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Meromorphic Functions That Share Two Values
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
209, 542]550 Ž1997.
AY975329
Meromorphic Functions That Share Two Values* Lian-Zhong Yang Department of Mathematics, Shandong Uni¨ ersity, Jinan, Shandong, 250100, People’s Republic of China Submitted by Bruce C. Berndt Received June 12, 1996
In this paper, we give some uniqueness theorems for meromorphic functions that share two values. Particularly, a positive answer to a question posed by Gross is derived. Q 1997 Academic Press
1. INTRODUCTION Let f and g be two nonconstant meromorphic functions in the complex plane. If f and g have the same a-points with the same multiplicities, we say that f and g share the value a CM Žsee w1x.. It is assumed that the reader is familiar with the basic notations and the fundamental results of Nevanlinna’s theory of meromorphic functions, as found in w2x. Nevanlinna proved the following well-known theorem Žsee w3x.. THEOREM A. Let f and g be two nonconstant meromorphic functions. Let a j Ž j s 1, 2, 3, 4. be four distinct shared ¨ alues CM by f and g. Then either f ' g or f is a linear fractional transformation of g. In 1995, Hong-Xun Yi proved the following Žsee w4x. THEOREM B. Let f and g be two nonconstant meromorphic functions such that f and g share 0, 1, and ` CM, and let a Ž/ 0, 1. be a finite ¨ alue. If
ž
N r,
1 fya
/
/ T Ž r , f . q SŽ r , f . ,
* This research was supported by a grant from NSF of Shandong Province and a grant from Shandong University. 542 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
543
MEROMORPHIC FUNCTIONS
then f and g satisfy exactly one of the following relations: Ži. Ž f y a.Ž g q a y 1. ' aŽ1 y a. Žii. f q Ž a y 1. g ' a Žiii. f ' ag. In 1976, Gross proved Žsee w5x. THEOREM C. Let f and g be nonconstant entire functions, and let S1 s 14 , S2 s y14 , and S3 s a, b4 where a and b are arbitrary constants such that Si l S j s B for i / j. Suppose that fy1 Ž Si . s gy1 Ž S i . for i s 1, 2, 3 with the same multiplicities. Then f and g satisfy one of the following relations: Ži. f s g, Žii. fg s 1, or Žiii. Ž f y 1.Ž g y 1. s 4. In w5x, Gross suggested the following open question: QUESTION 1. Can one find two Ž possibly e¨ en one. finite sets S j Ž j s 1, 2. such that any two entire functions f and g satisfying fy1 Ž S j . s gy1 Ž S j . counting multiplicity for j s 1, 2 must be identical? In this paper, we will establish a uniqueness theorem for meromorphic functions that share two values and give a positive answer to Gross’s questions.
2. LEMMAS LEMMA 1. Let f and g be nonconstant meromorphic functions and let a / 0 be a finite complex number. If f n q a and g n q a share the ¨ alues 0, ` CM, then for any integer n G 2 nq1 TŽ r, f . F T Ž r , g . q SŽ r , f . ny1 and nq1 T Ž r, g. F T Ž r , f . q SŽ r , g . . ny1 Proof. By the second fundamental theorem nT Ž r , f . s T Ž r , f n q a . q O Ž 1 . F NŽ r , f . q N r ,
1
q N r,
ž / ž ž / ž
F NŽ r , g . q N r ,
f
1 f
q N r,
1 f qa n
1 g qa n
/ /
qS Ž r , f . q SŽ r , f .
F Ž n q 1. T Ž r , g . q T Ž r , f . q S Ž r , f . .
544
LIAN-ZHONG YANG
In the same manner, we also have nT Ž r , g . F Ž n q 1 . T Ž r , f . q T Ž r , g . q S Ž r , g . . Lemma 1 is proved. LEMMA 2. Let F and G be two meromorphic functions, and let f s F9rF y G9rG. If z1 is a common simple zero of F and G, then
f Ž z1 . s
1
ž
2
F0 Ž z1 . F9 Ž z1 .
y
G0 Ž z1 . G9 Ž z1 .
/
.
