### Journal Menu >> Advances in Pure Mathematics, 2011, 1, 95-98 doi:10.4236/apm.2011.13021 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM Normality of Meromorphic Functions Family and Shared Set by One-way Yi Li School of Science, Southw est Universi ty of Scie nce a nd Te chn olo gy, Mianyang, Chin a, E-mail: liyi@swust.edu.cn Received March 16, 201 1; revised April 5, 2011; accepted April 10, 2011 Abstract We studied the normality criterion for families of meromorphic functions which related to One-way sharing set, and obtain two normal criterions, which improve the previous results. Keywords: Meromorphic Function, Normality Criterion, Shared Values, Shared Set by One-way 1. Introduction For Shared values, Schwick proved the following result : Theorem A Let F be a family of meromorphic func-tions in the domain D, 1a, 2a and 3a be three finite complex numbers. If for every ,1,2,3fi fifFEa Eai then F is normal in D. In 2000, Pang Xue-cheng and Zalcman generalized the Schwick’s result : Theorem B Let F be meromorphic functions family in the domain D, and 1a, 2a be two complex number. If for every  ,1,2fi fifFEa Eai then F is normal in D. Definition For ,ab are two distinct complex values, we have set ,Sab and  ,:0,ffES Eabzfzafzbz D If  fgES ES, we call that f and g share S in D; If  fgES ES, we call that f and g share S by One-way in D. For shared set, W. H. Zhang obtained impor tant results : Theorem C Let F be a family of meromorphic func-tions in the unit di sc , a and b be two distinct nonzero complex value, ,Sab, If for every fF, all of whose zeros is multiple, ffES ES, then F is normal on . W. H. Zhang continued considering the relation be-tween normality and the shared set, and proved the next result : Theorem D Let F be meromorphic functions family in the unit disk , a and b be two distinct nonzero com-plex values. If for every fF, all of whose zeros is multiplicity 1k at least (k is a positive integer), kffESES, then F is normal in . For shared set by One-way, Lv Feng-jiao got follow-ing theorem in : Theorem E Let F be a family of meromorphic func-tion in the unit disk , a and b is two distinct nonzero complex values, kis positive integer, ,Sab. If for every fF, all of whose zeros have multiplicity 1k at least, kffESES, then F is normal in . In 2007, Pang Xue-cheng proved the following im-portant results in : Theorem F Let F be meromorphic functions family in D, 123,,Saaa. If for everyfF ffES ES, then F is normal on D. To promote the results of Pang Xue-cheng, we con-tinue to discuss about normality theorem of meromorphic functions families concerning shared set and shared set by one-way, and obtain our main results as follow. Theorem 1 Let F be meromorphic functions fami-lies in D, 123 4,, ,iSaaaaa (i = 1, 2, 3). If for every fF,  ffES ES, and 4fa whenever 4fa, then F is normal on D. Theorem 2 Let F be meromorphic functions fami-lies in D, 12 3,,SaaaC. If for every fF, ffES ES, and 3fa, whenever 3fa, then Y. LI Copyright © 2011 SciRes. APM 96 F is normal on D. 2. Lemmas Lemma 1  Let F be meromorphic functions families in the unit disk , all of whose zeros have multiplicity kat least, and 0A. If for every fF, fzA whenever 0.fz If Fis not normal in , then for every 01, there exists 1) a positive number ,0 1rr, 2) complex sequence ,nnzz r, 3) Functions sequence nfF, 4) and positive sequence 0n, such that nnnnngfz converges locally and uniformly to a noncontant meromorphic function g, and  ##01ggkA. Where  #2.1ggg Lemma 2  Let fbe meromorphic function with finite order on the open plane C, and 123,,aaa be three finite complex values. If fz have only finite zero, and  1230,,fzfzSaaa  then f is a rational function. 