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![]() 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 [() n f Pf f ] where P(f) is a polynomial which generalize some result due to Abhijit Banerjee and Sonali Mukherjee [1]. 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 [2]. For any nonconstant meromorphic function f z 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 (, )Srf r )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,.., k ii Eazf zfzaziikstfz 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 g f zand () g z 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(,() k f Eaz(), k Eaz f za and () g za have same zeros with the same multiplicities k. Moreover, we also use the following notations. We denote by ), k Nrf the counting function for poles of f(z) with multiplicities and by ,k ), k Nrf 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 (, k Nrf be the corresponding one for which the multiplicity is not counted. Set (2 ( ,, ,, kk NrfrfrNNrNf .f Similarly, we have the notations ))(( 1111 ,, ,, ,, ,,, 1 kk kkk Nrr Nrr Nr fff NN . f f Let f(z) and g(z) be two nonconstant meromorphic functions and 1,1, .EE f g We denote by 1 ,1 L Nr f the counting function for 1-points of both () f z and () g z about which () f z has larger multi- plicity than () g z, with multiplicity is not being counted, and denote by 1] 1 ,1 Nr f 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 ,. 1 L Nr g In 2002 Fang and Fang [3] and in 2004 Lin-Yi [4] 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)n 2 (1) n f f f and 2 )(1 n g g g share 1 CM, then f ≡ g. In 2004 Lin-Yi [5] improved Theorem A by general- izing it in view of fixed point. Lin-Yi [5] proved the fol- lowing result. Theorem B ([5]). Let f and g be two transcendental meromorphic functions and be an integer. If (13)n 2 (1) n f f f and 2 )(1 n g g g share z CM, then .fg With the notion of weighted sharing of value recently the first author [6] improved Theorem A as follows. Theorem C ([6]). Let f and g be two nonconstant meromorphic functions and 12 2Θ;2Θ;minΘ;,Θ;, n fg fg 2 is an integer. If n 1 f ff and n 12 g gg share then“ 1, 2” fg. In the mean time Lahiri and Sarkar [7] 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 ([7]). Let f and g be two nonconstant meromorphic functions such that 2 1 n f ff and 2 1 n g g g share “”, where is an in- teger then either or . If n is an even integer then the possibility of does not arise. 1, 2 g n13 fg fg f In 2008, Banerjee and Murkherjee [1] proved the fol- lowing theorem. Theorem E ([1]). Let f and g be two transcendental meromorphic functions such that 2n f afbfc f and 2n g agbgcg where and 0a0bc share “”. Then the following holds: ,2 1) If and 0, 0bc max 122Θ;2Θ;minΘ;,Θ;, 42 Θ;Θ; n f gf fg g 0 be an integer, where , then Θ;Θ;fg . f g 2) If and 0, 0bc max 122Θ;2Θ; min Θ;,Θ; nf fg 2 g , 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; nf fg 2 g and the roots of the equation coincides, then 0azbz c . f g 4) If 0, 0cb and max 122Θ;2Θ; min Θ;,Θ; nf fg g , then either f g or f g . If n is an even integer then the possibility f g does not arise. Here, we obtain unicity theorem when [() n f Pf f ] and [() n g Pgg ] share a small function. Theorem 1. Let f and g be two transcendental meromorphic functions. Let 11 0 and 1mm f (),(0), mm m Pfafaf a a (0,1,,) i ai ma is the first nonzero coefficient from the right, and n, m, k be a positive integer with 10 2Θ;2Θ; min Θ;;Θ;. nmfg fg n If [() f Pf f ] and [() n g Pgg ] share “ ” then ,2 . f g 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. 