- Email: [email protected]
MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES
2004,24B(4):529-535
MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES 1 Huang Bin ( jfiA; ) Du Jinyuan ( #.~ Jt ) The College of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected] Abstract The uniqueness of meromorphic functions that share four values is investigated. A necessary condition to the case is acquired, and some partial results for question "lCM+3IM=4CM" are obtained. Key words
Meoromorphic functions, shared-value
2000 MR Subject Classification
1
30D35
Introduction
Throughout the paper, we use standard notations and fundamental results of Nevanlinna's theory (see W.K.Hayman [1]). Let j(z) be a meromorphic function in the complex plane, we denote by S(r, I) any quantity satisfying S(r, I) = o(T(r, I)) for r -t 00 except possibly a set of r of finite linear measure. We say that two nonconstant meromorphic functions j and 9 share the value c(c = 00 is allowed) provided that j(z) = c if and only if g(z) = c. Usually, we will state whether a shared value is by CM (counting multiplicities) or 1M (ignoring multiplicities).We denote by N E(r, j = c = g) or N E(r, c) the counting function of those c-points where j(z) and g(z) have same multiplicity (counting each point only once), while by N D(r, j = c = g) or N D (r, c) the counting function of those c-points where j and 9 have different multiplicities (counting each point only once).. R.Nevanlinna (see [3]) gave the following theorem: Theorem A If j and 9 are distinct nonconstant merom orphic functions that share four distinct values al,a2,a3,a4 CM,then j is a Mobius transformation of g;two of the values ,say, al and a2, are Picard values,and the cross ratio (al' a2, a3, a4) = -l. In 1976 L.Rubel asked such a question: whether CM can be replaced by 1M in Theorem A with the same conclusion or not? A counterexample given by G.G.Gundersen (see ref.[4]) showed that the answer to this question is negative, i.e., generally "41Mi:4CM". Meanwhile, on the other hand, he proved that "3CM + lIM=4CM". Furthermore, he showed by the following theorem (See [5]) that "2CM+21M=4CM" is also valid. Theorem B If two nonconstant meromorphic functions share two values 1M, and share two other values CM, then j and 9 share all four values CM. [Received January 21, 2002; revised May 10, 2003. Project supported by the National Natural Science Foundation of China(19971052) and the Programme of Hunan Education Foundation(02C095)
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Theorem B can be generalized slightly to the follows: Theorem B*[6] If two nonconstant meromorphic functions share two values 1M, and share two other values CM*(where the terminology "two nonconstant meromorphic functions share the value a CM*" means a is shared by f and 9 and, furthermore, N(r, a) = N E(r, a) + S(r, I)), then f and 9 share all four values CM. Let f and 9 be nonconstant meromorphic functions sharing the value a 1M. Define · . f N- E(r,a) , 1Imm
r(a) =
r-too
{ 1,
N(r,a)
if N(r, a)
¥ 0,
if N(r, a) == 0.
For the open question "whether 1CM+3IM=4CM?" i.e., "If two nonconstant meromorphic functions share three values 1M and share a fourth value Clvl.then do the functions necessarily share all four values CM ?", G.G.Gundersen (see [6]) proved the following partial result. Theorem C Let f and 9 be nonconstant meromorphic functions that share al, a2, a3 1M and a4 CM.Suppose that there exist some real constant A > 4/5 and some set t c: (0,00) that has infinite linear measure such that
N(r, a4, I) > A T(r, I) -
(1.1)
for all rEI. Then f and 9 share all four values CM. In this paper, a necessary condition for two meromorphic functions sharing four values is discovered as follows: Theorem 1 Let f and 9 be nonconstant meromorphic functions that share four distinct values al, a2, a3, a4 1M.Then either the functions share all four values CM or else for every i E {I, 2, 3, 4}, the relation
N E(r, ai)
~
2ND(r, ai)
+ 2ND(r, ak) + S(r, I)
holds for k E {1,2,3,4}\{i}. Moreover, by Theorem 1 we obtain the following results Theorem 2 Let f and 9 be nonconstant meromorphic functions that share al, a2, a3 1M and a4 CM. If N(r, al, I) + N(r, a2, I) ~ {IT(r, I) + S(r, I) holds for some {l < 2/3 ,then f and 9 share all four values CM. Furthermore, by Theorem 1, an improvement of Theorem C is obtained below. Theorem 3 Let f and 9 be nonconstant meromorphic functions that share three distinct values al, a2, a3 1M and a fourth value a4 CM. Suppose that there exist some real constant
A > 2~r' where r = I
c (0,00)
3
L:
j=l
l-r\a)(r
= 00 if r(aj) = 1 for some j E {1,2,3}), and some set
J
which has infinite linear measure such that
N(r, a4, I) > A T(r, I) -for all rEI. Then f and 9 share all four values CM. Remark 1 It is obvious that 2~r ~ so Theorem 3 improves Theorem C.
