THE THEOREM OF TUMURA-CLUNIE FOR
MEROMORPHIC FUNCTIONS
E. MUES AND N. STEINMETZ
1. Introduction and main results
Let / be a nonconstant meromorphic function in the complex plane. It is
assumed that the reader is familiar with the notations of Nevanlinna theory (see, for
example, [3], [4]). We denote by S(r,f), as usual, any function satisfying
S(r,f) = o{T(r,f))
as r -> +oo, possibly outside a set of finite Lebesgue measure. Throughout this
paper we denote by a, b, c, a0, a l 5 ... meromorphic functions (or constants) of smaller
growth than / , that is,
T(r, a) = S(r, / ) , . . .
asr->+oo.
(1)
Clunie [1] proved the following theorem on the zeros of
V = ao + aJ + ... + anf",
an#0,
(2)
where / is a given meromorphic function.
THEOREM
A. Let f and g be entire functions, and assume that
¥(z) = b(z)e9{z).
Then
This result was first stated by Tumura [5]. His proof, however, was incomplete.
The assumptions of Theorem A can be weakened (see, for example Hayman
[3; p. 69]), but it is always required that the logarithmic derivative ¥ ' / ¥ is a function
of small growth compared with / in the sense defined above. Since
T{r, V/V) ^ N(r, ¥ ) ^ N(r,f) + S(r,f), the logarithmic derivative does not have
this property if / is an arbitrary meromorphic function. For example, if
f(z) = tanz
and V = 1 + / 2 ,
we have V/W = 2/, and *¥ has no zeros. Thus, Clunie's theorem cannot hold for
arbitrary meromorphic functions.
In the present paper, we shall prove two theorems, which include Clunie's result
Received 3 January, 1980.
[J. LONDON MATH. SOC. (2), 23 (1981), 113-122]
114
E. MUES AND N. STEINMETZ
as a special case. We shall also give necessary and sufficient conditions so that
N(r, l/¥) = S(r,f).
THEOREM 1. Let f and a0, a l5 ..., an_2 be meromorphic functions, where at least
one of the coefficients a-t is not identically zero. If
V = ao + aJ + ... + an-2r-2 + r
(4)
satisfies
then there are exactly two cases as follows.
(a)
There exist a meromorphic function ao(z) ^ 0 and an integer pL such that
n = 2\i and
^ = (a o +/ 2 r.
(5)
Also, f is a solution of the Riccati differential equation
(6)
where c(z) ^ 0 is a meromorphic function satisfying T(r, c) = S(r,f).
(b)
There exist a meromorphic function ao(z) ^ 0, positive integers [it and \i2, and
distinct complex numbers K1 and K2 such that ^i+fi2 = n,pilKx-\-pL2K2 = 0,
and
(/-K2a0f .
(7)
Also, f is a solution of the Riccati differential equation
w'
=
- ^ a z
L o( )
-
(K1+K2)C(Z)OC0(Z) \W + C(Z)W2
J
(8)
On the other hand, iff satisfies equation (6) respectively (8) and if*? is given by (5)
respectively (7), then we have
N(r, l/¥) ^ N(r, a > 0 ) +JV(r, c) = S(r,f).
THEOREM
assume that
2. Let f be a meromorphic function. Suppose that *F is given by (2) and
THE THEOREM OF TUMURA-CLUNIE FOR MEROMORPHIC FUNCTIONS
Then we have the following three possibilities I b0 = -^-1
na
n
V
115
:
(a) V =
(b)
x
¥ = an{)
Here the quantities fj., fil} ju2, KX, K2, ct0
nave
the same meaning as in Theorem 1.
As a corollary, we obtain Clunie's result.
COROLLARY.
If, in addition to the assumptions of Theorem 2,
N(r,f) =
S(r,f),
then
nan
2. Notations and preliminary results
We recall the notation of Section 1. Suppose that / is a transcendental
meromorphic function and that a,ap... are meromorphic functions (or constants)
satisfying T(r, a) = S(r,f),....