22 11 f fgg hff gg 1111 11 11 22 and . 11 FFGG HFG FG Lemma 2.1. ([1]) If for a positive integer k, ,0; 0 k Nr 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;,. k pk Nr fNfrfrf fp f NS N rfr Lemma 2.2. ([1]) Let , f g be share “ ” and 1, 2 0h . Then 22 2 23 ,,0;,;,0; ,;,0;,, . TrfN rfN rfN rg g Nrgrp SrfNS g rg Copyright © 2011 SciRes. AM ![]() H. P. WAGHAMORE ET AL. 1366 ,. Lemma 2.3. ([8]) Let f be a nonconstant meromorphic function and , where are constants and . Then 01 n n Pfa afaf 0 n a 01 ,, , n aa a ,,TrPfnTrf Srf Lemma 2.4. Let 1 () n f Pf f F and 1 () n g Pgg G , 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,,. TrFnmTrfTrfSrf nmTrfSrf And 1 ,, 1 ,,',, n nmTrf TrfPfO TrFTrfSrf that is, 1 ,2,TrFnmTrfSrf ,,.Srf ,. Hence 1 In the same way we can prove SrF 1Srg,SrG,. This proves the Lemma. Lemma 2.5. ([9]) If and 0h ,0;,;,0;,; lim sup1, r NNNrfr frgrg Tr N rI then or . fg1fg ,fg Lemma 2.6. Let be two nonconstant mero- morphic functions. Then 2, nn fPffgPgg where is an integer. 6nm Proof. Let 2. nn fPffgPgg (2.1) Let be a 1-point of f with multiplicity . Then is a pole of g with multiplicity such that i.e., 0 z 0 z np (1)p 1)(q 11 pnqqmq , 2( 1)() mqnpq (2.2) From (2.2) we get 1 n qm and again from (2.2) we obtain 1( 1) 1 2 1 nm nnm pnm 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 1 z z()Pf(1)p 1) 1(q 1 2111(2pnmqnm 1 (3 2 nm p) . Sin f ce a pole of () n g Pg is either a zero of or a zero of , g we have 0 0 0 ,0; ,0;' ,, 2 ,;,0; ,0; 13 ,0;,, 2 ,,0; 13 , ,; ,0; m m NNfrg rg Srf Srg m rfNrg Nrg nm nm rgSrf Srg mm Trgr g nm nm N Sr N f N ,,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 1mm Pfafafafa 11 mm 0 where a10 ,,a are m distinct complex num nd fundal , mm abers. Then ament theorem of Nevanlinna we get by seco 0 ,,;,0; ,0;mTrfr frfrfNNN 1 0 0 ,; , ,0;,;,; ,0;( ,) 2,(,) 13 ,0; ,0; ,,. m j j m ra fSr f rfr fraf rf Srf mm Trg Trf nm nm rg N NNN N Nrf Srf Srg N (2.3) Similarly, we have 00 2 , , 13 ,0;,0;,,.N m mTr g N Trf nm nm rf rgSrfSrg (2.4) Adding (2.3) and (2.4) we obtain , m Tr g 24 1Tr , (,) 13 ,, g Trf nm nm Srf Srg which is a contradiction. This proves the Lemma. mero-Lemma 2.7. Let f and g be two transcendental morphic function and 1n m a Ff 1 10 11 mm m a a f f mn mnn 11 10 11 nmm mm aa Ggg g mn mnn a Copyright © 2011 SciRes. AM ![]() H. P. WAGHAMORE ET AL.1367 where is an integer. Then n( 2)m '' F G im lies that p F G. Proof. Let '' F G, then F Gc c is a constant 0. Then by second fundtal theo- rem w where t. Leamen c e get ; ,;,0;,; 1 ,0;, ;, 1 2,,,, ,. m m m m F a rf rfrf mn a rgrg Srf mn NNN N TrfmTrfTrg mTrg Srf N Hence we get ,, ,0;,;(,)TrFr rFrcFSrFNNN 1, 2, 1,, m nTrfmTrf mTrgSrf . (2.5) Similarly, we have (2.6) Adding (2.5) and (2.6) we