Proof. By the Taylor expansion of F and G at z1 , F9 F
s
1 z y z1
q
F0 Ž z1 . 2 F9 Ž z1 .
q O Ž Ž z y z1 . .
and G9 G
1
s
z y z1
q
G0 Ž z1 . 2G9 Ž z1 .
q O Ž Ž z y z1 . . ,
establishing Lemma 2. LEMMA 3. Let f and g be two meromorphic functions, a / 0 a complex number, and n G 4 an integer. If f n q a and g n q a share the ¨ alues 0, ` CM and f n k g n, then NŽ r , f . F
1 ny1
1
1
½ ž / ž /5 N r,
q N r,
f
q SŽ r , f . .
g
Proof. Set
ws
f9 f Ž f q a. n
y
g9 g Ž g q a. n
.
By Lemma 1 and the well-known lemma on the logarithmic derivative, we have mŽ r , w . s S Ž r , f . . Since a zero z1 of f n q a is a simple pole of f 9rŽ f Ž f n q a.. and g 9rŽ g Ž g n q a.. with Res
zsz 1
f9 f Ž f q a. n
s Res
zsz 1
g9 g Ž g q a. n
sy
1 na
,
545
MEROMORPHIC FUNCTIONS
we know that the poles of w only occur at the zeros of f and g. It follows that NŽ r , w . F N r ,
1
q N r,
1
ž / ž / f
g
and T Ž r, w. F N r,
1
q N r,
1
ž / ž / f
g
q SŽ r , f . .
We suppose that w k 0 and notice that ` is a value shared CM by f and g. Since it is easily seen that a pole of order p of f is a zero of w with order at least np y 1, we obtain 1
nN Ž r , f . y N Ž r , f . F N r ,
ž / ž / ž / 1
F N r,
w 1
q N r,
f
g
F T Ž r , w . q O Ž 1. q SŽ r , f . ,
establishing Lemma 3 if w k 0. We claim w k 0. If w ' 0, then n
f9 f
yn
g9
q
g
ng ny 1 g 9
y
gn q a
nf ny 1 f 9 fnqa
' 0,
and so f n Ž g n q a.
' c,
g n Ž f n q a.
where c / 0 is a constant. We rearrange this equation to obtain c f
n
s
1 g
n
1yc
q
a
,
and notice that c / 1 since f n k g n. We conclude that T Ž r , f . s T Ž r , g . q O Ž 1. . By the second fundamental theorem and Ž1., nT Ž r , f . F N Ž r , f . q N r ,
1
ž / f
F 3T Ž r , f . q S Ž r , f . .
q NŽ r , g . q SŽ r , f .
Ž 1.
546
LIAN-ZHONG YANG
This is incompatible with n G 4, and we conclude w k 0. Lemma 3 is proved.
3. THEOREMS AND THEIR PROOF THEOREM 1. Let f and g be two meromorphic functions such that f n q a and g n q a share 0, ` CM where a / 0 is a finite complex number and n ) 5 is an integer. Then f n s g n or f n g n s a2 , and so f s cg or fg s d for some constant c and d satisfying c n s 1 and d n s a2 . Proof. Since f n q a and g n q a share the value 0 and ` CM, we have fnqa gn q a
s eh,
where h is an entire function. Set
d s 2 h9 q
ž
g0 g9
y
f0 f9
/
q Ž n y 1.
ž
g9 g
y
f9 f
/
.
Then from Lemma 1 and the well-known lemma of logarithmic derivative, we have mŽ r , d . s S Ž r , f . and NŽ r , d . F N r ,
1
q N r,
1
q N0 r ,
1
q N0 r ,
1
ž / ž / ž / ž / f
g
f9
g9
,
where N0 Ž r, 1rf 9. and N0 Ž r, 1rg9. denote the counting functions of zeros of f 9 and g 9 which are not the zeros of f Ž f n q a. and g Ž g n q a., respectively. We get T Ž r, d . F N r,
1
q N r,
1
q N0 r ,
1
q N0 r ,
1
ž / ž / ž / ž / f
g
f9
g9
q SŽ r , f . .
Ž 2.
547
MEROMORPHIC FUNCTIONS
We claim d ' 0. Suppose the contrary and consider a simple zero z1 of f n q a. By Lemma 2, h9 Ž z1 . s s
½
Ž f n q a. 9 fnqa
1
½
2
f9
Ž n y 1.
