 Applied Mathematics, 2011, 2, 1364-1368 doi:10.4236/am.2011.211191 Published Online November 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Uniqueness of Meromorphic Functions with Their Nonlinear Differential Polynomials Share a Small Function Harina P. Waghamore, Tanuja Adaviswamy Department of Mat hematics, Ba ng al ore University, Bangalore, India E-mail: pree.tam@rediffmail.com, a.tanuja1@gmail.com Received August 22, 2011; revised September 27, 2011; accepted October 2, 2011 Abstract In this paper we deal with the uniqueness of meromorphic functions when two nonlinear differential poly-nomials generated by two meromorphic functions share a small function. We consider the case for some general differential polynomials [()nfPf f] where P(f) is a polynomial which generalize some result due to Abhijit Banerjee and Sonali Mukherjee . Keywords: Entire Functions, Meromorphic Functions, Nonlinear Differential Polynomials, Uniqueness 1. Introduction In this paper, we use the standard notations and terms in the value distribution theory . For any nonconstant meromorphic function fz on the complex plane C, we denote by any quantity satisfying = o as , except possibly for a set of r of finite linear measures. A meromorphic function is called a small function with respect to f(z) if . Let (, )Srfr)f(, )Srf(, ),Trfaz(, )(,TraSrSf()az be the set of meromorphic function in the complex plane C which are small func-tions with respect to f. Set , where a zero point with multiplicity m is counted m times in the set. If these zero points are only counted once, then we denote the set by (, )Eaz f: ()zf z0, ()()azS f(, ).fEa Let k be az a positive integer. Set  ),:0,,1,.., kiiEazfzfzaziikstfz az0 where a zero point with multiplicity m is counted m times in the set. Let f(z) and g(z) be two transcendental meromorphic functions, If , then we say that ()()( ).azS fSg()((),)((), )EazfEaz gfzand ()gz share the value CM, especially, we say that f(z) and g(z) have the same fixed points when . If ()az()az z(,)(, ),Eaf Eag then we say that f(z) and g(z) share the IM. If we say that ()az()))) (),g(,()kf Eaz(),kEazfza and ()gza have same zeros with the same multiplicities k. Moreover, we also use the following notations. We denote by ),kNrf the counting function for poles of f(z) with multiplicities and by ,k),kNrf the corresponding one for which the multiplicity is not counted. Let ,Nrf(k be the counting function for poles of f(z) with multiplicities and let ,k(,kNrf be the corresponding one for which the multiplicity is not counted. Set (2 (,, ,,kkNrfrfrNNrNf .f Similarly, we have the notations ))((1111,, ,, ,, ,,,1kk kkkNrr Nrr NrfffNN.ff Let f(z) and g(z) be two nonconstant meromorphic functions and 1,1, .EEfg We denote by 1,1LNrf the counting function for 1-points of both ()fz and ()gz about which ()fz has larger multi-plicity than ()gz, with multiplicity is not being counted, and denote by 1]1,1Nrf the counting function for common simple 1-points of both f(z) and g(z) where multiplicity is not counted. Similarly, we have the nota- tion 1,.1LNrg In 2002 Fang and Fang  and in 2004 Lin-Yi  in- H. P. WAGHAMORE ET AL.1365 dependently proved the following result. Theorem A ([3,4]). Let f and g be two nonconstant meromorphic functions and be an integer. If (13)n2(1)nfff and 2)(1nggg share 1 CM, then f ≡ g. In 2004 Lin-Yi  improved Theorem A by general-izing it in view of fixed point. Lin-Yi  proved the fol-lowing result. Theorem B (). Let f and g be two transcendental meromorphic functions and be an integer. If (13)n2(1)nfff and 2)(1nggg share z CM, then .fgWith the notion