CC BY
59
2000 MSC 32A15, 32A20, 32A22, 32H30
UNIQUENESS THEOREMS OF MEROMORPHIC FUNCTIONS IN SEVERAL COMPLEX VARIABLES
Pei-Chu Hu1), Chung-Chun Yang2)
^ Department of Mathematics, Shandong University,
Jinan 250100, Shandong, P. R. China e-mail: pchu@sdu.edu.cn 2) Department of Mathematics, The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong e-mail: chungchun.yang@gmail.com
Abstract. In the survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of homogeneous linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Bessel functions and prove in general that meromorphic solutions that grow much faster than the coefficient have zero Nevanlinna’s deficiency for each non-zero complex value. It’s well-know result that if a nonconstant meromorphic function f on C and its l-th derivative f(1) have no zeros for some I > 2, then f is of the form f (z) = exp(Az + B) or f (z) = (Az + B)-n for some constants A, B. We have extended this result to meromorphic functions of several variables, by first extending the classic Tumura-Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables by utilizing Nevanlinna theory.
Keywords: meromorphic functions, homogeneous linear partial differential equation, holomorphic coefficients, Nevanlinna’s value distribution theory.
Analytic properties or characterizations of meromorphic (or entire) solutions of some partial differential equations (or system) of the first order have been exhibited clearly by several authors (cf. [2], [13], [18], [19]). In this survey, we introduce a few results on meromorphic solutions of homogeneous linear partial differential equations of the second order in two independent complex variables
d2u d2u d2u du du
a°~Qf2 dtdz a2~d'z2 a3~dt a4~dz a&U =
where ak = ak(t,z) are holomorphic functions for (t,z) E E, where E is a region on C2. Basic idea comes from S. N. Bernstein [3], H. Lewy [17], I. G. Petrovskii[20]. For more detail, see [15]. To prove these results, we used some methods in [5], [7], [11], [14], [21], [23] and [26].
First of all, we examine the following special differential equation:
2 d2 u 2 d2u du du 2
tW-z3^ + tm-zlk+tu = ^ (L2)
The work of Chung-Chun Yang was partially supported by Natural Science Foundation of China and second author was partially supported by a UGC Grant of Hong Kong: Project 604106.
Theorem 1.1 The differential equation (1.2) has an entire solution f (t,z) on C2 if and only if f is an entire function expressed by the series
f (t, z) = ^ n!cn Jn(t)zn (1.3)
n=0
such that
limsup |cn|1/n = 0, (1.4)
U—tt
where JU(t) is the first kind of Bessel’s function of order n. Moreover, the order ord(f) of the entire function f satisfies
p < ord(f) < max{1,p},
where
2logn (1.5)
p = limsup------ , . (1.5)
U—tt log |cu| /
By definition, the order of f is defined by
V log+log+M(r,/)
ord(/) = limsup-----------------------,
r—tt log r
where
and
, + | log x, if x > 1;
log +x = < n
'0, if x < 1,
M(r,f) = . |f (t,z)|-|t|<r,|z|<r
G. Valiron [25] showed that each transcendental entire solution of a homogeneous linear ordinary differential equation with polynomial coefficients is of finite positive order. However, Theorem 1.1 shows that Valiron’s theorem is not true for general partial differential equations. Here we exhibit another example that the following equation
2 d2u d2u du
tW~d? + tm=0
has an entire solution exp(tez) of infinite order.
If 0 < A = ord(f) < to, we define the type of f by
log +M(r, f)
typ(/) = limsup
r
A
For the type of entire solutions of the equation (1.2), we have an analogue of Lindelof-Pringsheim theorem, its proof is essentially the same as that of the determining of the type for Taylor series of entire functions of one complex variable.
Theorem 1.2 If f (t,z) is an entire solution of (1.2) defined by (1.3) and (1.4) such that 1 < A = ord(f) < to, then the type a = typ(f) satisfies
eAa = 2-a/2 lim sup 2n|cU|A/(2n).
r—
U^tt
Brosch [4] proved that if two nonconstant meromorphic functions f and g on C share three distinct values c1; c2, c3 counting multiplicities, and if f is a solution of the differential equation
U 2U
( -rr ) = ^Hz)wj := p(z’ w)
' ' j=0
such that b0, b1, ■ ■ ■ , b2n (b2n ^ 0) are small functions of f (grow slower than f), furthermore if P(z, ci) ^ 0 for i =1, 2, 3, then f = g. To state a generalization of Brosch’s result to PDE, we abbreviate
du d2u d2 u
= di1 Utz = dilh1 Utt = W1
and so on, and set
Du = aout + 2aiutuz + a2u^,
Lu = aoutt + 2aiutz + a2u^z + a3ut + a4u.