Ž g n q a. 9
y
f0
q
f
5
gn q a
5
f9
zsz 1
y zsz 1
1 2
½
Ž n y 1.
g9 g
q
g0 g9
5
, zsz 1
implying d Ž z1 . s 0. Thus
ž
N 1. r ,
1 f qa n
1
F N r,
/ ž / d
F T Ž r , d . q SŽ r , f . ,
Ž 3.
where N 1. Ž r, 1rŽ f n q a.. is the counting function of the simple zeros of f n q a. By the second fundamental theorem, we have nT Ž r , f . q nT Ž r , g . s T Ž r , f n q a. q T Ž r , g n q a. q O Ž 1. 1
F NŽ r , f . q N r ,
ž
f qa 1
q NŽ r , g . q N r ,
ž
y N0 r ,
1
ž / g9
f
q N r,
1
1
q SŽ r , f .
f9
1 g
q N r,
1
q N r,
1
/ ž / ž /
f qa n
y N0 r ,
1
f
g
q SŽ r , f . q SŽ r , g . .
ž / ž / f9
y N0 r ,
q SŽ r , g .
ž
1
1
/ ž / ž / / ž /
g qa n
s 2 NŽ r , f . q 2 N r , y N0 r ,
q N r,
n
g9
Notice that
ž
2N r,
1 f qa n
/
F N 1. r ,
ž
1 f qa n
q N r,
/ ž
1 f qa n
/
.
We have from Ž2., Ž3., and Lemma 1 that n T Ž r , f . q T Ž r , g . 4 F 2 q
ž
2 ny1
q N r,
ž
N r,
1 g qa n
1
1
/½ ž / ž /5
/
f
q N r,
q SŽ r , f . ,
g
548
LIAN-ZHONG YANG
and thus nT Ž r , f . F 2 q
ž
2
1
1
/½ ž / ž /5 N r,
ny1
q N r,
f
g
q SŽ r , f .
and nT Ž r , g . F 2 q
ž
2 ny1
1
1
/½ ž / ž /5 N r,
f
q N r,
g
q SŽ r , f . .
Adding, we obtain 4
n T Ž r , f . q T Ž r , g . 4 F 4 q
ž
ny1
/Ž
T Ž r , f . q T Ž r , g . . q SŽ r , f . .
Since n ) 5, this is a contradiction, establishing d ' 0. Thus g ny 1 g 9 f ny 1 f 9
s cey2 h ,
and hence g ny 1 g 9
Ž g n q a.
2
sc
f ny 1 f 9
Ž f n q a.
2
and 1 g qa n
sc
1 f qa n
q c1 ,
where c and c1 are constants. Consequently, T Ž r , f . s T Ž r , g . q SŽ r , f . . If c1 s 0, then c / 0 and f n q a s cg n q ca Ž i.e., f n s cg n q aŽ c y 1... If c / 1, by the second fundamental theorem nT Ž r , f . F N Ž r , f . q N r ,
1
q N r,
ž / ž ž / ž /
s NŽ r , f . q N r ,
f
1 f
q N r,
1
f y aŽ c y 1. n
1
gn
/
q SŽ r , f .
q SŽ r , f .
F 2T Ž r , f . q T Ž r , g . q S Ž r , f . . In view of Lemma 1, this is a contradiction for n ) 5; hence c s 1 and we get f n ' g n.
MEROMORPHIC FUNCTIONS
549
If c1 / 0, then c1 f n s e h y c y c1 a. By the second fundamental theorem applied as above, we have c q c1 a s 0. Also g n s 1rc1 y a q aeyh . By the same reasoning, 1rc1 s a, and we get f n g n s a2 . Theorem 1 is thus proved. COROLLARY. Let f and g be two nonconstant entire functions, a / 0 be a finite complex number, and S s v N v n q a s 04 be a set of nŽ) 4. elements. If fy1 Ž S . s gy1 Ž S . with the same multiplicities, then f n s g n or f n g n s a2 , and so f ' cg or fg s d for some constants c and d. Proof. Since N Ž r, f . s N Ž r, g . s 0, the corollary follows from the proof of Theorem 1. From the above results, we immediately obtain the following theorem which answers Question 1 posed by Gross. THEOREM 2. Let f and g be two nonconstant entire functions, n ) 4 be an integer, and a, bŽ ab / 0, a2 nq2 / b 2 n . be finite complex numbers. Set S1 s v N v n q a s 0 4
S2 s v N v nq 1 q b s 0 4 .
