if $ f ( z) $ has finite lower order $ \lambda $, then for any $ \alpha $, $ 1/3 \leq \alpha \leq 1 $,

$$ \tag{2 } \sum _ { ( } a) \delta ^ \alpha ( a, f ) \leq K ( \lambda , \alpha ), $$

where the constant $ K ( \lambda , \alpha ) $ depends only on $ \lambda $ and $ \alpha $. On the other hand, there are meromorphic functions of finite lower order such that the series on the left-hand side of (2) diverges when $ \alpha < 1/3 $. For a meromorphic function $ f ( z) $ of lower order $ \lambda \leq 0.5 $, the existence of a deficient value $ a $ such that $ \delta ( a, f ) \geq 1 - \cos \pi \lambda $ influences its asymptotic properties: such a function cannot have other deficient values.

The inverse problem of value-distribution theory.

In a somewhat simplified form it is possible to formulate the inverse problem of value-distribution theory in any class $ {\mathcal K} $ of meromorphic functions in the following way. Every point of a certain sequence $ \{ a _ {k} \} $ in the extended complex plane is assigned a number $ \delta ( a _ {k} ) $, $ 0 < \delta ( a _ {k} ) < 1 $, in such a way that $ \sum _ {k} \delta ( a _ {k} ) \leq 2 $. It is required to find a meromorphic function $ f ( z) \in {\mathcal K} $ such that $ \delta ( a _ {k} , f ) = \delta ( a _ {k} ) $, $ k = 1, 2 \dots $ and $ \delta ( a, f ) = 0 $ for each $ a \neq a _ {k} $, $ k = 1, 2 \dots $ or to prove that $ {\mathcal K} $ contains no such function. The inverse problem has been completely solved in the affirmative in the class of entire functions of infinite lower order and in the class of meromorphic functions of infinite lower order. In the solution of the inverse problem in the class of meromorphic functions of finite lower order there arise specific difficulties, due to the fact that in this case the defects satisfy further relations (like (2)) in addition to (1).

The growth of meromorphic functions.

Given a meromorphic function $ f ( z) $, let

$$ L ( r, \infty , f ) = \max _ {| z | = r } \mathop{\rm ln} ^ {+} | f ( z) |, $$

$$ L ( r, a, f ) = L \left ( r, \infty , { \frac{1}{f - a } } \right ) ,\ a \neq \infty , $$

$$ \beta ( a, f ) = \lim\limits _ {\overline{ {r \rightarrow \infty }}\; } \frac{L ( r, a, f ) }{T ( r, f ) } . $$

$ \beta ( a, f ) $ is called the deviation of $ f ( z) $ from $ a $, and the set $ \Omega ( f ) = \{ {a } : {\beta ( a, f ) > 0 } \} $ is called the set of positive deviations of $ f ( z) $; $ D ( f ) \subseteq \Omega ( f ) $. It is known that if $ g ( z) $ is an entire function of finite order $ \rho $, then

$$ \beta ( \infty , g) \leq \ \left \{ Thus, there is the following result: If a meromorphic function $ f ( z) $ has finite lower order $ \lambda $, then a) $ \Omega $ is at most countable; b) for each $ a $, $$ \beta ( a, f ) \leq \ \left \{

c) for any $ \alpha $, $ 0.5 < \alpha \leq 1 $,

$$ \sum _ { ( } a) \beta ^ \alpha ( a, f ) \leq K ( \lambda , \alpha ), $$

where the constant $ K ( \lambda , \alpha ) $ depends only on $ \lambda $ and $ \alpha $; and d) $ \Omega ( f ) \subseteq V ( f ) $.

Moreover, there exist meromorphic functions of infinite lower order for which the set $ \Omega ( f ) $ has the cardinality of the continuum. For any meromorphic function $ f ( z) $, the set $ \Omega ( f ) $( like $ V ( f ) $) has zero logarithmic capacity. The following theorem characterizes the differences between $ \delta ( a, f ) $ and $ \beta ( a, f ) $: For any $ \lambda $, $ 0 \leq \lambda < \infty $, there is a meromorphic function $ f _ \lambda ( z) $ of lower order $ \lambda $ such that for some $ a $,

$$ \delta ( a, f ) = 0 \ \ \textrm{ and } \ \ \beta ( a, f ) \geq 1. $$

Exceptional values of meromorphic functions in the sense of Picard and Borel.