t,
(1.2)
Huang & Du: MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES
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531
Remark 2 Theorem 3 improves Theorem B while Theorem C does not. In fact, suppose two meromorphic functions share two values al,a2 CM,and share other two values a3,a4 1M, then by Theorem 3,we can deduce that either all four values are shared CM, or else N(r, ad = 0,N(r,a2) = 0, from which it follows N(r,a3) = NE(r,a3) + S(r,/),N(r,a4) = NE(r,a4) + S(r, I) by Nevanlinna's second fundamental theorem and 4IM theorem (see Lemma 1 in the next section).Thus f and 9 share all four values CM.
2
Lemmas
Some lemmas below are needed in the proofs of the theorems. Lemma 1[2,4,5,8] Let f and 9 be distinct nonconstant meromorphic functions that share four values al, a2,a3,a4 1M. Then the following statements hold: (i) T(r, I) = T(r, g) + S(r, I), T(r, g) = T(r, I) + S(r, g); 4
E N(r, f!a.) = 2T(r, I) + S(r, I); j=l (iii) No(r"j;) = S(r, I), No(r, -]r) = S(r,g), where No(r, f;) and No(r, -]r) are respectively counting functions of I' only to those points such that f(z) t ai and g(z) t ai for i = 1,2,3,4. 4 (iv) E N*(r,aj) = S(r,l), (ii)
J
°and g' = 0 refer
=
j=l
where N*(r,aj) is the counting function for common multiple zeros of f(z) - aj and g(z) - aj, counting the smaller one of the two multiplicities at each of the points. Lemma 2(see [4][5][6] or see p.247 in ref.[8]) Let f be a nonconstant meromorphic function and let bi, bz, ... ,bq be q constants. Then for any polynomial PU) of degree p(p < q) in f with constant coefficients, the equality
PU)f' m(r U - bl)U - b2) ... U _ bq ) )
= S(r, I)
holds. Lemma 3[4,5,6] Let f and 9 be distinct nonconstant meromorphic functions that share four values al, a2,a3, 00 1M. Then the function
1jJ(z)
/,g'U - g)2
= -:-U.,----al...,..).,--U---a-2.,--)(""';f:"'--:::"a":':3"""H-g:;";--a--:-l)"""(g---a2-'--H"--g---a--:-3)
is an entire function and satisfies
T(r,1jJ(z)) = S(r, I). Lemma 4 Let f and 9 be nonconstant merom orphic functions that share four distinct values al , a2,a3,a4 1M. Then either the functions share all four values CM or else the inequality
holds for distinct i,j,k,m E {l,2,3,4}. Proof Assume f -:t 9 and, without loss of generality, a4
= 00.
For i
= 1,2,3, put (2.1)
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where j::j; k,{j,k} E {1,2,3} \ {i}. If 1]i == 0, then aI, a2, a3, a4 are shared CM by I and g. Therefore we assume From (2.1), Lemma 1(i) and Lemma 2, for distinct i,j,k,E {1,2,3} we have
N(r, ai) ~ N(r, ~
1]11]21]3 ~
i) ~ T(r, 1]i) + 0(1) ~ N(r,1]d + S(r, I) + S(r, g)
N oir, aj)
+ N D(r, ak) + N D(r, I) + S(r, I).
