 Advances in Pure Mathematics, 2011, 1, 322-324 doi:10.4236/apm.2011.16058 Published Online November 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM The Normal Meromorphic Functions Family Concerning Higher Order Derivative and Shared Set by One-Way Yi Li School of Science, Southw est Universi t y of Scie nce a nd Te chn olo gy, Mianyang, China E-mail: liyi@swust.edu.cn Received October 7, 2011; revised October 28, 2011; accepted November 7, 2011 Abstract Let F be a meromorphic functions family on the unit disc Δ, If for every fF (the zeros of f is a multi-plicity of at least k) and if 0fz thenkfzc and __ __kffESES (,Sab), then F is nor-mal on Δ. Keywords: Meromorphic Function, Normality Criterion, Shared Set by One-Way, Higher Order Derivative 1. Introduction and Results First, we introduce the following definition: Definition: For are two distinct complex values, we have set and ,ab,bSa__ __, :)0,ffES Eabzfza fzbzD 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 the normal meromorphic functionsfamilyconcern-ing one-way sharing set, W. H. Zhang proved the fol-lowing result : Theorem A. Let F be a family of meromorphic func-tions in the unit disc , a and b be two distinct nonzero complex values. , If for every ,SabfF, all of whose zeros is multiple, fS'fES E, 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 B. Let F be meromorphic functions fam-ily in the unit disk , a and b be two distinct nonzero complex values. If for every , all of whose zeros is multiplicity at least (k is a positive integer), 1fFk SkffIn 2008, F. J. Lv got following theorem in : ESE, then F is normal in . Theorem C. Let F be a family of meromorphic function in the unit disk , a and b is two distinct non-zero complex values, is positive integer, k,Sab . If for every fF, all of whose zeros have multiplicity 1k at least, kffESES, then F is normal in . In this paper,we continue to discuss about normality theorem of meromorphic functions families concerning higher order derivative and shared set by one-way, and obtain main results as follow. Theorem. Let F be a meromorphic functions family on the unit disc Δ, a and b is two distinct nonzero com-plex values, is positive integer, , If for every kF,bSaf, all of whose zeros have multiplicity at least, If k1) exists ， 0csuch that 0kfzfz c; 2) ;  __ __kffESESthen F is normal in . Remark: Through the following example, all of whose zeros have multiplicity at least is necessary. kExample: Let = 1,. 1 , and 1, kiSi12 1, 2. a family of meromorphic function in the unit disk  be nFfz there 12,1,2,3zznfz nneekn,. Obvously  nfzf z and Y. LI323 11111ee, 1,2,3,,lzzllnfznl k , Hence satisfy 1) and 2) and all of whose zeros of 0znfz have multiplicity 1k at most, Since   #1220010ffnfn, F is not normal at by Marty Theorem. 0z 2. Lemmas Lemma 1 . Let F be meromorphic functions fami-lies in the unit disk , all of whose zeros have multi-plicity at least. If for every kfF, there exists , such that 0AkfzA when eve0fz. If F is not normal in , then for every 0k, there exists , 01rnzr, nfF, 0n, such thatnn nnnfzg converges locally and uni- formly to a noncontant meromorphic function g, and . where  ##0ggkA1 #21ggg is said to be Spherical derivative of g. Lemma 2 . Let fz be nonconstant meromor-phic function in C   _____1,,1lim, ,rNrfaaf aTrf ,  _____ ,,1lim,rNrffTrf then .  ,,aCaf f2 3. Proof of Theorem Suppose that F be not normal in , then by Lemma 1 we get that there exists , and nznfF 0n such that nn nnknfzg converges locally and uniformly to a noncontant meromorphic function g , and  ##01ggkc. We claim that the following conclusions hold. 1) zeros of g have multiplicity at least, and k ()k0ggc ; 2) ,kkgag b; In fact, suppose that there exists 0, such that 00g, there exists 0n, 0nnn nknfznng for sufficiently large n, Hence nn nnfz0. Therefore, the following conclusions is obviously j, 2,3,,1nnnfz k0, 1nj  and knnnnfz ck, Hence 0, 1,2,3,,1jjjknn nngf jnnnz  and knngc, So 0lim 1,2,3,,1ngg j0,jjnnk  and 0kgc, Hence, zeros of g have multiplicity at least, kand kg0gc, Therefore, conclusion (1) is hold In what follow, we complete the proof of the claim 2): Suppose that there exists 0 Such that 0kga, by Hurwitz theorem, there exists 0n such that  nn nnnnkkgfza for sufficiently large n. By conditions:  __ __kffESES, or nn nnfzab, Hence   0limlim nn nnnn knnnfzgg  , this is a contradictions for 0kga． If kga, then g is polynomial of degree k. Because, zeros of g have multiplicityat least, kthen0!kagk and  101!kagk Obviously  100#00,11! 20,111!!kkakkgaakk  Because 0kgac, then Copyright © 2011 SciRes. APM Y. LI Copyright © 2011 SciRes. APM 324 ,, ,21kk kkag bggk11! 1!ackckk 2  1c, Suppose is not general, this is a contradictions for Lemma 2. The proof of Theo-rem is completed. therefore 12kkc, hence #0 1gkc, this is a con- contradictions for conditions of g. Hence kga. Similar to prove kgb. 4. References Therefore  W.-H. Zhang, “The Normality of Meromorphic Functions Concerning One-Way Sharing Set,” Journal of Nanhua University, Vol. 18, 2004, pp. 36-38. _____1,,1lim,kkkrNrgaag Trg 1 and  W.-H. Zhang, “The Normality of Meromorphic Func-tions,” Journal of Nanhua University, Vol. 12, No. 6, 2004, pp. 709-711. _____1,,1lim,kkkrNrgbbg Trg F.-J. Lv and J.-T. Li, “Normal Families Related to Shared Sets,” Journal of Chongqing University: English Edition, Vol. 7, No. 2, 2008, pp. 155-157. 1,  X. C. Pang and L. Zalcman, “Normal Families and Shared Values,” Bulletin of London Mathematical Society, Vol. 32, No. 3, 2000, pp. 325-331. doi:10.1112/S002460939900644X Since  __ __,,,,1kkTrg NrgkNrg NrgkNrg  W. K. Hayman, “Meromorphic Function,” Oxford Uni-versity Press, London, 1964. So  J. Clunie and W. Hayman, “The Spherical Derivative of Integral and Mer-Functions,” Commentarii Mathematici Helvetici, Vol. 40, 1966, pp. 117-148. doi:10.1007/BF02564366 _____ ,1,1lim 111,kkrNrg kgkkTrgk   Therefore