obtain 1, 2, , ,. m nTrgmTrg rfSrg 1mT 1,,32 , , mnTrf TrgmTrf rg 32 (,),mTrg SrfS i.e., (2), ,,,nmTrfT rgS rfSrg . which is a contradiction. So and the Lemma is ed. a no 0c prov Lemma 2.8. ([10]) Let f benconstant meromor- phic function. Then ,0;, ;,0;,. k Nrfk rfNrfNfSr Lemma 2.9. Let F and be given as in Lemma 2.7 and G 1 F , 1 G ben byma 2.4. Then give Lem where are roots of the algebraic equation 11 1 , 0 ,; ,;,0;' , m m r b f Nrc fNrfSrf 1) ,,;,; ,; TrFTrFNrfNrbf NNrc f 11 1 2) ,,,0;,; ,; ,; ,;,0;', m m T rGT rGN rgN rbg Nrb gNrcg Nrc gNrgSrg 12 ,,, m bb b 1 10 0 11 mn n mm mm aa a zz mn and are roots of the algebraic equat on oof. By therem and Lemmas 2.3 we obtain 12 ,,, m cc c i 1 10 0. mm mm aza za Pr Nevanlinna’s first fundamental theo 11 1 1 ,,1 1 ,0;,,0;'1 ,,0;,0;, ,,0;,; ,; ,;, ;,0;',. m m TrFTrO F F Nr Fmrmr FO F TrFNr FNr FSrF TrFNrfNrbfNrbf Nrc fNrcfNrfSrf Similarly, we have 1 ,0 ; , F Nr FmrO 11 12 ,, ,0;,; ,; ,; ,; ,0; , m m T rGT rGNrgN rb g NrbgNrcgNrc g r gSrg ,;NrcgN where are roots of the algebraic equation 12 ,,, m bb b 1 10 0 11 mm mm aa a zz mn mnn and 12 ,,cre roots of the algebraic equation , m c c a 1 10 0. mm mm aza za Th is proves the Lemma. 3. Proofs of the Theorems rPoof of Theorem 1. Let , F G be defined as in Lemma 2.7 and 11 and F G be defined as ollows that and in Lemma 2.4. Then it f F G share “ ,2 ” and hence 1 F and G1 share “ ,2 ”. Suppose 0H. Then by Lem- ma 2.2, 2.4 and (2.3) we get 1 1 1 1 ,0; ,; ,;,0;' 2,0;2,; ,;, ;,0; ,,. m m r FF Sr Nrcf Nrc fNrfrgrg NrcgN rcgN rg Srf Srg NN (3.1) Now from Lemma 2.3, 2.8 and 2.9 we can obtain from (3.1) for 1 2221 21 (,), ;,0; ,;, , 2,0; 2,; TrFNN rN rG Nr Gf Srg rfrfN N 0 1, ,0;2,0;2, ; (, ),0; 3, (,)2,; 3,0; ,,. mn Trf rfr grg mT r gNrg mTrfTrg rf rgSrf Srg NN N N (3.2) 2, 0;2,;(,)rfr fmTrf N NN Copyright © 2011 SciRes. AM ![]() H. P. WAGHAMORE ET AL. Copyright © 2011 SciRes. AM 1368 .Sr In a similar manner we can obtain .Sr (3.3) From (3.2) and (3.3) we get 1, 2113Θ;2Θ;2 mn Trf mgfTr 1, 2113Θ;2Θ;2 mn Trg mfgTr 10 2Θ;2Θ; 2 . nmf g minΘ;;Θ; f g T Sr r (3.4) Since is arbitrary, (3.4) implies a contrad- tion. He. Since for we have 0 nce H ic 0 0 ( ,1) ,,;,;(,1) (, ) 2Θ;,,1, rf mrfNrfrfmr f Srf ,0; ,rfTrf m . f Trfmr fSrf N N te that We no 1 1 1 , ,; ,0;,;,; ,; ,0;,0; ,;, ;,;,0;. 282Θ;2Θ;2() ,0; . m m r G rf rcfrcf rf rfrg rc grcgrgrg mfgTr NN N NNN r NN Sr NN g ,0 ;mr fm 11 1 ,; 0;,0;rFF rrGNNNN (3.5) Also using Lemma 2.3 we get ,,1 ,,1 (, ;) ,(,;) ,,(1). n n nn n TrFmrfmrfP ffmrf Nrf Pff mrfP fNrfP f TrfPfnmTrfO (3.6) Similarly ,,1 ,TrGmrgnmTrgO 1. (3.7) From (3.6) and (3.7) we get 11 max,,,,1TrFTrGn mTrmrf .(3. By (3.5) and (3.8) applying Lemma 2.5 we get either 11 F G or 111.FG Now from Lemma 2.6 it follows111.FG that Again 11 F G implies . F G So from Lem . 4. Acknowledgements Researce first y, Bangalore under the project U.O. No.: 1. . References nction,” Archivum Mathe- p. 41-56. ue Distribution Theory,” Springer, Berlin, . 44, No. 5-6, ma 2.7 the theo- rem follows h work of th author is supported by Banga- lore Universit DEV:D2:YRB-BUIRF:2010-1 5 [1] A. Banerjee and S. 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