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ANNALES MATHÉMATIQUES
BLAISE PASCAL
Henna Jurvanen
An ultrametric Nevanlinna’s second main theorem for
small functions of a special type
Volume 17, no 2 (2010), p. 425-431.
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Annales mathématiques Blaise Pascal 17, 425-431 (2010)
An ultrametric Nevanlinna’s second main
theorem for small functions of a special type
Henna Jurvanen
Abstract
In ultrametric Nevanlinna theory, the Nevanlinna’s second main theorem for
small functions has only been proved in the case of at most three small functions.
In this paper, we prove a second main theorem for q small functions of a special
type when the residue characteristic of the field is zero.
Le theoréme de Nevanlinna ultramétrique pour petites fonctions
Résumé
En théorie de Nevanlinna ultramétrique, le second théorème fondamental de
Nevanlinna pour des petites fonctions a seulement été établi pour trois petites
fonctions. Dans cet article, on montre un second théorème fondamental pour q
petites fonctions d’un certain type quand la caractéristique résiduelle du corps est
zero.
1. Introduction
The Nevanlinna Second Main Theorem on three small functions is well
known in complex meromorphic functions. It is easily proved by using the
so-called bi-ratio technique. However, there is no way to generalize such a
technique to n small functions. In 2004, Yamanoi proved the Second Main
Theorem on n small functions in [6] by a method that has no link with
the elementary proof of the theorem on three small functions.
Now, let K be an algebraically closed field of characteristic zero, com-
plete with respect to an ultrametric absolute value. The Nevanlinna Sec-
ond Main Theorem does exist for meromorphic functions in the whole
field and also for meromorphic functions in an open disk, see [1], [2], [3].
In the same way as for complex functions, an ultrametric version of Nevan-
linna Second Main Theorem on three small functions can be proved by
Keywords: Ultrametric Nevanlinna theory, Second main theorem.
Math. classification: 12H99.
425
H. Jurvanen
the technique of bi-ratio. But it is impossible to adapt to p-adic anal-
ysis Yamanoi’s work that mainly uses typically archimedean properties
in the complex field, particularly integrations on paths. Therefore it re-
mains open how to obtain a similar result in an ultrametric field and also,
whether it can be obtained.
This is why, here, we will try to obtain a kind of Nevanlinna Second
Main Theorem on n small functions in ultrametric analysis, in a case where
such a solution appears: we place ourselves in a field of residue character-
istic zero, consider just analytic functions inside an open disk and assume
that the ”small functions” actually are bounded and satisfy an additional
condition: intersections of two global images are empty. The hypothesis of
zero residue characteristic is due to the behaviour of derivatives in a field
of residue characteristic p 6= 0, which is explained after the proof of the
theorem.
Let K be an algebraically closed field of characteristic zero and of
residue characteristic zero, complete for an ultrametric absolute value | |.
Let d(0, R−) denote a disk d(0, R−) = {x ∈ K| |x − 0| < R} and let
d(0, R) = {x ∈ K| |x − 0| ≤ R}. Moreover, let C(0, R) denote the circle
C(0, R) = {x ∈ K| |x− 0| = R}.
For each r ∈]0, R[, let η(r, f) denote the number of zeros of f in the
circle C(0, r) counting multiplicity, and let η(r, f) be the number of zeros
of f in C(0, r) ignoring multiplicity.
Let A(d(0, R−)) denote the analytic functions in d(0, R−). Moreover,
let Ab(d(0, R−)) denote the bounded analytic functions in d(0, R−) and
let Au(d(0, R−)) = A(d(0, R−))\Ab(d(0, R−)) be the unbounded analytic
functions in d(0, R−).
Notation 1.1. For f ∈ A(d(0, R−)), let
φa,r(f) = lim
|x−a|→r,|x−a|<r
|f(x)|.
Moreover, given f ∈ Ab(d(0, R−)), let
||f || = sup{|f(x)| | x ∈ d(0, R)}.
Let f ∈ A(d(0, R−)). Let (an)n∈N∗ be the sequence of zeros of f with
0 < |an| ≤ |an+1|, with a0 whenever f(0) = 0 and let kn denote the order
of the zero an. Moreover, if f(0) = 0, we denote by k0 the order of the
origine as a zero and we set l0 = 1. And if f(0) 6= 0 we set k0 = l0 = 0.
426
Second main theorem for small functions
Then we define the counting function of zeros of f as
Z(r, f) = k0 log r +
∑
|an|≤r
kn(log r − log |an|).
Respectively, let the counting function ignoring multiplicities be defined
as Z(r, f) = l0 log r +
∑
|an|≤r(log r − log |an|).
Remark 1.2. As far as analytic functions are involved, it is now easy to
state the second main theorem by only using the function Z(r, f), this
function being equivalent to the characteristic function T (r, f) for analytic
functions.
Definition 1.3. We call a function ω ∈ A(d(0, R−)) a small function with
respect to f ∈ A(d(0, R−)), if
lim
r→R
Z(r, ω)
Z(r, f)
= 0.
Acknowledgement. The paper was begun during a short stay at Uni-
versité Blaise Pascal.
2. Second main theorem for small functions
The following lemma is a special case of Theorem 2.6.1 in [5].
Lemma 2.1. Let f ∈ A(d(0, R−)) and ω ∈ A(d(0, R−)) and let ω be a
small function of f . Then we have
Z(r, f − ω) = Z(r, f) +O(1).
Lemma 2.2. For f ∈ A(d(0, R−)) we have
Z(r, f ′) ≤ Z(r, f)− log r.
Proof. For analytic functions, it is obvious that Z(r, fg) = Z(r, f) +
Z(r, g). Hence Z(r, f) = Z(r, f/g) +Z(r, g). The lemma now follows from
Corollary 2.4.14 in [5], which says that Z(r, f ′/f) ≤ − log r. 