If fy1 Ž Si . s gy1 Ž S i . for i s 1, 2 with the same multiplicities, then f ' g. Proof. By fy1 Ž S1 . s gy1 Ž S1 . and the corollary of Theorem 1, we have f n s gn
or
f n g n s a2 .
Ž 4.
f nq 1 g nq 1 s b 2 .
Ž 5.
Similarly by fy1 Ž S2 . s gy1 Ž s2 ., we get f nq 1 s g nq 1
or
From Ž4. and Ž5., we discuss the following four cases. Ži. If f n s g n and f nq1 s g nq1, it is easily seen that f ' g. Žii. The equations f n s g n and f nq1 g nq1 s b 2 clearly cannot hold simultaneously for any sequence z n such that < f Ž z n .< ª `. Žiii. The equations f n g n s a2 and f nq1 s g nq1 clearly cannot hold simultaneously for any sequence z n such that < f Ž z n .< ª `. Živ. If f n g n s a2 and f nq1 g nq1 s b 2 , we have a2 nq2 s b 2 n which contradicts the condition of Theorem 2. Combining Ži. ] Živ., Theorem 2 is proved.
REFERENCES 1. G. G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc. 20 Ž1979., 457]466.
550
LIAN-ZHONG YANG
2. W. K. Hayman, ‘‘Meromorphic Functions,’’ Clarendon, Oxford, 1964. 3. R. Nevanlinna, ‘‘Le Theorems de Picard-Borel et la Theorie des Fonctions Meromorphes,’’ Gauthier]Villars, Paris, 1929. 4. Hong-Xun Yi, Unicity theorem for meromorphic functions that share three values, Kodai Math. J. 18, No. 2Ž1995., 300]314. 5. F. Gross, Factorizations of meromorphic function and some open problems, in ‘‘Complex analysis,’’ pp. 51]69, Lecture Notes in math., Vol. 599, Springer-Verlag, Berlin, 1977.
209, 542]550 Ž1997.
AY975329
Meromorphic Functions That Share Two Values* Lian-Zhong Yang Department of Mathematics, Shandong Uni¨ ersity, Jinan, Shandong, 250100, People’s Republic of China Submitted by Bruce C. Berndt Received June 12, 1996
In this paper, we give some uniqueness theorems for meromorphic functions that share two values. Particularly, a positive answer to a question posed by Gross is derived. Q 1997 Academic Press
1. INTRODUCTION Let f and g be two nonconstant meromorphic functions in the complex plane. If f and g have the same a-points with the same multiplicities, we say that f and g share the value a CM Žsee w1x.. It is assumed that the reader is familiar with the basic notations and the fundamental results of Nevanlinna’s theory of meromorphic functions, as found in w2x. Nevanlinna proved the following well-known theorem Žsee w3x.. THEOREM A. Let f and g be two nonconstant meromorphic functions. Let a j Ž j s 1, 2, 3, 4. be four distinct shared ¨ alues CM by f and g. Then either f ' g or f is a linear fractional transformation of g. In 1995, Hong-Xun Yi proved the following Žsee w4x. THEOREM B. Let f and g be two nonconstant meromorphic functions such that f and g share 0, 1, and ` CM, and let a Ž/ 0, 1. be a finite ¨ alue. If
ž
N r,
1 fya
/
/ T Ž r , f . q SŽ r , f . ,
* This research was supported by a grant from NSF of Shandong Province and a grant from Shandong University. 542 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
543
MEROMORPHIC FUNCTIONS
then f and g satisfy exactly one of the following relations: Ži. Ž f y a.Ž g q a y 1. ' aŽ1 y a. Žii. f q Ž a y 1. g ' a Žiii. f ' ag. In 1976, Gross proved Žsee w5x. THEOREM C. Let f and g be nonconstant entire functions, and let S1 s 14 , S2 s y14 , and S3 s a, b4 where a and b are arbitrary constants such that Si l S j s B for i / j. Suppose that fy1 Ž Si . s gy1 Ž S i . for i s 1, 2, 3 with the same multiplicities. Then f and g satisfy one of the following relations: Ži. f s g, Žii. fg s 1, or Žiii. Ž f y 1.Ž g y 1. s 4. In w5x, Gross suggested the following open question: QUESTION 1. Can one find two Ž possibly e¨ en one. finite sets S j Ž j s 1, 2. such that any two entire functions f and g satisfying fy1 Ž S j . s gy1 Ž S j . counting multiplicity for j s 1, 2 must be identical? In this paper, we will establish a uniqueness theorem for meromorphic functions that share two values and give a positive answer to Gross’s questions.