of weighted sharing of value recently the first author  improved Theorem A as follows. Theorem C (). Let f and g be two nonconstant meromorphic functions and  12 2Θ;2Θ;minΘ;,Θ;,nfg fg 2 is an integer. If n1fff and n12ggg share then“1, 2” fg. In the mean time Lahiri and Sarkar  also studied the uniqueness of meromorphic functions corresponding to nonlinear differential polynomials which are different from that of previously mentioned and proved the fol-lowing. Theorem D (). Let f and g be two nonconstant meromorphic functions such that 21nfffand 21nggg share “”, where is an in- teger then either or . If n is an even integer then the possibility of does not arise. 1, 2gn13fgfgfIn 2008, Banerjee and Murkherjee  proved the fol-lowing theorem. Theorem E (). Let f and g be two transcendental meromorphic functions such that 2nfafbfc f and 2ngagbgcg where and 0a0bc share “”. Then the following holds: ,21) If and 0, 0bcmax 122Θ;2Θ;minΘ;,Θ;,42Θ;Θ;nfgffg g0 be an integer, where , then Θ;Θ;fg.fg 2) If and 0, 0bcmax 122Θ;2Θ;min Θ;,Θ;nffg2g, the roots of the equation are distinct and one of f and g is nonentire meromorphic function having only multiple poles, then f ≡ g. 0azbz c3) If 0, 0bc and max 122Θ;2Θ;min Θ;,Θ;nffg2g and the roots of the equation coincides, then 0azbz c.fg 4) If 0, 0cband max 122Θ;2Θ;min Θ;,Θ;nffgg, then either fg or fg. If n is an even integer then the possibility fg does not arise. Here, we obtain unicity theorem when [()nfPf f] and [()ngPgg] share a small function. Theorem 1. Let f and g be two transcendental meromorphic functions. Let 110 and 1mmf(),(0),mm mPfafaf a a (0,1,,)iai ma is the first nonzero coefficient from the right, and n, m, k be a positive integer with 10 2Θ;2Θ;min Θ;;Θ;.nmfgfgn If [()fPf f] and [()ngPgg] share “” then ,2.fg 2. Lemmas In this section we present some lemmas which will be needed in the sequel. Let f, g, F1, G1 be four nonconstant meromorphic functions. Henceforth we shall denote by h and H the following two functions. 2211ffgghff gg   1111111122and .11FFGGHFGFG  Lemma 2.1. () If for a positive integer k, ,0; 0kNr ff denotes the counting function of those zeros of f which are not the zeros of f, where a zero of f with multiplicity m is counted m times if mk and k times if then mk1,0;0,0;,; ,0;,. kpkNr fNfrfrffpfNSNrfr Lemma 2.2. () Let ,fg be share “” and 1, 20h. Then   22 223,,0;,;,0;,;,0;,, .TrfN rfN rfN rggNrgrp SrfNSg rg Copyright © 2011 SciRes. AM H. P. WAGHAMORE ET AL. 1366 ,.Lemma 2.3. () Let f be a nonconstant meromorphic function and , where are constants and . Then 01nnPfa afaf 0na01,, ,naa a,,TrPfnTrf Srf Lemma 2.4. Let 1()nfPf fF and 1()ngPggG, where is a small function of f and g. Then and . 0,1,F Srf,,Sr1,,SrG SrgProof. Using Lemma 2.3 we see that   1,,, 2,,. TrFnmTrfTrfSrfnmTrfSrf And  1,, 1 ,,',, nnmTrf TrfPfOTrFTrfSrf that is,  1,2,TrFnmTrfSrf ,,.Srf,. Hence 1In the same way we can prove SrF1Srg,SrG,. This proves the Lemma. Lemma 2.5. () If and 0h,0;,;,0;,;lim sup1,rNNNrfr frgrgTrN rI then or . fg1fg,fgLemma 2.6. Let be two nonconstant mero-morphic functions. Then 2,nnfPffgPgg where is an integer. 