0.
(2.2)
= _,1 ,G = _1_. Then F and 9 share b1,b3,b3,b4 1M, where b1 = 00,b2 = -1-,b 3= -at g-at a2-at _1_, b4 = 0. Consider the following auxiliary function aa-at
Let F
(2.3)
If 1]4 == 0, then bi, bz, b3, b4 are shared CM by F and G. Thus aI, a2, a3, a4 are shared CM by I and g. Now we suppose 1]4 ~ 0. Since T(r, F) = T(r, I) + 0(1), T(r, G) = T(r, g) + 0(1), from (2.3), Lemma 1(i) and Lemma 2 we deduce that N(r, a4, I)
= -N(r, i; F)
~
1 N(r, -) 1]4
~
T(r, 1]4)
+ 0(1)
(2.4)
By (2.2) and (2.4), we complete the proof of Lemma 4.
3
Proof of Theorem 1
Now we come to prove Theorem 1. The proof below has some similarities to the proof of Theorem C as in [6]. If I == g, then there is nothing to prove. So we assume I ~ g. Picking an integer i E {1,2,3,4}, say, i = 4, we shall estimate N E(r,a4) by considering two cases. Case 1. a4 = 00. Put
1/;(z)
I"
1]
= (f l'
= f - 1- a1
-
/,g'(f - g)2 )( ( , ad(f - a2)(f - a3 9 - ad 9 - a2)(g - a3)
l' /' s" 1- a2 - 1- a3 - (9' Hj
g' = -I--I' aj - -, 9 - aj
-
g'
g'
(3.1) g'
9 - a1 - 9 - a2 - 9 - a3)'
. J = 1,2,3.
(3.2) (3.3)
By Lemma 3 we know 1/; is an entire function and satisfies
T(r,1/;) = S(r, I).
(3.4)
By considering residues in (3.2), we deduce that 1] is analytic at any a-point (a E {a1,a2,a3}) as well as such a pole at which I(z) and g(z) have the same multiplicity. And it is obvious that 1] has a simple pole when I = a4 and 9 = a4 with different multiplicities. Thus from (3.3), (3.2), Lemma 1 and the fundamental estimate of the logarithmic derivative we have
T(r,3Hj
+ 1]) = N D(r, aj) + N D(r, a4) + S(r, I), j = 1,2,3.
(3.5)
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Huang & Du: MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES
Let Zl be a common simple pole of
533
f and g.Assume that
An elementary calculation gives that
= -bbl - -ClCo
Hl(zd 1/(zd
o
~ = -3(b
o
~
-) Co
'I/J(zd
+ (al
= (b~
1
1
-al(- - -), bo Co 1
(3.6) 1
+a2 +a3)(- - -), bo Co -
~ )2.
(3.7) (3.8)
From (3.6),(3.7) and (3.8) we obtain
(3H l(zd + 1/(Zl))2
= (2al -
a2 - a3)2'I/J(zl).
(3.9)
If (3H l +1/)2 == (2al -a2 -a3)2'I/J, then 3Hl +1/ has no poles since 'I/J is an entire function. Thus N(r,3H l + 1/) = N D(r, al) + N D(r, a4) + S(r, f) = 0, which implies al and a4(= 00) must be shared CM by f and g. Thus f and g share all four values CM by Theorem C. Now we suppose (3H l + 1/)2 ~ (2al - a2 - a3)2'I/J. Then from Lemma l(iv), (3.9) and (3.4), we can deduce that (3.10)
Similarly, considering H 2 and H 3 , we can obtain that either or else
f and
g share all four values CM
(3.11)
and (3.12)
hold. It is shown from (3.10),(3.11) and (3.12) that in the case a4 Theorem 1 is valid. Case 2. a4 =I 00.