3. Proof of Theorem 1 Suppose that F be no t normal in , then by Lemma 1 we have that there exists ,and0nn nfFz , such that 14nnnnngfzag  con- verges locally and uniformly to a noncontant meromor-phic function g. We claim that the following con-clusions hold.  0410gga ;  0## 4201gga; 03gS; It is not difficult to prove claims 001,2, in what fol-low, we complete the proof of the claim 03. Suppose that there exists 0C such that 0iga. Obvi-ously, iga, in fact, if 0igac, it is a contradictions for 01. Thus from Hurwitz Theorem, we know that there exists a point sequence 0n, such that 0nn iga, for sufficiently large n, that is nn nnifza. Obviously, 14,nnn igaa as n. Thus 0g, this is a contradiction. Hence, claim 03 holds. From claim 03 we have that  1,2,3igai So g is identical in nonconstant. Again because claim 01, we know 40ga and 404#2040101112aaga. Clearly, this is a contradictions for claim 02. There-fore, F is normal in D. The proof of Theorem 1 is com-pleted. 4. Proof of Theorem 2 Suppose that F is not normal in , by Lemma 1 there exists ,nnfFz and 0n, such that ng nn nfz g converges locally and uni-formly to a noncontant meromorphic function g with finite orders, there  ##0gg. We asserts that 0gSg. In fact, suppose that there exists 0C, such that 0gS, thus there exists 1, 2iai such that 0iga. From Hurwitz Theor em and iga, we have there exists 0n such that nniga, that is nng nn nifza for sufficiently large n. Thus in contrast with conditions of Theorem, we get nn nfzS. Obviously, nn nnfzA. So we get 00g. Since g is a nonconstan t entire function, without loss of generality, we assume that 1ga have zero on C for 1a, and consider function sequence nG: 11nnnnnnnfzagaG Obviously, nG is not normal in zero of 1ga. In fact，if 0 is zero of 1ga, then 0nG 000nn nfz. With conditions of Theorem, we get 0nn nfzA and 0nGA. There-fore, nG is not normal in zero of 1ga. So there exists nG, n and 0n, such that 111nnnnnnnnnFGgaF   converges locally and uniformly to a noncontant and meromorphic function F with finite order, and 01 the number of zeros of F is finite, Y. LI Copyright © 2011 SciRes. APM 97 0320FFSa ,  030FSF , 000410FF In fact, suppose that 0 is the zero of 1ga with order k. If there exists 1k distinct 121,k  at least, such that 0,1,2, ,1jFjk . By Hurwitz Theorem, it is certainly that there exist a positive integer N, such that 0, 1,2, ,1jnnFjk  as nN. Thus, 10jnn nnga. Since 0jnnn n , 1, 2,,1jk, we deduce that 0 is a zero of 1ga with 1k orders, this is a contradictions for suppose. Therefore zeros numbers of F is finite. Suppose that 0 is a zero of 00F. For 0F and Hurwitz theorem, we know that there exists sequence 0n, such that  110,nnnn nnnn nnnnnn nnfz aFfz a     Thus, we get 3nnnn nnfzS a   and subsequence nfF such that nnnn nnifz a  , thus  03lim nnnn nnnFfzS a , for 3iaS a. If there exists 0 such that 0FS, that is, there exists iaS such that 0iFa. Since iFa, by Hurwitz theorem, there exists 0n such that  nnn nnnnniFfz a. Hence, nnnn nnfz S  , 0FS. If there exists N such that 1for ,nnnn nnfza nN   we get 10lim nnnnnnnnnfz aF   This contradicts 0iFa. Thus exists subsequence nf, such that 1nnnn nnfz a   for every n. Therefore,  10lim 0nnnnnnnnnfz aF   Now we prove that 0010FF  . Since 31 3113110()nnnnnnnnnnn nFaaGaagaaa   there exists 0n, such that 3110nnnFaa, we get 3nnnn nnfza , thus 3nnnn nnfza , that is, 3nnFa. Therefore, 000221lim 0nnnnFFFFF   . So far, we give complete proofs of all assertion. Next we will complete the proof of theorem 2 using assertion 001~4. By Lemma 2 and assertion 02, we get that F is a rational function. Again by assertion 04, it is clear that the pole of F be multiple. If nG is not normal at 0, thus 0 be zero of 1ga. By the isolation of zero, we have that nG are holomorphic functions at 0 for sufficiently large n. We get that 1nnnnnFG  are holomorphic functions in R for sufficiently large R, thus F be nonconstant holomorphic functions in C. Therefo re F be a polynomial. Let it, s order is 0pp. Thus,  ,1lnTrF pr , ,lnand, 1NrFprSrFO Therefore, 21lnln 1,prprOr. We get 02p easily. If 1p, thus 0100Fccc, by 02 and 03, we find that there exists ia for every , such that iFa. Therefore  be an zero of F. But F have only a zero, this is a contradiction. If 2p, thus Y. LI Copyright © 2011 SciRes. APM 98 00 10010,Fc c   As a result, 0012Fc. Obviously ze-ros of iFa are 00 0102iac cc . Hence we get that F have three zeros, this still is a con-tradiction from 00 10010,Fc c  and the proof of theorem 2 is completed. 5. References  W. 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