 Applied Mathematics, 2011, 2, 718-724 doi:10.4236/am.2011.26095 Published Online June 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Mer omorphic Functions Sharing Thr ee Values* Changjun Li, Limei Wang School of Mathemat i cal Sci ences, Ocean University of China, Qingdao, Chin a E-mail: changjunli7921@hotmail.com, wang-83limei@163.com Received March 26, 2011; revised April 10, 2011; accepted April 13, 2011 Abstract In this paper, we prove a result on the uniqueness of meromorphic functions sharing three values counting multiplicity and improve a result obtained by Xiaomin Li and Hongxun Yi. Keywords: Uniqueness, Meromorphic Functions, Sharing Three Values 1. Introductionand Main Results Let f and g be two non-constant meromorphic func- tions in the complex plane. It is assumed that the reader is familiar with the standard notations of Nevanlinna’s theory such as , ,Trf,mrf , ,Nrf,Nrf and so on, which can be found in . We use to denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. The notation denotes any quantity satisfying . A meromorphic function E,Srf,rf rSrf,r E ,=Tb is called a small function with respect to f provided that Tr . A meromorphic function  ,=b Sr,fb is called a exceptional function of f provided that ,=,Nr Srffb1 . Let be a complex number, we say that af and g share the value a CM provided and af ag have the same zeros counting multiplicities (see ). We say that and fg share CM provided that 1f and 1g share 0 CM. Xiaomin Li and Hongxun Yi prove the following theorem: Theorem A (). Let and fg be two distinct nonconstant meromorphic functions sharing three values and CM, if there exists a finite complex number such that a is not a Picard value of , and 0,1f0,1a11,,NrTrfSrffa ,, then 112,=, ,kNrTrf Srffa k and one of the following cases will hold: 1)  11e1e=, =e1e 1kkssfg1, with  111111=1ks ksskkasksak and 1kas; 2)  11e1e 1=, =e1esskkfg1, with  11111=1ksskssksk saa k and 1sak; 3)  11e1e1=, =e1essks ksfg 1, with 1111=11sksskkasksak and 1saks  ; 4) e1e 1=, =e11e 1kkssfg, with and 0,1k 1=ks ksskk kasksak; 5) e1e 1=, =e11e 1sskkfg, with and 0,1s 1=kssksssksksaa k; 6) e1e1=, =e11essks ksfg 1, with *The Project-sponsored by SRF for ROCS, SEM. C. J. LI ET AL.719 s0,1 and =1sksskkaskska. where  is a nonconstant entire function, s and are positive integers such that 2ks and 1k are mutually prime and ks1 in 1), 2), 3), s and are mutually prime and k11ks in 4), 5), 6). Xinhou Hua and Mingliang Fang proved the following theorem: Theorem B (). Let f and g be two non- constant meromorphic functions sharing three values and  CM, if 0,1 ,,,TrfNrbz fSrf,  0,1,bz  is a small function of , then one of the following holds: f1) fg; 2) fbg, and are exceptional functions of , 1bf; 3) , and are exceptional functions of f11 1g b, 0bf; 4)  11fbgb bbf, and are ex- ceptional functions of . , bAs we all know, many results on constants are also valid for small functions, although some times they are more difficult. In this paper, we improve the above theorems and obtain the following result. Theorem 