By Theorem 23.7 in [4], we have
Lemma 2.3. Let f, g ∈ A(d(0, R−)), let r > 0 and assume that φa,r(f) >
φa,r(g). Then f and f + g have the same number of zeros (counting mul-
tiplicity) in d(a, r).
427
H. Jurvanen
Theorem 2.4. Let f ∈ Au(d(0, R−)) and let ωj ∈ Ab(d(0, R−)) be of the
form ωj = aj + θj, j = 1, . . . , q, where aj ∈ K and
||θj || < min
j 6=k
|aj − ak|.
Then
(q − 1)Z(r, f) ≤
q∑
j=1
Z(r, f − ωj) +O(1). (2.1)
Proof. Let ωj = aj + θj , 1 ≤ j ≤ q and let
||θj || < min
k 6=j
|aj − ak| =: tj . (2.2)
Let f ∈ Au(d(0, R−)) and let α ∈ d(0, R−) be such that f(α) = ωj(α) for
some 1 ≤ j ≤ q.
Since φα,r(f) is an increasing function of r, and tends to infinity as r
tends to R, there exists a unique radius ρα,j for which
φα,ρα,j (f − aj) = tj .
By hypothesis (2.2), it follows that f − ωk has no zeros in d(α, ρ−α,j) for
k 6= j. On the other hand, since φα,ρα,j (f − aj) > ||θj ||, the number of
zeros of f − aj in d(α, ρ−α,j) is equal to this of f − ωj , see Lemma 2.3.
Now consider the circle C(0, r) for fixed r, and consider the possibly
several points αm,j ∈ C(0, r), 1 ≤ m ≤ uj , 1 ≤ j ≤ q, such that f(αm,j) =
ωj(αm,j). For each indices m and j, let σm,j denote the number ραm,j ,j
previously defined as ρα,j for α.
Now the disks d(αm,j , σ−m,j) are pairwise disjoint.
For every m = 1, . . . , uj and j = 1, . . . , q, let νm,j be the number of
zeros of f−ωj in d(αm,j , σ−m,j) counting multiplicities, and let νm,j be this
number ignoring multiplicities. Hence νm,j is also the number of zeros of
f − aj in d(αm,j , σ−m,j) counting multiplicities.
Since K has residue characteristic zero, it follows from the above that
the number of zeros of f ′ in each disk d(αm,j , σ−m,j) is exactly νm,j − 1.
428
Second main theorem for small functions
Hence for any r ∈]0, R[, we have
η(r, f ′) ≥
q∑
j=1
( uj∑
m=1
(νm,j − 1)
)
≥
q∑
j=1
( uj∑
m=1
(νm,j − νm,j)
)
≥
q∑
j=1
(η(r, f − ωj)− η(r, f − ωj)).
Adding all the inequalities together, this finally yields
q∑
j=1
(Z(r, f − ωj)− Z(r, f − ωj)) ≤ Z(r, f ′).
Since by Lemma 2.1 we have Z(r, f −ωj) = Z(r, f)+O(1) and by Lemma
2.2 we have Z(r, f ′) ≤ Z(r, f) +O(1), we obtain
(q − 1)Z(r, f) ≤
q∑
j=1
Z(r, f − ωj) +O(1). (2.3)

Writing (2.3) in the form
(q − 1)Z(r, f)
Z(r, f)
≤
q∑
j=1
Z(r, f − ωj)
Z(r, f)
+ o(1),
we can easily see that if, e.g., all the zeros of f −ωj and f −ωk are double
for some j 6= k, then almost all the zeros of f −ωi must be simple for any
i 6= j, k.
Corollary 2.5. At most two of the functions f − ωi may be such that all
their zeros are double.
Remark 2.6. In any algebraically closed complete field, we have this im-
portant property on meromorphic functions:
|f ′|(r) ≤ |f |(r)
r
.
Now, the equality holds when the residue characteristic is 0, provided
the number of zeros and this of poles in d(0, r) are not equal. However, such
429
H. Jurvanen
a property does not hold when the residue characteristic is p 6= 0. In the
proof of the theorem, the equality is crucial. This is why we presently can’t
generalize the theorem to fields with a non-zero residue characteristic.
On the other hand, we have used the fact that |f |(r) is an increasing
property for analytic functions inside a disk. This is no longer true for
meromorphic functions. Therefore, generalizing the theorem to meromor-
phic functions does not seem possible in this way.
Acknowledgement. The author would like to thank Alain Escassut for
useful conversations.
References
[1] A. Boutabaa – “Théorie de Nevanlinna p-adique”, Manuscripta
Math. 67 (1990), p. 251–269.
[2] A. Boutabaa & A. Escassut – “An improvement of the p-adic
Nevanlinna theory and application to meromorphic functions”, Lecture
Notes in Pure and Appl. Math. 207 (1999), p. 29–38.
[3] , “Applications of the p-adic Nevanlinna Theory”, Lecture
Notes in Pure and Appl. Math. 222 (2001), p. 49–61.
[4] A. Escassut – Analytic Elements in p-adic Analysis,, World Scien-
tific, Singapore, 1995.
[5] , p-adic value distribution, Some topics on value distribution
and differentiability in complex and p-adic analysis, Science Press, Bei-
jing, 2008.
[6] K. Yamanoi – “The second main theorem for small functions and
related problems”, Acta Math. 192 (2004), p. 225–294.
430
Second main theorem for small functions
Henna Jurvanen
Department of Physics
and Mathematics
University of Eastern Finland
P.O.Box 111
Joensuu, FI-80101
FINLAND
henna.jurvanen@uef.fi
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