NjoJi, •••Jk a r e nonnegative integers, we call
a
differential
monomial
in
/
of
degree
yM=j0
+ -.-+jk
and
of
weight
r M = ./o + 2/i + ... + (/c + i)jk. If M l 5 . . . , Mp are differential monomials and if ax,...,
are meromorphic functions (fl/z) ^ 0), we call
ap
pin = i ajizWjtn
a differential polynomial in / , and we define the degree yp and the weight FP of P by
p
p
yp = maxy M and F P = max VM.. If P is a differential polynomial, then P' denotes the
differential polynomial which satisfies P'[j(z)~] = -rP[_j{z]] for any meromorphic
function / . Note that yF = yP. The following result on differential polynomials is
essentially due to Clunie [1].
LEMMA. Let Q and Q* be differential polynomials in f having coefficients a} and
af. Suppose that m(r, aj) = S(r,f) and m(r, af) = S(r,f), but that it is not necessarily
the case that T(r, a;) = S(r,f) or that T(r, af) = S{r,f). IfyQ^n
and
/ n Q*[/] = QU1,
then
116
E. MUES AND N. STE1NMETZ
Remark. Clunie proved his lemma under the stronger hypothesis that
T{r, aj) = S{r,f) and T{r, aj) = S{r,f). His proof, however, does also work under
the weaker assumptions stated above. In particular, there might be coefficients of the
form / ' / / or, more generally, T'/H1 where ¥ is given by (2).
In order to prove Theorem 1, we need some lemmata. It is always assumed that
¥ is given by (4) and satisfies
N(r, 1/T) = S(r,f).
LEMMA
1. There exists a meromorphic function a{z) ^ 0 such that
T{r, a) = S(rJ) and
1 ¥'
r = *+--& fn T
LEMMA
2. (a) m{r,f) = S{r,f).
(b) m(r, 1//) = S(r,f).
(c) N^rJ)
= S(r,f).
(d) N^r, \/f) = S(r,f).
LEMMA 3. There exist meromorphic functions b(z) and c(z) ^ 0 such that
T{r,b)+T{r,c) = S{rJ) and
4"/¥ = n(b + cf).
Combining Lemma 1 with Lemma 3, we obtain the following.
LEMMA
4. The function f is a solution of the Riccati differential equation
w' = a(z) + b(z)w + c(z) w2 .
LEMMA
5. Any solution w = rj(z) of the equation
satisfies the differential equation (10).
3. Proof of lemmata
Proof of Lemma 1. The differential polynomial
is not identically zero by assumption. By differentiating the equation
we get
(10)
THE THEOREM OF TUMURA-CLUNIE FOR MEROMORPHIC FUNCTIONS
where Q and Q are given by Q [ / ] = nf
117
- ^ - / and Q [ / ] = ^ - P [ / ] - / " [ / ]
respectively. Since ye ^ n — 2, the assumptions of Clunie's lemma are satisfied. We
define
1 ¥'
n ¥
hence this lemma yields
m(r,fl) = S(r,/)
(13)
Since a(z) = 0 would imply ¥ = c/" (c =/= 0 a constant), and
a(z) cannot vanish identically.
Now, let z0 be a pole of order pi of a. If / is regular at z = z 0 , we have ¥(z 0 ) = 0
or ¥(z 0 ) = oo, and a}(z0) = oo for some ; and, therefore, n = 1. If / has a pole at
z = z 0 of order p, and if the coefficients a} have poles of order not greater than /,
then it follows from (11) that (n-l)p + n < l + ( n - 2 ) p + / or ju < l-p + l < /.
Thus,
N(r, a) ^ N(r, l/¥)+ " ^ N(r, aj) = S(r,f)
(14)
by hypothesis, which together with (13) proves Lemma 1.
Proof of Lemma 2. By dividing equation (11) by Q [ / ] = na, we get
Applying Clunie's lemma to equation (15), where Q*\_f~\ = f, assertion (a) of Lemma
2 follows.
In order to prove (b), we divide equation (12) by af and deduce that
/ iff
/ 1\
m\r,-)
= m\r,-\-
IT'
—
by Nevanlinna's lemma on the proximity function of the logarithmic derivative. It is
easily seen that a vanishes with multiplicity p — 1 at any zero of / of order p. Thus,
we have
JV^r, 1//) ^ N(r, I/a) < T(r, a) + 0(1) = S(r,f),
which proves (d).