$ a $ is called an exceptional value of a meromorphic function $ f( z) $ in the sense of Picard if the number of $ a $- points of $ f ( z) $ in $ \{ | z | < \infty \} $ is finite. The value $ a $ is called an exceptional value of $ f ( z) $ in the sense of Borel if $ n ( r, a, f ) $ increases more slowly (in a certain sense) than $ T ( r, f ) $ as $ r \rightarrow \infty $. A non-constant meromorphic function cannot have more than two Borel (and hence Picard) exceptional values.

The value-distribution theory of holomorphic mappings of complex manifolds is being successfully developed as a higher-dimensional analogue of Nevanlinna theory (see [6], [7]), as is the value-distribution theory of minimal surfaces (see [9], [10]).

The distribution of values of functions meromorphic in a disc.

The value-distribution theory of meromorphic functions in the open complex plane has been described above; this is the parabolic case. A theory of growth and value distribution can also be set up in the hyperbolic case, that is, when $ f ( z) $ is a function meromorphic in the unit disc $ \{ | z | < 1 \} $( see [1], [8]). In this case, the functions $ N( r, a, f ) $, $ m ( r, a, f ) $, $ L ( r, a, f ) $, and $ T ( r, f ) $ are defined for $ 0 \leq r < 1 $, just as in the parabolic case. The Nevanlinna and Valiron defects of $ f ( z) $ at a point $ a $ are thus defined as follows:

$$ \delta ( a, f ) = \ \lim\limits _ {\overline{ {r \rightarrow 1 }}\; } \ \frac{m ( r, a, f ) }{T ( r, f ) } , $$

$$ \Delta ( a, f ) = \overline{\lim\limits}\; _ {r \rightarrow 1 } \frac{m ( r, a, f ) }{T ( r, f ) } . $$

The quantity

$$ \beta ( a, f ) = \ \lim\limits _ {\overline{ {r \rightarrow 1 }}\; } \ \frac{L ( r, a, f ) }{T ( r, f ) } $$

is called the deviation of $ f ( z) $ with respect to $ a $.

Let $ D ( f ) = \{ {a } : {\delta ( a, f ) > 0 } \} $, $ V ( f ) = \{ {a } : {\Delta ( a, f ) > 0 } \} $ and $ \Omega ( f ) = \{ {a } : {\beta ( a, f ) > 0 } \} $.

The main properties of $ \delta ( a, f ) $, $ \Delta ( a, f ) $ and $ \beta ( a, f ) $ and of the structure of the sets $ D ( f ) $, $ V ( f ) $ and $ \Omega ( f ) $ in the parabolic case carry over to the hyperbolic case, but only for those functions for which $ T ( r, f ) $ increases rapidly (in a certain sense) as $ r \rightarrow 1 $.

References

[1]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2]	W.K. Hayman, "Meromorphic functions" , Clarendon Press (1964)
[3]	N.U. Arakelyan, "Entire functions of infinite order with an infinite set of deficient values" Soviet Math. Dokl. , 7 : 5 (1966) pp. 1303–1306 Dokl. Akad. Nauk SSSR , 170 : 5 (1966) pp. 999–1002
[4]	A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)
[5a]	V.I. Petrenko, "The study of the structure of the set of positive deviations of meromorphic functions" Math. USSR Izv. , 3 : 6 (1969) pp. 1251–1270 Izv. Akad. Nauk SSSR, Ser. Mat. , 33 : 6 (1969) pp. 1330–1348
[5b]	V.I. Petrenko, "A study of the structure of the set of positive deviations of meromorphic functions" Math. USSR Izv. , 4 : 1 (1970) pp. 31–57 Izv. Akad. Nauk SSSR, Ser. Mat. , 34 : 1 (1970) pp. 31–56
[6]	P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" Acta Math. , 130 (1975) pp. 145–220
[7]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[8]	V.P. Petrenko, "The growth of meromorphic functions" , Khar'kov (1978) (In Russian)
[9]	V.P. Petrenko, "On the growth and distribution of values of minimal surfaces" Dokl. Akad. Nauk SSSR , 255 : 1 (1981) pp. 40–42 (In Russian)
[10]	E.F. Beckenbach, G.A. Hutchison, "Meromorphic minimal surfaces" Pacific J. Math. , 28 : 1 (1969) pp. 17–47