2. LEMMAS LEMMA 1. Let f and g be nonconstant meromorphic functions and let a / 0 be a finite complex number. If f n q a and g n q a share the ¨ alues 0, ` CM, then for any integer n G 2 nq1 TŽ r, f . F T Ž r , g . q SŽ r , f . ny1 and nq1 T Ž r, g. F T Ž r , f . q SŽ r , g . . ny1 Proof. By the second fundamental theorem nT Ž r , f . s T Ž r , f n q a . q O Ž 1 . F NŽ r , f . q N r ,
1
q N r,
ž / ž ž / ž
F NŽ r , g . q N r ,
f
1 f
q N r,
1 f qa n
1 g qa n
/ /
qS Ž r , f . q SŽ r , f .
F Ž n q 1. T Ž r , g . q T Ž r , f . q S Ž r , f . .
544
LIAN-ZHONG YANG
In the same manner, we also have nT Ž r , g . F Ž n q 1 . T Ž r , f . q T Ž r , g . q S Ž r , g . . Lemma 1 is proved. LEMMA 2. Let F and G be two meromorphic functions, and let f s F9rF y G9rG. If z1 is a common simple zero of F and G, then
f Ž z1 . s
1
ž
2
F0 Ž z1 . F9 Ž z1 .
y
G0 Ž z1 . G9 Ž z1 .
/
.
Proof. By the Taylor expansion of F and G at z1 , F9 F
s
1 z y z1
q
F0 Ž z1 . 2 F9 Ž z1 .
q O Ž Ž z y z1 . .
and G9 G
1
s
z y z1
q
G0 Ž z1 . 2G9 Ž z1 .
q O Ž Ž z y z1 . . ,
establishing Lemma 2. LEMMA 3. Let f and g be two meromorphic functions, a / 0 a complex number, and n G 4 an integer. If f n q a and g n q a share the ¨ alues 0, ` CM and f n k g n, then NŽ r , f . F
1 ny1
1
1
½ ž / ž /5 N r,
q N r,
f
q SŽ r , f . .
g
Proof. Set
ws
f9 f Ž f q a. n
y
g9 g Ž g q a. n
.
By Lemma 1 and the well-known lemma on the logarithmic derivative, we have mŽ r , w . s S Ž r , f . . Since a zero z1 of f n q a is a simple pole of f 9rŽ f Ž f n q a.. and g 9rŽ g Ž g n q a.. with Res
zsz 1
f9 f Ž f q a. n
s Res
zsz 1
g9 g Ž g q a. n
sy
1 na
,
545
MEROMORPHIC FUNCTIONS
we know that the poles of w only occur at the zeros of f and g. It follows that NŽ r , w . F N r ,
1
q N r,
1
ž / ž / f
g
and T Ž r, w. F N r,
1
q N r,
1
ž / ž / f
g
q SŽ r , f . .
We suppose that w k 0 and notice that ` is a value shared CM by f and g. Since it is easily seen that a pole of order p of f is a zero of w with order at least np y 1, we obtain 1
nN Ž r , f . y N Ž r , f . F N r ,
ž / ž / ž / 1
F N r,
w 1
q N r,
f
g
F T Ž r , w . q O Ž 1. q SŽ r , f . ,
establishing Lemma 3 if w k 0. We claim w k 0. If w ' 0, then n
f9 f
yn
g9
q
g
ng ny 1 g 9
y
gn q a
nf ny 1 f 9 fnqa
' 0,
and so f n Ž g n q a.