= 00 the
conclusion of
Set F = '!.a4,G = g!.a4. Then F and G share bl,b2,b3,b4 1M, where bj = aj~a4,j = 1,2,3;b4 = 00. Since T(r,F) = T(r,f) + O(l),NE(r,b j) = NE(r,aj), and ND(r,b j) = ND(r,aj),j = 1,2,3,4, treating NE(r,b 4) in the same way as in Case 1, we still obtain that either f and g share all four values CM or else (3.10), (3.11) and (3.12) hold. Thus Theorem 1 is proved.
4
The Proof of Theorem 2
Assume that each of the values al,a2,a3 is not shared CM by f(z) and g(z), then the following inequalities hold by Lemma 4 and Theorem 2:
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+ S(r, f),
N(r, a4) ~ 2N D(r, al)
N(r, a4) ~ 2N D(r, a2) + S(r, f).
It follows that 2T(r,f)
=
4
LN(r,aj) j=1
+ S(r,f)
< N(r, ad + N(r, a2) + 2(Noir, ad + N D(r, a2)) + S(r, f) < 3(N(r, ad + N(r, a2)) + S(r, f). This is a contradiction since N(r, al) is thus proved.
5
+ N(r, a2) ~ f..lT(r, f) + S(r, f),
and f..l < 2/3. Theorem 2
The Proof of Theorem 3
Assume that each of the values aI, a2, a3, is not shared CM by f (z) and g( z). Notice that a4 is shared CM by f(z) and g(z), by Theorem 2 we have N(r, a4) ~ 2N D(r, ai), If r(ai)
= 1,2,3.
i
(5.1)
> O(i = 1,2,3), then we take 0 < f..li < r(ai)' The inequality N D(r, ai) ~ (1 - f..li)N(r, ai)
(5.2)
(k = 1,2,3)
holds for sufficiently large r. If r(ai) = 0, then we take f..li = O. The inequality (5.2) still holds. So from (5.1) and (5.2), it follows that N(r, a4) ~ 2(1 - f..li)N(r, ai)
(k = 1,2,3).
(5.3)
Hence, {(1 - f..ll)(l - f..l2) where A
= (1 -
+ (1 -
f..l2)(1 - f..l3)
+ (1 -
3
f..l3)(1 - f..ll)}N(r, a4) ~ 2A L N(r, aj), j=1
f..ll)(l - f..l2)(1 - f..l3)' That is, 4
{(1- f..ld(l- f..l2) + (1- f..l2)(1- f..l3) + (1- f..l3)(1- f..ld + 2A}N(r,a4) ~ 2A LN(r,aj). j=1 From this and Lemma l(ii), we derive that 4
. N(r,a4) lim sup T( f) ~ r-oc> r,
3
2+
rrtE
1
L ~ j=1 1-',
where E is a set of r of finite linear measure. This leads to . N(r, a4) hmsup T( T, f) ~ r_oc rrtE
4 3
2
+L
)=1
1 I-T(aj)
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535
which contradicts the condition (1.2). Thus at least one of al,aZ,a3 must be shared CM by f(z) and g(z). As a4 is also shared CM, aI, az, a3 are all shared CM by f(z) and g(z) according to Theorem B. This completes the proof of Theorem 3. References 1 Hayman W K. Meromorphic Function. Oxford: Clarendon Press, 1964 2 Nevanlinna R. Einige Eindeutigkeitssdtze in der Theorie der meromorphen Funktionen. Acta Math, 1926, 48: 367-391 3 Nevanlinna R. Le thereme de Picard-Borel et la theorie des fonctions meromorphes. Paris: GauthierVillars, 1929 4 Gundersen G G. Meromorphic functions that share three or four values. J London Math Soc, 1979, 20(2): 457-466 5 Gundersen G G. Meromorphic functions that share four values. Trans Am Math Soc, 1983, 277: 545-567. Correction: 1987, 304: 847-850 6 Gundersen G G. Meromorpic functions that share three values 1M and a fourth value CM. Complex Variables, 1992, 20: 99-106 7 Mues E. Meromorphic functions sharing four values. Complex Variables, 1989, 12: 169-179 8 Yi H X, Yang C C. Unicity theory of meromorphic functions(in Chinese). Beijing: Science Press, 1995. 248-250
MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES 1 Huang Bin ( jfiA; ) Du Jinyuan ( #.~ Jt ) The College of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected] Abstract The uniqueness of meromorphic functions that share four values is investigated. A necessary condition to the case is acquired, and some partial results for question "lCM+3IM=4CM" are obtained. Key words
Meoromorphic functions, shared-value
2000 MR Subject Classification
1
30D35
Introduction