118
E. MUES AND N. STEINMETZ
In order to prove (c), we use the notation of the proof of Lemma 1. Let z0 be
a pole of f of order p ^ 2, and let the coefficients a-} have poles at z0 of order
not greater than /. At z0 the function a has a pole of order [i(fi > 0) or vanishes
(— ^)-times (/( ^ 0). From (n — l)p + n ^ (n — 2)p + l + l we conclude that
p—1 ^ l — fi ^ l + \n\ and, therefore,
N1(r,/)<N(r,l/a)+ £
N(r,aj)
^T(r,a)+"X N{r,aj) = S{r,f)
because of Lemma 1.
Proof of Lemma 3. Let z0 be a simple pole of / which is neither a pole of one of
the coefficients ai nor a pole or a zero of a. Dividing equation (15) by fn~x and
letting z -+• z0 we can compute the residue
R
r
IO
e
s
/
^ % .
n
a(z0)
Since, by Lemma 2 (a), (c), / has infinitely many simple poles with the required
properties, an_2 cannot vanish identically. Now we define meromorphic functions
n a
c=2an_2
and b =
IV
-—-c.
n ¥
Obviously, c(z) j= 0, T(r, c) = S{r,f), and w(r, b) = S(r,/) by Lemma 2 (a). By
definition, 6(z0) =/= oo at every simple pole of / which is neither a pole of one of the
coefficients nor a pole or a zero of a. Thus, we have
which proves Lemma 3.
Proof of Lemma 5. From Lemma 3 and Lemma 4, it follows that
H(z, w): = Qz(z, w) + Qw(z, w){a(z) + b(z)w + c(z)w2)-n{b(z) + c(z)w)Q(z, w)
= 0
(16)
if w is replaced by f(z). However, H(z,f(z)) can only vanish identically if H(z, w) does
and if (16) holds for arbitrary complex z and w.
Now let z0 be neither a pole of one of the coefficients of Q{z, w) nor a zero of the
w-discriminant of Q(z, w). Then any solution w = rj(z) of Q(z, w) = 0 is a
holomorphic function in a neighbourhood A of z 0 , and we have
Q(z, w) = {w-rj(z)YQ*(z, w), Q*(z, r,(z)) # 0
(17)
THE THEOREM OF TUMURA-CLUNIE FOR MEROMORPHIC FUNCTIONS
119
for z 6 A and arbitrary w. The integer fi is uniquely determined. The equations (16)
and (17) yield
H(-rj' + a + bw + cw2){w-r\f~'Q*(z,
w)
+ {w-rjY[Q*(z,w) + Qt(z,w){a + bw + cw2)-n(b + cw)Q*(z,wy\ = 0 .
Dividing by {w — rjf~] and letting w -> rj(z), z e A, we get the desired result that
-r,'(z) + a(z) + b(z)r,(z) + c(z)rj2(z) = 0
in A and then in the whole plane by analytic continuation.
Remark. In general, rj(z) is not a meromorphic function. If rj is a rational
function and if the coefficients in (10) are also rational, Lemma 5 is essentially due to
Wittich [6].
4. Proof of Theorem 1
First, we prove that the equation Q(z, w) = 0 has exactly two distinct solutions. If
there were at least three solutions, then the Riccati equation (10) would have four
distinct solutions
f,rj1,rj2,ri3.
(18)
By a well-known fact (see, for example, [2; p. 45]) the cross ratio of any four distinct
solutions of a Riccati differential equation is constant. This would imply that / is
expressible as a rational function of ^ l 5 r\2 and rj3 which is impossible.
On the other hand, there must be at least two distinct solutions of the equation
Q(z, w) = 0. For, otherwise,
= w"-nrj(z)w"~i + ... + ( - l ) V ( z ) .
which contradicts the hypothesis that an_x{z) = 0, but at least one a^z) ^ 0. Thus,
(19)
where Hi+fa ~n a n ^ A^i*7i +^2^2 — ~an-i = 0 . The last condition gives
r\x = K^ and r\2 = K2rl^ where /q =f= K2 are constants Q U 1 K 1 + \X2K2 = 0) and where
r\(z) is an algebroid function. From (19) it is easily seen that
dj(z) = Cjr}n~j(z), Cj constant
(20)
(cn = 1, cn_1 = 0). At least one coefficient, say ap is not identically zero. Therefore,
T(r, ,7"-;) = S(r,/) and
T(r,r,) = S(r,f),
if rj is a meromorphic function.
