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.
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]
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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
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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).
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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,/) 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 = 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/