' c,
g n Ž f n q a.
where c / 0 is a constant. We rearrange this equation to obtain c f
n
s
1 g
n
1yc
q
a
,
and notice that c / 1 since f n k g n. We conclude that T Ž r , f . s T Ž r , g . q O Ž 1. . By the second fundamental theorem and Ž1., nT Ž r , f . F N Ž r , f . q N r ,
1
ž / f
F 3T Ž r , f . q S Ž r , f . .
q NŽ r , g . q SŽ r , f .
Ž 1.
546
LIAN-ZHONG YANG
This is incompatible with n G 4, and we conclude w k 0. Lemma 3 is proved.
3. THEOREMS AND THEIR PROOF THEOREM 1. Let f and g be two meromorphic functions such that f n q a and g n q a share 0, ` CM where a / 0 is a finite complex number and n ) 5 is an integer. Then f n s g n or f n g n s a2 , and so f s cg or fg s d for some constant c and d satisfying c n s 1 and d n s a2 . Proof. Since f n q a and g n q a share the value 0 and ` CM, we have fnqa gn q a
s eh,
where h is an entire function. Set
d s 2 h9 q
ž
g0 g9
y
f0 f9
/
q Ž n y 1.
ž
g9 g
y
f9 f
/
.
Then from Lemma 1 and the well-known lemma of logarithmic derivative, we have mŽ r , d . s S Ž r , f . and NŽ r , d . F N r ,
1
q N r,
1
q N0 r ,
1
q N0 r ,
1
ž / ž / ž / ž / f
g
f9
g9
,
where N0 Ž r, 1rf 9. and N0 Ž r, 1rg9. denote the counting functions of zeros of f 9 and g 9 which are not the zeros of f Ž f n q a. and g Ž g n q a., respectively. We get T Ž r, d . F N r,
1
q N r,
1
q N0 r ,
1
q N0 r ,
1
ž / ž / ž / ž / f
g
f9
g9
q SŽ r , f . .
Ž 2.
547
MEROMORPHIC FUNCTIONS
We claim d ' 0. Suppose the contrary and consider a simple zero z1 of f n q a. By Lemma 2, h9 Ž z1 . s s
½
Ž f n q a. 9 fnqa
1
½
2
f9
Ž n y 1.
Ž g n q a. 9
y
f0
q
f
5
gn q a
5
f9
zsz 1
y zsz 1
1 2
½
Ž n y 1.
g9 g
q
g0 g9
5
, zsz 1
implying d Ž z1 . s 0. Thus
ž
N 1. r ,
1 f qa n
1
F N r,
/ ž / d
F T Ž r , d . q SŽ r , f . ,
Ž 3.
where N 1. Ž r, 1rŽ f n q a.. is the counting function of the simple zeros of f n q a. By the second fundamental theorem, we have nT Ž r , f . q nT Ž r , g . s T Ž r , f n q a. q T Ž r , g n q a. q O Ž 1. 1
F NŽ r , f . q N r ,
ž
f qa 1
q NŽ r , g . q N r ,
ž
y N0 r ,
1
ž / g9
f
q N r,
1
1
q SŽ r , f .
f9
1 g
q N r,
1
q N r,
1
/ ž / ž /
f qa n
y N0 r ,
1
f
g
q SŽ r , f . q SŽ r , g . .
ž / ž / f9
y N0 r ,
q SŽ r , g .
ž
1
1
/ ž / ž / / ž /
g qa n
s 2 NŽ r , f . q 2 N r , y N0 r ,
q N r,
n
g9
Notice that
ž
2N r,
1 f qa n
/
F N 1. r ,
ž
1 f qa n
q N r,
/ ž
1 f qa n
/
.
We have from Ž2., Ž3., and Lemma 1 that n T Ž r , f . q T Ž r , g . 4 F 2 q
ž
2 ny1
q N r,
ž
N r,
1 g qa n
1
1
/½ ž / ž /5
/
f
q N r,
q SŽ r , f . ,
g
548
LIAN-ZHONG YANG
and thus nT Ž r , f . F 2 q
ž
2
1
1
/½ ž / ž /5 N r,
ny1
q N r,
f
g
q SŽ r , f .
and nT Ž r , g . F 2 q
ž
2 ny1
1
1
/½ ž / ž /5 N r,
f
q N r,
g
q SŽ r , f . .