(21)
120
E. MUES AND N. STEINMETZ
Case A, when rj is many-valued. Since an_2 — cn-1r\1 ^ 0 (see the proof of Lemma
3), any coefficient an_2j_l must vanish identically. If n were odd, this would imply
that a0 = 0, which contradicts Lemma 2 (b). Indeed, a0 = 0 yields
JV(r, 1//) ^ N(r, \m + S(r,f) = S(r,f).
Thus, n = 2\i is even and we obtain
(22)
on using the transformation / = f2
H— an_2- Here, the a,- are certain meromorphic
functions satisfying T{r,aj) = S(r,/) = S(r,/). Since N ^ r J ) = | N ( r , / ) + S(r,/),
the hypothesis of Lemma 2 cannot hold when / is replaced by / . Therefore, either
pi = 1 or fj, > 1, and the a,- vanish identically. In both cases,
2
where a 0 = - an_2- Next, by Lemma 3 and Lemma 4, we have
which gives (6) by comparing coefficients.
In order to prove the sufficiency of condition (a), we show that *P can only have
zeros at zeros or poles of <x0 or at poles of c. If ao(zo) =£ 0, oo and c{z0) ^= oo, we can
choose a holomorphic branch of ^/a^ in a neighbourhood of z 0 . By the
transformation w = y/oLgV, equation (6) will change into
Since v =/= ±i, the uniqueness theorem for differential equations of the first order
gives v{z0) j= i and, therefore, ^(ZQ) ± 0.
Case B, when rj is one-valued. Consider the meromorphic functions
and
THE THEOREM OF TUMURA-CLUNIE FOR MEROMORPHIC FUNCTIONS
121
By (21) we have T(r,F) = T(r,f) + S{r,f). Obviously,
m(r,X) = S(r,F) = S(r,f).
Since x n a s o n ty simple poles at zeros
vj> = rj"(F -K2Y^f-Klf\
it follows that
of {F — K1)(F — K2)
(24)
and, since
r,
which yields, together with (24), that T(r, x) — S{r,f). Substituting / = rjF in (23), we
get the desired differential equation (8), where c = xM a n d aQ = rj.
If / is a meromorphic solution of (8), then F — f/<x0 satisfies
It follows from
that
= N(r, l/ao) + N(r,cao) ^ N(r, a'0/a0)+ N(r, c)
which proves the sufficiency of condition (b).
5. Proof of Theorem 2 and the Corollary
In order to prove Theorem 2, we substitute
¥ = a n 0 and / = / + "
nan
Obviously, T{rJ) = T(r,f) + S(r,f), S(r,f) = S(r,f), and
N(r, I/O) ^ N(r, l/*) + N(r, an) = S(rJ).
Since
either Theorem 1 is applicable, or we have a0 = ... = an_2 = 0. The latter case leads
to
= an<l> = aj"
= an [ f + ^ nan
122
THE THEOREM OF TUMURA-CLUNIE FOR MEROMORPHIC FUNCTIONS
as stated in (a). In the first case, application of Theorem 1 gives
or
Substituting back we get the desired result.
In order to prove the Corollary, we note that in both cases (b) and (c) of Theorem
2 the assumptions of Lemma 2 are satisfied if / is replaced by /. Thus, if
N(r,f) = N(r,f) + S(r,f) = S(r,f), only case (a) of Theorem 2 can hold. This proves
the Corollary and also Theorem A.
References
1. J. Clunie, "On integral and meromorphic functions", J. London Math. Soc, 37 (1962), 17-27.
2. W. W. Golubew, Vorlesungen uber Differentialgleichungen im Komplexen (Dt. Verlag d. Wiss., Berlin,
1958).
3. W. K. Hayman, Meromorphic functions (Oxford University Press, Oxford, 1964).
4. R. Nevanlinna, Eindeutige analytische Funktionen (Springer, Berlin, 1936).
5. Y. Tumura, "On the extensions of Borel's theorem and Saxer-Csillag's theorem", Proc. Phys.-Math. Soc.
Japan (3), 19 (1937), 29-35.
6. H. Wittich, "Eindeutige Losungen der Differentialgleichung w' = R(z,w)'\ Math. Z., 74 (1960),
278-288.
Universitat Hannover,
Institut fur Mathematik,
Welfengarten 1,
D-3000 Hannover 1,
West Germany.
Universitat Karlsruhe,
Mathematisches Institut I,
KaiserstraBe 12,
D-7500 Karlsruhe 1,
West Germany.