1.1. Let and fg be two distinct nonconstant meromorphic functions sharing three values and CM, if there exists a small function of such that is a exceptional function of , and 0,1bz 1,f 0,fbz1,,, ,NrbzfTrf Srf (1.1) then 12,,=, ,kNrbzfTrf Srfk (1.2) and one of the following cases will hold: 1) 11e1e=, =e1e 1kkssfg1, with  1111111ks ksskkbsksbk  and 1kbs; 2) 11e1e 1=, =e1esskkfg1, with  11111ksskssksk sbb k and 1sbk; 3) 11=,11=)1()1( sksskseegeef, with 1111)()1()(1)(ksksksksksbb and sksb1; 4) 1))(1(1/1=,1)(11=skskecbegecbef, with and 0,1k21112 1121 ;kskskb bsbcbbcb cbbcb bks bcb  5)  e1e 1=, =1e1 11e1sskkfgcb cb , with and 0,1s21112 1121 ;sksksb bkbcbbcb cbbcb bskbcb   6) e1e 1=, =e111essks ksfgcb 1, with and 0,1s21112 1121 .sskksb bsk bcbbcbcbbcb bkbcb   where  is a nonconstant entire function, s and 2k are positive integers such that s and 1k are mutually prime and ks 1 in 1), 2), 3), s and are mutually prime and in 4), 5), 6), and k1k1 sc are constants. 2. Some Lemmas Lemma 2.1 (). Let and fg be two nonconstant meromorphic functions sharing three values and 0,1 CM. If gf, then for any small function 0,1,bz we have 33,, ,,=,NrbzfNrbzg Srf. Lemma 2.2 (). Let be a nonconstant mero- rphic function, 1 and be three distinct small functions of , if faa 2a3f1111,,=Nr NrSrffa fa,, Copyright © 2011 SciRes. AM C. J. LI ET AL. 720 then 131,=, ,NrTrfSrffa  . Using the same method of  in Lemma 2.2, we get the following result: Lemma 2.3. Let and fg be two nonconstant meromorphic functions sharing three values and CM. If is a fractional linear transformation of 0,1fg, for any small function bz0,1,f, then either is a exceptional function of , or bz11,=,,NrTrf Srffb  . Lemma 2.4. Let s and t are two integers, and  be a nonconstant meromorphic function and bz is a small function of , if , then 1sb0,1, =,stNrb Sr, where denotes the reduced coun- ting function of the common zero of and 0,1,stNr b1stb. Proof. If is a zero of and , then we have 0z1stb0=1,sz (2.1) and 0=tzbz0., (2.2) From (2.1) and (2.2) we get , thus 0=1sbz0,1, =,stNrb Sr since . 1sbLemma 2.5. Let =nmPa ,b (2.3) where =e,  is a nonconstant entire function, a and b are two small functions of , and m are positive integers such that . n>nm1)  31,=,.Nr SrP (2.4) 2) If  ,nbab mabamna amna     m (2.5) then  21,=,Nr SrP. (2.6) 3) If and are mutually prime, and n m ,nmbab mabamnaamna     (2.7) then  21,=2, ,NrTr SrP.  (2.8) Proof. 1) Differentiating P two times and eli- minating n and m from the three equations we obtain 12=1,PhP hP  (2.9) with ,=, =1,2iTrh Sri. Thus (2.4) holds. 2) Suppose  21,NrSrP, , and let be 0za zero of P with multiplicity , then from (2.3) we have 20000=0,nmzaz zbz (2.10) and    000000000(=0.nmnz zazzaz zzbzz (2.11) From (2.10) and (2.11) we get   00000000=,mbz nbzzzzaz mnazzz 0 (2.12) and  00 000000=.nma zb zzzzaz mnazzz 0 (2.13) Since =e,  is a nonconstant entire function, we have ,=,TrSr . (2.14) From (2.7) (2.12) (2.13) and (2.14), we get (2.6) holds. 