Throughout the paper, we use standard notations and fundamental results of Nevanlinna's theory (see W.K.Hayman [1]). Let j(z) be a meromorphic function in the complex plane, we denote by S(r, I) any quantity satisfying S(r, I) = o(T(r, I)) for r -t 00 except possibly a set of r of finite linear measure. We say that two nonconstant meromorphic functions j and 9 share the value c(c = 00 is allowed) provided that j(z) = c if and only if g(z) = c. Usually, we will state whether a shared value is by CM (counting multiplicities) or 1M (ignoring multiplicities).We denote by N E(r, j = c = g) or N E(r, c) the counting function of those c-points where j(z) and g(z) have same multiplicity (counting each point only once), while by N D(r, j = c = g) or N D (r, c) the counting function of those c-points where j and 9 have different multiplicities (counting each point only once).. R.Nevanlinna (see [3]) gave the following theorem: Theorem A If j and 9 are distinct nonconstant merom orphic functions that share four distinct values al,a2,a3,a4 CM,then j is a Mobius transformation of g;two of the values ,say, al and a2, are Picard values,and the cross ratio (al' a2, a3, a4) = -l. In 1976 L.Rubel asked such a question: whether CM can be replaced by 1M in Theorem A with the same conclusion or not? A counterexample given by G.G.Gundersen (see ref.[4]) showed that the answer to this question is negative, i.e., generally "41Mi:4CM". Meanwhile, on the other hand, he proved that "3CM + lIM=4CM". Furthermore, he showed by the following theorem (See [5]) that "2CM+21M=4CM" is also valid. Theorem B If two nonconstant meromorphic functions share two values 1M, and share two other values CM, then j and 9 share all four values CM. [Received January 21, 2002; revised May 10, 2003. Project supported by the National Natural Science Foundation of China(19971052) and the Programme of Hunan Education Foundation(02C095)
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Theorem B can be generalized slightly to the follows: Theorem B*[6] If two nonconstant meromorphic functions share two values 1M, and share two other values CM*(where the terminology "two nonconstant meromorphic functions share the value a CM*" means a is shared by f and 9 and, furthermore, N(r, a) = N E(r, a) + S(r, I)), then f and 9 share all four values CM. Let f and 9 be nonconstant meromorphic functions sharing the value a 1M. Define · . f N- E(r,a) , 1Imm
r(a) =
r-too
{ 1,
N(r,a)
if N(r, a)
¥ 0,
if N(r, a) == 0.
For the open question "whether 1CM+3IM=4CM?" i.e., "If two nonconstant meromorphic functions share three values 1M and share a fourth value Clvl.then do the functions necessarily share all four values CM ?", G.G.Gundersen (see [6]) proved the following partial result. Theorem C Let f and 9 be nonconstant meromorphic functions that share al, a2, a3 1M and a4 CM.Suppose that there exist some real constant A > 4/5 and some set t c: (0,00) that has infinite linear measure such that
N(r, a4, I) > A T(r, I) -
(1.1)
for all rEI. Then f and 9 share all four values CM. In this paper, a necessary condition for two meromorphic functions sharing four values is discovered as follows: Theorem 1 Let f and 9 be nonconstant meromorphic functions that share four distinct values al, a2, a3, a4 1M.Then either the functions share all four values CM or else for every i E {I, 2, 3, 4}, the relation
N E(r, ai)
~
2ND(r, ai)
+ 2ND(r, ak) + S(r, I)
holds for k E {1,2,3,4}\{i}. Moreover, by Theorem 1 we obtain the following results Theorem 2 Let f and 9 be nonconstant meromorphic functions that share al, a2, a3 1M and a4 CM. If N(r, al, I) + N(r, a2, I) ~ {IT(r, I) + S(r, I) holds for some {l < 2/3 ,then f and 9 share all four values CM. Furthermore, by Theorem 1, an improvement of Theorem C is obtained below. Theorem 3 Let f and 9 be nonconstant meromorphic functions that share three distinct values al, a2, a3 1M and a fourth value a4 CM. Suppose that there exist some real constant
A > 2~r' where r = I
c (0,00)
3
L:
j=l
l-r\a)(r
= 00 if r(aj) = 1 for some j E {1,2,3}), and some set
J
which has infinite linear measure such that
N(r, a4, I) > A T(r, I) -for all rEI. Then f and 9 share all four values CM. Remark 1 It is obvious that 2~r ~ so Theorem 3 improves Theorem C.