Adding, we obtain 4
n T Ž r , f . q T Ž r , g . 4 F 4 q
ž
ny1
/Ž
T Ž r , f . q T Ž r , g . . q SŽ r , f . .
Since n ) 5, this is a contradiction, establishing d ' 0. Thus g ny 1 g 9 f ny 1 f 9
s cey2 h ,
and hence g ny 1 g 9
Ž g n q a.
2
sc
f ny 1 f 9
Ž f n q a.
2
and 1 g qa n
sc
1 f qa n
q c1 ,
where c and c1 are constants. Consequently, T Ž r , f . s T Ž r , g . q SŽ r , f . . If c1 s 0, then c / 0 and f n q a s cg n q ca Ž i.e., f n s cg n q aŽ c y 1... If c / 1, by the second fundamental theorem nT Ž r , f . F N Ž r , f . q N r ,
1
q N r,
ž / ž ž / ž /
s NŽ r , f . q N r ,
f
1 f
q N r,
1
f y aŽ c y 1. n
1
gn
/
q SŽ r , f .
q SŽ r , f .
F 2T Ž r , f . q T Ž r , g . q S Ž r , f . . In view of Lemma 1, this is a contradiction for n ) 5; hence c s 1 and we get f n ' g n.
MEROMORPHIC FUNCTIONS
549
If c1 / 0, then c1 f n s e h y c y c1 a. By the second fundamental theorem applied as above, we have c q c1 a s 0. Also g n s 1rc1 y a q aeyh . By the same reasoning, 1rc1 s a, and we get f n g n s a2 . Theorem 1 is thus proved. COROLLARY. Let f and g be two nonconstant entire functions, a / 0 be a finite complex number, and S s v N v n q a s 04 be a set of nŽ) 4. elements. If fy1 Ž S . s gy1 Ž S . with the same multiplicities, then f n s g n or f n g n s a2 , and so f ' cg or fg s d for some constants c and d. Proof. Since N Ž r, f . s N Ž r, g . s 0, the corollary follows from the proof of Theorem 1. From the above results, we immediately obtain the following theorem which answers Question 1 posed by Gross. THEOREM 2. Let f and g be two nonconstant entire functions, n ) 4 be an integer, and a, bŽ ab / 0, a2 nq2 / b 2 n . be finite complex numbers. Set S1 s v N v n q a s 0 4
S2 s v N v nq 1 q b s 0 4 .
If fy1 Ž Si . s gy1 Ž S i . for i s 1, 2 with the same multiplicities, then f ' g. Proof. By fy1 Ž S1 . s gy1 Ž S1 . and the corollary of Theorem 1, we have f n s gn
or
f n g n s a2 .
Ž 4.
f nq 1 g nq 1 s b 2 .
Ž 5.
Similarly by fy1 Ž S2 . s gy1 Ž s2 ., we get f nq 1 s g nq 1
or
From Ž4. and Ž5., we discuss the following four cases. Ži. If f n s g n and f nq1 s g nq1, it is easily seen that f ' g. Žii. The equations f n s g n and f nq1 g nq1 s b 2 clearly cannot hold simultaneously for any sequence z n such that < f Ž z n .< ª `. Žiii. The equations f n g n s a2 and f nq1 s g nq1 clearly cannot hold simultaneously for any sequence z n such that < f Ž z n .< ª `. Živ. If f n g n s a2 and f nq1 g nq1 s b 2 , we have a2 nq2 s b 2 n which contradicts the condition of Theorem 2. Combining Ži. ] Živ., Theorem 2 is proved.
REFERENCES 1. G. G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc. 20 Ž1979., 457]466.
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2. W. K. Hayman, ‘‘Meromorphic Functions,’’ Clarendon, Oxford, 1964. 3. R. Nevanlinna, ‘‘Le Theorems de Picard-Borel et la Theorie des Fonctions Meromorphes,’’ Gauthier]Villars, Paris, 1929. 4. Hong-Xun Yi, Unicity theorem for meromorphic functions that share three values, Kodai Math. J. 18, No. 2Ž1995., 300]314. 5. F. Gross, Factorizations of meromorphic function and some open problems, in ‘‘Complex analysis,’’ pp. 51]69, Lecture Notes in math., Vol. 599, Springer-Verlag, Berlin, 1977.