3) Let 0 be a zero of zP with multiplicity , using proceeding as in 2) we can get (2.12) and (2.13). On the other hands, since and are mutually prime, there exist one and only one pair of integers 2n ms and t such that =1 0<<,0<0st such that 12 1ffst where 2 denotes the reduced counting function of and related to the common 1-point and 01,1; ,Nr ff1f2f 12=,,, = ,Tr TrfTrfSroTrrr E  only depending on and . 1 2Lemma 2.7 () Let be a nonconstant mero- f ffmorphic function and  =PfRf Qf , where  =1 =1= and =pqkjkkjPfafQfbfj are two mutually prime polynomials in . If the co- efficients fkaz, jbz are small functions of and , , then f0pa0qb,=max,, ,TrRfpqTrfSrf. 3. Proof of Theorem 1.1 If is a fractional transformation of fg, by Lemma 2.3 we have that either is a exceptional function bzof , or f11,=,,NrTrf Srffb  , which contradicts with the assumption of Theorem 1.1. Thus is not a fractional transformation of fg. By Theorem B we have 1,=,,. (3.1) NrTrf Srffb From (1.1) and (3.1) we obtain 21,Nr Srffb,. (3.2) By Lemma 2.1 we have 31,=,NrSrffbCombining (3.2) and (3.3) we get 21,Nr Srffb,. (3.4) Noting that and fg share 0,1 and CM, we have 1=e, =e.1ffgg (3.5) where  and  are two entire functions. From (3.5) we get e1e 1=, =e1e 1fg, (3.6) and ee1=e1bbfb. (3.7) Assume that ,=,Tre Srf, Noting 0 and  are Picard values of , from (3.6) we have e1e1 and  are exceptional functions of , by Lemma 2.2 we get f11,=,,NrTrf Srffb  , which contradicts with the assumption of Theorem 1.1. Thus ,,TreSrf. Similarly, we have ,Tre Srf, and ,,Tre Srfz.fb Let 0 be a multiple zero of , but not a zero of ,, and . From (3.7) we obtain 00001=0.zzebzebz (3.8) and   0000 000eezzzbzbz zbz  =0. (3.9) From (3.8) and (3.9) we have  2000 000000000000000000e=e= .zzbzbzzbzzbz bzz bzzbzz bzzbz bzz bzz , (3.10) Set 122=e, =bb bbb bffbbbb be,     (3.11) and   12=, ,,= ,Tr TrfTrfSroTrrr E . (3.12) . (3.3) Copyright © 2011 SciRes. AM C. J. LI ET AL. 722 .From (3.5) (3.11) and (3.12) we get ,=SrfSr (3.13) From (3.11) (3.12) and (3.13) we get 1,,=, =1,2iiNrfNrSr if . (3.14) From (3.10) and (3.11) we have . Thus 10=1,fz20=1fz01221,,1;,NrNrff Srffb,. (3.15) 01,1; ,Nr ff2 denotes the reduced counting function of the common 1-points of 1f and 2f. From (3.4) (3.13) and (3.15), we obtain 012,1; ,.Nr ffSr (3.16) From (3.16) and Lemma 2.5, we know there exist two integers and p>0qp q such that 12 1.pqff (3.17) Noting , and , from (3.11) and (3.17), we have , and , and ,e ,Tr Srf,Srf0pq,e ,TrSrf,eTr0pq2e=pqpq bbbbbbb bbb b      . (3.18) Let =bQz bbb., then from (3.18) we get e=11pqpq QbQ (3.19) Noting that is a small function of , we obtain that bz f,Qz c (3.20) where is a constant. From (3.19) and (3.20) we obtain c e=11.pqpq cbc (3.21) Without loss of generality, From (3.21) we may assume that and are mutually prime and . p q>0qLet =1pqc and q=, where  is an entire function. Then from (3.6) and (3.21) we obtain e1=,1e1e1=.11 e1qpqpfcbgcb (3.22) Noting that , and , We discuss the following three cases. 0p0qqp Case 1. Suppose that , we discuss the following two subcases. 