t,
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Huang & Du: MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES
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Remark 2 Theorem 3 improves Theorem B while Theorem C does not. In fact, suppose two meromorphic functions share two values al,a2 CM,and share other two values a3,a4 1M, then by Theorem 3,we can deduce that either all four values are shared CM, or else N(r, ad = 0,N(r,a2) = 0, from which it follows N(r,a3) = NE(r,a3) + S(r,/),N(r,a4) = NE(r,a4) + S(r, I) by Nevanlinna's second fundamental theorem and 4IM theorem (see Lemma 1 in the next section).Thus f and 9 share all four values CM.
2
Lemmas
Some lemmas below are needed in the proofs of the theorems. Lemma 1[2,4,5,8] Let f and 9 be distinct nonconstant meromorphic functions that share four values al, a2,a3,a4 1M. Then the following statements hold: (i) T(r, I) = T(r, g) + S(r, I), T(r, g) = T(r, I) + S(r, g); 4
E N(r, f!a.) = 2T(r, I) + S(r, I); j=l (iii) No(r"j;) = S(r, I), No(r, -]r) = S(r,g), where No(r, f;) and No(r, -]r) are respectively counting functions of I' only to those points such that f(z) t ai and g(z) t ai for i = 1,2,3,4. 4 (iv) E N*(r,aj) = S(r,l), (ii)
J
°and g' = 0 refer
=
j=l
where N*(r,aj) is the counting function for common multiple zeros of f(z) - aj and g(z) - aj, counting the smaller one of the two multiplicities at each of the points. Lemma 2(see [4][5][6] or see p.247 in ref.[8]) Let f be a nonconstant meromorphic function and let bi, bz, ... ,bq be q constants. Then for any polynomial PU) of degree p(p < q) in f with constant coefficients, the equality
PU)f' m(r U - bl)U - b2) ... U _ bq ) )
= S(r, I)
holds. Lemma 3[4,5,6] Let f and 9 be distinct nonconstant meromorphic functions that share four values al, a2,a3, 00 1M. Then the function
1jJ(z)
/,g'U - g)2
= -:-U.,----al...,..).,--U---a-2.,--)(""';f:"'--:::"a":':3"""H-g:;";--a--:-l)"""(g---a2-'--H"--g---a--:-3)
is an entire function and satisfies
T(r,1jJ(z)) = S(r, I). Lemma 4 Let f and 9 be nonconstant merom orphic functions that share four distinct values al , a2,a3,a4 1M. Then either the functions share all four values CM or else the inequality
holds for distinct i,j,k,m E {l,2,3,4}. Proof Assume f -:t 9 and, without loss of generality, a4
= 00.
For i
= 1,2,3, put (2.1)
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where j::j; k,{j,k} E {1,2,3} \ {i}. If 1]i == 0, then aI, a2, a3, a4 are shared CM by I and g. Therefore we assume From (2.1), Lemma 1(i) and Lemma 2, for distinct i,j,k,E {1,2,3} we have
N(r, ai) ~ N(r, ~
1]11]21]3 ~
i) ~ T(r, 1]i) + 0(1) ~ N(r,1]d + S(r, I) + S(r, g)
N oir, aj)
+ N D(r, ak) + N D(r, I) + S(r, I).
0.
