In this paper, we generalize two theorems on the uniqueness of nonlinear differential polynomials sharing 1-points, which improves a result of Lahiri and Pal 7. Â© 2017 Pushpa Publishing House, Allahabad, India

Rajeshwari, S.

Waghamore, H.P.

ePrints@Bangalore University

 Far East Journal of Mathematical Sciences (FJMS) © 2017 Pushpa Publishing House, Allahabad, India http://www.pphmj.com http://dx.doi.org/10.17654/MS101010001 Volume 101, Number 1, 2017, Pages 1-13 ISSN: 0972-0871 Received: March 28, 2016;  Revised: May 13, 2016;  Accepted: June 24, 2016 2010 Mathematics Subject Classification: 30D35. Keywords and phrases: uniqueness, meromorphic function, nonlinear differential polynomial. GENERALIZATION OF NONLINEAR DIFFERENTIAL POLYNOMIALS SHARING 1-POINTS Harina P. Waghamore and S. Rajeshwari Department of Mathematics Central College Campus Bangalore University Bangalore 560 001, India   Abstract In this paper, we generalize two theorems on the uniqueness of nonlinear differential polynomials sharing 1-points, which improves a result of Lahiri and Pal [7]. 1. Introduction, Definitions and Main Results Let f and g be two nonconstant meromorphic functions defined in the open complex plane .C  Let k be a positive integer or infinity and { }∞∈a  .C∪  We denote by ( )faEk ;)  the set of all a-points of f with multiplicities not exceeding k, where an a-point is counted according to its multiplicity. If for some { } ,C∪∞∈a  ( ) ( )gaEfa ;, )) ∞∞ =E  we say that f, g share the value a CM (counting multiplicities). In [4], the problem of uniqueness of meromorphic functions when two linear differential polynomials share the same 1-points was studied. Harina P. Waghamore and S. Rajeshwari 2 Regarding the nonlinear differential polynomials the following question was asked in [4]: What can be said if two nonlinear differential polynomials generated by two meromorphic functions share 1 CM? Some works have already been done in this direction [1, 2, 8, 9]. Recently Fang and Fang [2] and Lin and Yi [9] proved the following result. Theorem A. Let f and g be two nonconstant meromorphic functions and ( )13≥n  be an integer. If ( ) fff n ′− 21  and ( ) gggn ′− 21  share the value 1 CM, then .gf ≡  In 2006, Lahiri and Pal [7] investigated the uniqueness problem of meromorphic functions when two nonlinear differential polynomials share the value 1 and proved the following two theorems, the first of which improves Theorem A. Theorem B. Let f and g be two nonconstant meromorphic functions and ( )13≥n  be an integer. If ( ( ) ) ( ( ) ),1;11;1 2)32)3 gggEfffE nn ′−=′−  then .gf ≡  Theorem C. Let f and g be two nonconstant meromorphic functions and ( )14≥n  be an integer. If ( ( ) ) ( ( ) ),1;11;1 3)33)3 gggEfffE nn ′−=′−  then .gf ≡  In this paper, we generalize and improve Theorems A, B and C and obtain the following results. Theorem 1.1. Let f and g be two nonconstant meromorphic functions and ( )11+≥ mn  be an integer. If ( ( ) ) ( ( ) ),1;11;1 )3)3 gggEfffE mnmn ′−=′−  then .gf ≡  Theorem 1.2. Let f and g be two nonconstant meromorphic functions and ( )11+≥ mn  be an integer. If Generalization of Nonlinear Differential … 3 ( ( ) ) ( ( ) ),1;11;1 )3)3 gggEfffE mnmn ′−=′−  then .gf ≡  Remark. (1) If 2=m  in Theorem 1.1, then Theorem 1.1 reduces to Theorems A and B. (2) If 3=m  in Theorem 1.2, then Theorem 1.2 reduces to Theorem C. Though for the standard notations and definition of value distribution theory we refer [3], in the following definition we explain a notation used in the paper. Definition 1.1. Let f be a meromorphic function and { }.