We make the following assumption:
(A) All coefficients aj in (1.1) are polynomials and when a6 = 0 there are no nonconstant polynomials u satisfying the system
Du = 0,
Lu = 0.
For technical reason, here we study only meromorphic functions of finite orders. The order of a meromorphic function of several variables may be defined by using its Nevanlinna’s characteristic function (cf. [12], [22]).
Theorem 1.3 Assume that the assumption (A) holds. Let f (t, z) be a nonconstant meromorphic solution of (1.1) such that ord(f) < to and let g be a nonconstant meromorphic function of
finite order on C2. If f and g share 0, 1, to counting multiplicity, one of the following five cases
is occurred:
(a) g = f;
(b) gf = 1;
(c) a6 = °, gf = f + g;
(d) a6 = 0, and there exist a constant b E {0,1} and a polynomial ft such that
/ = 0.9 =
b 1 b 1
(e) a6 = 0; f 2g2 = 3fg - f - g.
When a6 = 0, the case (b) may happen. For example, we consider the differential equation
d2u d2u du
+ 7^ - 1S7 - « = 0, (1-6)
dt2 dz2 dt
which has an entire solution of order 1
f (t,z)= e‘+z.
Let’s compare f with the following entire function of order 1
g(t,z) = e-t-z.
Obviously, f and g share 0, 1, -1, to counting multiplicity, but g = f, gf =1. Now the differential equation
Lu + Du + a6 = 0 has a nonconstant polynomial solution
u(t, z) = t + z.
The condition (A) is meaningful. For example, Theorem 1.1 shows that the differential equation (1.2) has a lot of entire solutions of finite orders. Obviously, the condition (A) associated to the differential equation (2) holds, and hence we can obtain the fact:
Corollary 1.4 Let f (t,z) be a nonconstant meromorphic solution of (1.2) such that ord(f) < to and let g be a nonconstant meromorphic function of finite order on C2. If f and g share 0, 1, to counting multiplicity, then we have either g = f or gf =1 or f 2g2 = 3fg — f — g.
The case (b) in Theorem 1.3 may really happen for a6 = 0. For example, we consider the differential equation
d2u du
( ^
which has an entire solution f (t,z) = et+z of order 1 such that the assumption (A) holds obviously. The entire solution f and the function g = e-t-z share 0, 1, to counting multiplicity, and satisfy gf = 1, that is, the case (b) in Theorem 1.3 happens for the case a6 = 0.
For a real number x, let [x] denote the maximal integer < x. We give the following result that is an analogue of Anastassiadis’s theorem [1] on uniqueness of entire functions of one variable.
Theorem 1.5 Let f (t,z) and g(t,z) be transcendental entire solutions of (1.2) such that ord(f) < to, ord(g) < to, and
9 J/-(0, 0) = / y (0i o), j = 0,1,
dtj dzj dtj dzj
where
q = max{[ord(f)], [ord(g)]}.
If there exists a complex number a with (a, f(0, 0)) = (0, 0) such that f and g share a counting multiplicity, then we have f = g.
Theorem 1.3 shows that when a6 = 0, global solutions of the equation (1.1) can be quite complicated, however, when a6 = 0, these solutions have normal properties. Next result also supports this view. Theorem 1.6 extends a theorem (cf. Theorem 5.8 of [10]) on meromorphic solutions of linear ordinary differential equations.
Theorem 1.6 Assume that all ak in (1.1) are entire functions on C2 which grow slower than a meromorphic solution of equations (1.1) on C2. If a6 ^ 0, then the deficiency of the solution for each non-zero complex number is zero.
For example, the telegraph equation
d2u 2 d2u du 2
— C ——~ -j- 2q'— + Ol u = 0
dt2 dz2 dt
has entire solutions
u(t, z) = e-at{f (z + ct) + g(z — ct)},
where f and g are entire functions on C. If a = 0, Theorem 1.6 shows that the deficiency of a non-constant u(t, z) for each non-zero complex number a is zero, which means that the equation
f (z + ct) + g(z — ct) — aeat = 0
has zeros.