Comments

The solution of the inverse problem of value-distribution theory (in a form sharper than that stated above) is due to D. Drasin [a1]; the inverse problem for entire functions had been solved previously by W.H.J. Fuchs and W.K. Hayman (cf. [2], Chapt. 4). To Drasin [a2] is also due the characterization of functions of finite lower order for which $ \sum _ {(} a) \delta ( a, f ) = 2 $. That the sum in (2) is finite for $ \alpha = 1/3 $ was proved by A. Weitsman [a3]; earlier, Hayman had shown that this was true for $ \alpha > 1/3 $. On the other hand, there are meromorphic functions of finite order for which the sum in (2) diverges for every $ \alpha < 1/3 $. For entire functions the situation is different. Recently, J.L. Lewis and J.-M. Wu have shown [a4] that there exists an $ \alpha _ {0} < 1/3 $ such that the sum in (2) converges for all $ \alpha > \alpha _ {0} $ whenever $ f $ is an entire function of finite lower order. In fact, according to an old conjecture of N.U. Arakelyan, for such functions $ \sum _ {(} a) [ \mathop{\rm log} 1/ \delta ( a, f ) ] ^ {-} 1 < \infty $. This is perhaps the major open question concerning deficiencies.

For a detailed discussion of value-distribution theory in several variables, see the articles in [a5] and [a7].

Around 1986 P. Vojta [a6] found a remarkable analogy between the main theorems in value-distribution theory and theorems from Diophantine approximations. Let $ k $ be an algebraic number field of degree $ d $ and $ B \subset k $ an infinite subset. Let $ S $ be a finite set of (suitably normalized) valuations on $ k $ including the infinite ones. The guiding principle of the analogy is that the set of $ r $ from Nevanlinna theory is replaced by $ B $, the angles $ \theta $ become elements of $ S $ and $ | f( r e ^ {i \theta } ) | $ becomes $ \| b \| _ {v} $. See [a6] for a more complete dictionary. The analogue of $ T( r, f ) $ is $ h( b) = ( 1 / d) \sum _ {v} \mathop{\rm log} ^ {+} \| b \| _ {v} $, the analogue of $ m( r, a, f ) $ is $ m( a, b) = ( 1 / d) \sum _ {v \in S } \mathop{\rm log} ^ {+} \| 1 /( b- a) \| _ {v} $ and $ N( r , a , f ) $ is translated into $ N( a, b) = ( 1 / d) \sum _ {v \notin S } \mathop{\rm log} ^ {+} \| 1 /( b- a ) \| _ {v} $. The first main theorem then changes into $ N( a, b) + m ( a, b) = h( b) + O( 1) $ and this is a well-known property of heights in algebraic number theory. One can also introduce a defect, $ \delta ( a) = {\lim\limits \inf } _ {b \in B } {m( a, b) } / h( b) $. The statement $ \sum _ {a \in \overline{k}\; } \delta ( a) \leq 2 $ is precisely Roth's theorem on the approximation of algebraic numbers by elements from $ k $.

A similar translation of value distribution of meromorphic functions in several variables leads to a number of fascinating conjectures in the area of Diophantine approximations and Diophantine equations.

References

[a1]	D. Drasin, "The inverse problem of Nevanlinna theory" Acta. Math. , 138 (1977) pp. 83–151
[a2]	D. Drasin, "Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two" Acta Math. , 158 (1987) pp. 1–94
[a3]	A. Weitsman, "A theorem on Nevanlinna deficiencies" Acta Math. , 125 (1972) pp. 41–52
[a4]	J.L. Lewis, J.-M. Wu, "On conjectures of Arakelyan and Littlewood" J. d'Anal. Math. , 50 (1988) pp. 259–283
[a5]	I. Laine (ed.) S. Rickman (ed.) , Value distribution theory , Lect. notes in math. , 981 , Springer (1983)
[a6]	P. Vojta, "Diophantine approximation and value distribution theory" , Lect. notes in math. , 1239 , Springer (1987)
[a7]	P.A. Griffiths, "Entire holomorphic mappings in one and several variables" , Annals Math. Studies , 85 , Princeton Univ. Press (1976)