(2.2)
= _,1 ,G = _1_. Then F and 9 share b1,b3,b3,b4 1M, where b1 = 00,b2 = -1-,b 3= -at g-at a2-at _1_, b4 = 0. Consider the following auxiliary function aa-at
Let F
(2.3)
If 1]4 == 0, then bi, bz, b3, b4 are shared CM by F and G. Thus aI, a2, a3, a4 are shared CM by I and g. Now we suppose 1]4 ~ 0. Since T(r, F) = T(r, I) + 0(1), T(r, G) = T(r, g) + 0(1), from (2.3), Lemma 1(i) and Lemma 2 we deduce that N(r, a4, I)
= -N(r, i; F)
~
1 N(r, -) 1]4
~
T(r, 1]4)
+ 0(1)
(2.4)
By (2.2) and (2.4), we complete the proof of Lemma 4.
3
Proof of Theorem 1
Now we come to prove Theorem 1. The proof below has some similarities to the proof of Theorem C as in [6]. If I == g, then there is nothing to prove. So we assume I ~ g. Picking an integer i E {1,2,3,4}, say, i = 4, we shall estimate N E(r,a4) by considering two cases. Case 1. a4 = 00. Put
1/;(z)
I"
1]
= (f l'
= f - 1- a1
-
/,g'(f - g)2 )( ( , ad(f - a2)(f - a3 9 - ad 9 - a2)(g - a3)
l' /' s" 1- a2 - 1- a3 - (9' Hj
g' = -I--I' aj - -, 9 - aj
-
g'
g'
(3.1) g'
9 - a1 - 9 - a2 - 9 - a3)'
. J = 1,2,3.
(3.2) (3.3)
By Lemma 3 we know 1/; is an entire function and satisfies
T(r,1/;) = S(r, I).
(3.4)
By considering residues in (3.2), we deduce that 1] is analytic at any a-point (a E {a1,a2,a3}) as well as such a pole at which I(z) and g(z) have the same multiplicity. And it is obvious that 1] has a simple pole when I = a4 and 9 = a4 with different multiplicities. Thus from (3.3), (3.2), Lemma 1 and the fundamental estimate of the logarithmic derivative we have
T(r,3Hj
+ 1]) = N D(r, aj) + N D(r, a4) + S(r, I), j = 1,2,3.
(3.5)
No.4
Huang & Du: MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES
Let Zl be a common simple pole of
533
f and g.Assume that
An elementary calculation gives that
= -bbl - -ClCo
Hl(zd 1/(zd
o
~ = -3(b
o
~
-) Co
'I/J(zd
+ (al
= (b~
1
1
-al(- - -), bo Co 1
(3.6) 1
+a2 +a3)(- - -), bo Co -
~ )2.
(3.7) (3.8)
From (3.6),(3.7) and (3.8) we obtain
(3H l(zd + 1/(Zl))2
= (2al -
a2 - a3)2'I/J(zl).
(3.9)
If (3H l +1/)2 == (2al -a2 -a3)2'I/J, then 3Hl +1/ has no poles since 'I/J is an entire function. Thus N(r,3H l + 1/) = N D(r, al) + N D(r, a4) + S(r, f) = 0, which implies al and a4(= 00) must be shared CM by f and g. Thus f and g share all four values CM by Theorem C. Now we suppose (3H l + 1/)2 ~ (2al - a2 - a3)2'I/J. Then from Lemma l(iv), (3.9) and (3.4), we can deduce that (3.10)
Similarly, considering H 2 and H 3 , we can obtain that either or else
f and
g share all four values CM
(3.11)
and (3.12)
hold. It is shown from (3.10),(3.11) and (3.12) that in the case a4 Theorem 1 is valid. Case 2. a4 =I 00.
= 00 the
conclusion of
Set F = '!.a4,G = g!.a4. Then F and G share bl,b2,b3,b4 1M, where bj = aj~a4,j = 1,2,3;b4 = 00. Since T(r,F) = T(r,f) + O(l),NE(r,b j) = NE(r,aj), and ND(r,b j) = ND(r,aj),j = 1,2,3,4, treating NE(r,b 4) in the same way as in Case 1, we still obtain that either f and g share all four values CM or else (3.10), (3.11) and (3.12) hold. Thus Theorem 1 is proved.