Let Z+ denote the set of non-negative integers. For z = (z1, ...,zm) G Cm, i = (i1, ...,im) G Zm, we write
5 ....
dZk = k=1iщ 91 = d\ = dl\ ■ ■ ■ dlz; M = *! + ••• +
We have interesting in the following problem:
Conjecture 1.7 If f is a meromorphic function in Cm such that f and d1 f have no zeros for some l = (/1,...,/m) G Zm with > 2 (1 < k < m) and such that the set of poles of f is algebraic, then there exists a partition
{1,..., m} = 1o U /1 U ■ ■ ■ U /fc
such that /» H /j = 0 (i = j), and
f (z1,..., zm) = exp I ^2 Aizi + Bo I ]^[ I ^2 A»zi + Bj \ie/o / j=1 \ie/j
where Ai, Bj are constants with Ai = 0, and nj are positive integers.
This is open if m > 1. For detail discussion, see [16]. When m =1, the conclusion of Conjecture 1.7 was obtained by Tumura [24], and Hayman [8] gave a proof for the case l = /m = 2. Later, as a correction of the gap in Tumura’s proof, Clunie [6] gave a valid proof of the assertion for any l > 1 .
Let f be a meromorphic function in Cm which we shall assume to be not constant. We shall be concerned largely with meromorphic functions h which are polynomials in f and the partial derivatives of f with coefficients a of the form
|| T(r, a) = o(T(r,f)), (1.8)
where T(r, f) is the Nevanlinna’s characteristic function of f, and where the symbol “ || "means that the relation holds outside a set of r of finite linear measure. Such functions h will be called differential polynomials in f. To study Conjecture 1.7, the following result will play a crucial role.
Theorem 1.8 Suppose that f is meromorphic and not constant in Cm, that
g = fn + Pn-1(f), (1.9)
where Pn-1(f) is a differential polynomial of degree at most n — 1 in f, and that
II N(rJ)+N^r-^j=o(T(rJ)),
where N(r, f) is the Nevanlinna’s valence function of f for poles. Then
aN n
9 ' ' '
where a is a meromorphic function of the form (1.8) in Cm determined by the terms of degree n — 1 in Pn-1(f) and by g.
When m =1, Theorem 1.8 is due to Hayman ([9], Theorem 3.9, p.69). By using Theorem 1.8, we can give a proof of Conjecture 1.7, under a condition on non-vanishing of the partial derivatives of order > 1 that differs from the one posed in the conjecture, as follows:
Theorem 1.9 If f is a meromorphic function in Cm such that f, d^1 f, ..., dZ™ f have no zeros
for some lk > 2 (1 < k < m) and such that the set of poles of f is algebraic, then there exists a partition
{1,..., m} = /o U /1 U ■ ■ ■ U 4 such that /i H /j = 0 (i = j), and
f (z1, ...,2m) = exp £ Aizi + Bo I Aizi + Bj
\ie/o / j=1 \ie/j
where Ai, Bj are constants with Ai = 0, and nj are positive integers.
In particular, if f is entire, the function f in Theorem 1.9 has only an exponential form
f (zb ..., zm) = exp (A1z1 + ■ ■ ■ + Amzm + Bo) .
We shall utilize the methods developed in [9], [12] and [13] and generalized Clunie lemma to
prove the main results.
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ТЕОРЕМЫ ЕДИНСТВЕННОСТИ МЕРОМОРФНЫХ ФУНКЦИЙ НЕСКОЛЬКИХ КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ
Пей Чу Ху1), Чунг Чун Янг2)
1) Шаньдунский университет,
Жиньян 250100, Шаньдун, П. Р. Китай e-mail: pchu@sdu.edu.cn 2) Гонконгский университет науки и технологии Клеар Уотер Бэй, Ковлун, Гонконг e-mail: chungchun.yang@gmail.com
Аннотация. В работе исследуются вопросы существования, единственности и распределния значений мероморфных (или целых) решений линейных дифференциальных уравнений в частных производных второго порядка с полиномиальными коэффициентами.
Ключевые слова: мероморфные функции, однородные линейные дифференциальные уравнения в частных производных, теория Неванлинны распределения ценности.