6nmProof. Let 2.nnfPffgPgg (2.1) Let be a 1-point of f with multiplicity . Then is a pole of g with multiplicity such that i.e., 0z0znp(1)p1)(q11 pnqqmq ,2( 1)() mqnpq (2.2) From (2.2) we get 1nqm and again from (2.2) we obtain 1( 1)121nm nnmpnm m 1.)i.e., Let be a zero of with multiplicity 1. Then 1 is a pole of g with multiplicity , say. So from (2.1) we get 1zz()Pf(1)p1)1(q12111(2pnmqnm   1(32nmp). Sinfce a pole of ()ngPgis either a zero of or a zero of ,g we have   000,0; ,0;' ,,2,;,0; ,0;13 ,0;,,2 ,,0;13 ,   ,; ,0;  mmNNfrg rgSrf SrgmrfNrg Nrgnm nmrgSrf SrgmmTrgr gnm nmNSrNfN,,Srg where NNr rg0,0;Nr g denotes the reduced counting func- those zerotion ofs of g which are not the zeros of g()Pg. As 1mmPfafafafa11mm0 where a10,,a are m distinct complex numnd fundal,mmabers. Thenament theorem of Nevanlinna we get by seco0,,;,0; ,0;mTrfr frfrfNNN 100,; ,,0;,;,;,0;( ,)2,(,)13,0; ,0;,,.mjjmra fSr frfr frafrf SrfmmTrg Trfnm nmrgNNNNNNrfSrf SrgN  (2.3) Similarly, we have  002,,13 ,0;,0;,,.NmmTr gNTrfnm nmrf rgSrfSrg   (2.4) Adding (2.3) and (2.4) we obtain ,mTr g241Tr,(,)13,,g Trfnm nmSrf Srg  which is a contradiction. This proves the Lemma. mero-Lemma 2.7. Let f and g be two transcendentalmorphic function and 1nmaFf11011mmma af fmn mnn  111011nmmmmaaGgg gmn mnna Copyright © 2011 SciRes. AM H. P. WAGHAMORE ET AL.1367 where is an integer. Then n( 2)m ''FG im lies that pFG. Proof. Let ''FG, then FGc c is a constant 0. Then by second fundtal theo-rem w where t. Leamence get    ;,;,0;,;1,0;, ;,12,,,, ,.mmmmFarf rfrfmnargrg SrfmnNNNNTrfmTrfTrg mTrg SrfN    Hence we get ,, ,0;,;(,)TrFr rFrcFSrFNNN  1, 2, 1,,m nTrfmTrfmTrgSrf . (2.5) Similarly, we have  (2.6) Adding (2.5) and (2.6) we obtain  1, 2,, ,.m nTrgmTrgrfSrg1mT1,,32 ,,mnTrf TrgmTrfrg   32 (,),mTrg SrfS i.e., (2), ,,,nmTrfT rgS rfSrg  .which is a contradiction. So and the Lemma is ed. a no0c provLemma 2.8. () Let f benconstant meromor-phic function. Then ,0;, ;,0;,.kNrfk rfNrfNfSr Lemma 2.9. Let F and be given as in Lemma 2.7 and G 1F, 1G ben byma 2.4. Then give Lemwhere are roots of the algebraic equation111, 0,;,;,0;' ,mmrb fNrc fNrfSrf 1) ,,;,;,;TrFTrFNrfNrbfNNrc f  1112) ,,,0;,;,; ,;,;,0;',mmT rGT rGN rgN rbgNrb gNrcgNrc gNrgSrg  12,,,mbb b 110011mn n  mmmmaa azzmnand are roots of the algebraic equat on oof. By therem and Lemmas 2.3 we obtain 12,,,mcc c i1100.mmmmaza za Pr Nevanlinna’s first fundamental theo     1111,,11,0;,,0;'1,,0;,0;,,,0;,; ,;,;, ;,0;',.mmTrFTrOFFNr Fmrmr FOFTrFNr FNr FSrFTrFNrfNrbfNrbfNrc fNrcfNrfSrf   Similarly, we have 1,0; ,FNr FmrO 1112,, ,0;,;,; ,; ,;,0; ,mmT rGT rGNrgN rb gNrbgNrcgNrc gr gSrg,;NrcgN where are roots of the algebraic equation 12,,,mbb b 110011mmmmaa azzmn mnn   and 12,,cre roots of the algebraic equation ,mc c a1100.mmmmaza za  Th is proves the Lemma. 3. Proofs of the Theorems rPoof of Theorem 1. Let ,FG be defined as in Lemma 2.7 and 11andFG be defined asollows that and in Lemma 2.4. Then it fFG share “,2” and hence 1F and G1 share “,2”. Suppose 0H. Then by Lem- ma 2.2, 2.4 and (2.3) we get      1 111,0;,;,;,0;' 2,0;2,;,;, ;,0;,,.mmr FFSrNrcfNrc fNrfrgrgNrcgN rcgN rgSrf SrgNN  (3.1) Now from Lemma 2.3, 2.8 and 2.9 we can obtain from (3.1) for 1 222121(,), ;,0;,;, ,2,0; 2,;TrFNN rN rGNr Gf SrgrfrfN N 0  1,,0;2,0;2, ;(, ),0;3, (,)2,;3,0; ,,.mn Trfrfr grgmT r gNrgmTrfTrg rfrgSrf SrgNNNN   (3.2) 2,0;2,;(,)rfr fmTrfNNNCopyright © 2011 SciRes. AM H. P. WAGHAMORE ET AL. Copyright © 2011 SciRes. AM 1368 .SrIn a similar manner we can obtain .Sr(3.3) From (3.2) and (3.3) we get  1,2113Θ;2Θ;2mn TrfmgfTr  1,2113Θ;2Θ;2mn TrgmfgTr  10 2Θ;2Θ;2.nmf gminΘ;;Θ;fgTSr r (3.4) Since is arbitrary, (3.4) implies a contrad-tion. He. Since for we have 0 nce Hic00  ( ,1),,;,;(,1)(, )2Θ;,,1,rfmrfNrfrfmr fSrf,0; ,rfTrf m.fTrfmr fSrfNN te that We no 111, ,;,0;,;,;,; ,0;,0;,;, ;,;,0;.282Θ;2Θ;2(),0; .mmr Grf rcfrcfrf rfrgrc grcgrgrgmfgTrNN NNNNrNNSrNNg ,0;mr fm11 1,; 0;,0;rFF rrGNNNN  (3.5)Also using Lemma 2.3 we get  ,,1 ,,1(, ;),(,;),,(1).nnnnnTrFmrfmrfP ffmrfNrf PffmrfP fNrfP fTrfPfnmTrfO   (3.6) Similarly ,,1 ,TrGmrgnmTrgO1. 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