4
The Proof of Theorem 2
Assume that each of the values al,a2,a3 is not shared CM by f(z) and g(z), then the following inequalities hold by Lemma 4 and Theorem 2:
534
ACTA MATHEMATICA SCIENTIA
Vol.24 Ser.B
+ S(r, f),
N(r, a4) ~ 2N D(r, al)
N(r, a4) ~ 2N D(r, a2) + S(r, f).
It follows that 2T(r,f)
=
4
LN(r,aj) j=1
+ S(r,f)
< N(r, ad + N(r, a2) + 2(Noir, ad + N D(r, a2)) + S(r, f) < 3(N(r, ad + N(r, a2)) + S(r, f). This is a contradiction since N(r, al) is thus proved.
5
+ N(r, a2) ~ f..lT(r, f) + S(r, f),
and f..l < 2/3. Theorem 2
The Proof of Theorem 3
Assume that each of the values aI, a2, a3, is not shared CM by f (z) and g( z). Notice that a4 is shared CM by f(z) and g(z), by Theorem 2 we have N(r, a4) ~ 2N D(r, ai), If r(ai)
= 1,2,3.
i
(5.1)
> O(i = 1,2,3), then we take 0 < f..li < r(ai)' The inequality N D(r, ai) ~ (1 - f..li)N(r, ai)
(5.2)
(k = 1,2,3)
holds for sufficiently large r. If r(ai) = 0, then we take f..li = O. The inequality (5.2) still holds. So from (5.1) and (5.2), it follows that N(r, a4) ~ 2(1 - f..li)N(r, ai)
(k = 1,2,3).
(5.3)
Hence, {(1 - f..ll)(l - f..l2) where A
= (1 -
+ (1 -
f..l2)(1 - f..l3)
+ (1 -
3
f..l3)(1 - f..ll)}N(r, a4) ~ 2A L N(r, aj), j=1
f..ll)(l - f..l2)(1 - f..l3)' That is, 4
{(1- f..ld(l- f..l2) + (1- f..l2)(1- f..l3) + (1- f..l3)(1- f..ld + 2A}N(r,a4) ~ 2A LN(r,aj). j=1 From this and Lemma l(ii), we derive that 4
. N(r,a4) lim sup T( f) ~ r-oc> r,
3
2+
rrtE
1
L ~ j=1 1-',
where E is a set of r of finite linear measure. This leads to . N(r, a4) hmsup T( T, f) ~ r_oc rrtE
4 3
2
+L
)=1
1 I-T(aj)
No.4
Huang & Du: MEROMORPHIC FUNCTIONS THAT SHARE FOUR VALUES
535
which contradicts the condition (1.2). Thus at least one of al,aZ,a3 must be shared CM by f(z) and g(z). As a4 is also shared CM, aI, az, a3 are all shared CM by f(z) and g(z) according to Theorem B. This completes the proof of Theorem 3. References 1 Hayman W K. Meromorphic Function. Oxford: Clarendon Press, 1964 2 Nevanlinna R. Einige Eindeutigkeitssdtze in der Theorie der meromorphen Funktionen. Acta Math, 1926, 48: 367-391 3 Nevanlinna R. Le thereme de Picard-Borel et la theorie des fonctions meromorphes. Paris: GauthierVillars, 1929 4 Gundersen G G. Meromorphic functions that share three or four values. J London Math Soc, 1979, 20(2): 457-466 5 Gundersen G G. Meromorphic functions that share four values. Trans Am Math Soc, 1983, 277: 545-567. Correction: 1987, 304: 847-850 6 Gundersen G G. Meromorpic functions that share three values 1M and a fourth value CM. Complex Variables, 1992, 20: 99-106 7 Mues E. Meromorphic functions sharing four values. Complex Variables, 1989, 12: 169-179 8 Yi H X, Yang C C. Unicity theory of meromorphic functions(in Chinese). Beijing: Science Press, 1995. 248-250