0>>pq Subcase 1.1. If 1qcb1. Setting 1=kq and ps=, let =1cb and e=ess. From (3.22) and (3.7) we get  11δδe1e=, =e1e 1kkssfgδ1. (3.23) And 1δδδee=.e1kssbbfb 1 (3.24) Since in this subcase b is a constant, let =, (3.23) assume the form (1) in Theorem 1.1. From the proof of Theorem A we know (1.2) holds with  1111111ks ksskkbsksbk and 1kbs Subcase 1.2. If 1qcb1. Setting and qk ==sp. From (3.22) we get e1=,1e1e1=.11e 1ksksfcbgcb (3.25) which assume the form (iv) in Theorem 1.1. We have from (3.25) and (3.7) e1e=1e1kssbcb bfb cb11 (3.26) Since 1kcb, from (3.25) (3.26) Lemma 2.4 and Lemma 2.7 we get ,= ,,Trf kTreSrf. (3.27)  0,0;e1e 1,1e=,ksNrbcbbcbSrf1s (3.28) where 0,0;e1e 1,1eksNrbcbbcb1s denotes the reduced counting function of common zeros of e1eksbcb b1 1 and . 1escbIf Copyright © 2011 SciRes. AM C. J. LI ET AL.723 21112 112 1kskskb bsbcbbcb cbbcb bks bcb  by Lemma 2.5 (2), we get a contradiction with (3.4). Thus From (3.27) (3.28) and Lemma 2.5 (3) we obtain (1.2) holds with 21112 1121 .kskskb bsbcbbcb cbbcb bks bcb  Case 2. Suppose that , we discuss the following two subcases. >>0pqSubcase 2.1. If . Setting and 11qcb=1=kpsq=1, let and cbke=ek. From (3.22) and (3.7) we get  δδ1δ1δe1e 1=, =e1esskkfg1 (3.29) and δδδee=e1skkbbfb 1 (3.30) Since in this subcase is a constant, let b=δ, (3.29) assume the form of 2) in Theorem 1.1. By the proof of Theorem A we know (1.2) holds with  111111ksskssksk sbb k and .1sbk Subcase 2.2 If . Setting 1qcb1p=k and =sq, from(3.22) we get e1=,1e1e1=.11e1skskfcbgcb (3.31) Which assume the form 5) in Theorem 1.1. We have from (3.31) and (3.7) e1e=.1e1skkbcbbfb cb 1 (3.32) In the same manner as Subcase 1.2 we know (1.2) holds with 21112 1121 .sksksb bsbcbbcb cbbcb bskbcb   Case 3. Suppose that , we discuss the follow- ing two subcases. >0pSubcase 3.1. If 1qcb1. Setting 1=kpq and =sq, let and =1cbδe=kseks . From (3.22) and (3.7) we get  δδ1δ1δe1e 1=, =e1essks ksfg .1 (3.33) and 1δδ1δee 1=.e1kssksbbfb  (3.34) Since in this subcase b is a constant, let =δ, (3.33) assume the form (3) in Theorem 1.1. By the proof of Theorem A we know (1.2) holds with 111111sksskkbsksbk and 1sbks Subcase 3.2. If 1qcb1. Setting =kpq and =sq, we have from (3.22) e1=,1e1e1=.11 e1skssksfcbgcb (3.35) which assume the form (6) in Theorem 1.1. From (3.35) and (3.7) we get e1e=.1e1kssksbcbbfb cb1 (3.36) In the same manner as Subcase 1.2 we get (1.2) holds with  21112 1121 .skksb bsbcbbcb cbbcb bkbcb Theorem 1.1 is thus completely proved. Copyright © 2011 SciRes. AM C. J. LI ET AL. Copyright © 2011 SciRes. AM 724 4. Acknowledgements The authors want to express theirs thanks to the anony- mous referee for his valuable suggestions and professor Qizhi Fang for her support. 5. References  W. K. Hayman, “Meromorphic Functions,” Clarendon Press, Oxford, 1964.  H. X. Yi and C. C. Yang, “Uniqueness Theory of Mero-morphic Functions,” Science Press, Beijing, 1995.  X. M. Li and H. X. Yi, “Meromorphic Function Sharing Three Values,” Journal of the Mathematical Society of Japan, Vol. 56, No. 1, 2004, pp. 147-167.  X. H. Hua and M. L. 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