∞∈ ∪Ca  For a positive integer p we denote by ( )farN p ;,  the counting function of a-points of f, where an a-point of multiplicity m is counted m times if pm ≤  and is counted p times if .pm >  2. Lemmas In this section, we present some lemmas which will be needed in the sequel. Lemma 2.1. Let f and g be two nonconstant meromorphic functions. Then ( ) ( ) ,111 ≡/′−′− gggfff mnmn  where n is an integer. Proof. If possible let ( ) ( ) .111 ≡′−′− gggfff mnmn  Let 0z  be an 1-point of f with multiplicity ( ).1≥p  Then 0z  is a pole of g with multiplicity ( )1≥q  such that ( ) 2111 ++≥+++=−+ mnqmnpmp  and so .13+++≥ mmnp  Let 1z  be a zero of f with multiplicity ( )1≥p  and it be a pole of g with multiplicity ( ).1≥q  Then Harina P. Waghamore and S. Rajeshwari 4 ,11 +++=−+ qmqnqpnp  i.e., ( ) ( ) .21 +=−+ mqqpn  Hence .1mmnp −+≥  Since a pole of f is either a zero of ( )1−gg  or a zero of ,g′  we get ( ) ( ) ( ) ( )grNgrNgrNfrN ′++≤∞ ;0,;1,;0,;, 0  ( ) ( ) ( )grNgrNmnmgrNmnm ′+++++−+≤ ;0,;1,31;0,1  ( ) ( ),;0,,311 0 grNgrTmnmmnm ′+⎟⎠⎞⎜⎝⎛++++−+≤  where ( )grN ′;0,0  is the reduced counting function of those zeros of g′  which are not the zeros of ( ).1−gg  By the second fundamental theorem, we obtain ( ) ( ) ( ) ( ) ( ) ( )frSfrNfrNfrNfrNfrT ,;0,;1,;,;0,, 0 +′−+∞+≤  ( ) ( )frNmnmfrNmnm ;1,31;0,1 ++++−+≤  ( )grTmnmmnm ,311 ⎟⎠⎞⎜⎝⎛++++−++  ( ) ( ) ( ),,;0,;0, 00 frSfrNgrN +′−′+  i.e., ( ) ( )grTmnmmnmfrTmnmmnm ,311,3111 ⎟⎠⎞⎜⎝⎛++++−+≤⎟⎠⎞⎜⎝⎛+++−−+−  ( ) ( )frNgrN ′−′+ ;0,;0, 00  ( )., frS+  (2.1) Generalization of Nonlinear Differential … 5 Similarly, we get ( ) ( )frTmnmmnmgrTmnmmnm ,311,3111 ⎟⎠⎞⎜⎝⎛++++−+≤⎟⎠⎞⎜⎝⎛+++−−+−  ( ) ( )grNfrN ′−′+ ;0,;0, 00  ( )., grS+  (2.2) Adding (2.1) and (2.2), we get ( ) ( ) ( ){ } ( ) ( ),,,,,312121 grSfrSgrTfrTmnmmnm +≤+⎟⎠⎞⎜⎝⎛+++−−+−  which is a contradiction. This proves the lemma. Lemma 2.2 [10]. Let f be a nonconstant meromorphic function and ( )fP  ,2210nn fafafaa ++++= "  where naaaa ...,,,, 210  are constants and .0≠ma  Then ( )( ) ( ) ( ).,,, frSfrnTfPrT +=  Lemma 2.3. Let ( ) ,111111 10 ⎥⎦⎤⎢⎣⎡+−+++−++=−++nfmnmfmnmfF mmncmcn "  ( ) ,111111 10 ⎥⎦⎤⎢⎣⎡+−+++−++=−++ngmnmgmnmgG mmncmcn "  where ( )3+> mn  is an integer. Then GF ′≡′  implies .GF ≡  Proof. Let .GF ′≡′  Then ,cGF +≡  where c is a constant. If possible, let .0≠c  Then by the second fundamental theorem, we get ( ) ( ) ( ) ( ) ( )FrSFrNFcrNFrNFrT ,;,;,;0,, +∞++≤  ( ) ( ) ⎟⎟⎠⎞⎜⎜⎝⎛+++∞+≤mc fmnmrNfrNfrN ;1,;,;0,0  Harina P. Waghamore and S. Rajeshwari 6 ( ) ( )frSgmnmrNgrN mc ,;1,;0,0 +⎟⎟⎠⎞⎜⎜⎝⎛++++  ( ) ( ) ( ) ( ) ( ).,,,,,2 frSgrmTgrTfrmTfrT ++++≤  Since by Lemma 2.2, ( ) ( ) ( ) ( ),,,1, frSfrTmnFrT +++=  it follows that ( ) ( ) ( ) ( ) ( ) ( ) ( ).,,1,2,1 grSgrTmfrTmfrTmn ++++≤++  (2.3) Similarly, we get ( ) ( ) ( ) ( ) ( ) ( ) ( ).,,1,2,1 frSfrTmgrTmgrTmn ++++≤++  (2.4) Adding (2.3) and (2.4), we obtain ( ) ( ) ( ){ } ( ) ( )grSfrSgrTfrTmn ,,,,2 +≤+−−  which is a contradiction. So 0=c  and the lemma is proved. Lemma 2.4 [9]. Let F and G be given as in Lemma 2.3. Then GF ≡  implies .gf ≡  Lemma 2.5 [6]. Let f, g are nonconstant meromorphic functions and ( ) ( )gEfE ;1;1 )3)3 =  then one of the following cases holds:   (i) ( ) ( ) { ( ) ( ) ( )frNgrNfrNgrTfrT ;,;0,;0,2,, 222 ∞++≤+  ( )} ( ) ( );,,;,2 grSfrSgrN ++∞+   (ii) ;gf ≡  (iii) .1≡fg  Lemma 2.6 [5]. Let f be a nonconstant meromorphic function and k be a positive integer. Then ( ( ) ) ( ) ( ) ( ).,;0,;,;0, 22 frSfrNfrNkfrN kk ++∞≤ +  Generalization of Nonlinear Differential … 7 Lemma 2.7. Let F and G be given as in Lemma 2.3. Then  (i) ( ) ( ) ( ) ( ) ( )fbrNfbrNfrNFrTFrT ;,;,;0,,, 21 +++′≤  ( ) ( ) ( )fcrNfcrNfbrN m ;,;,;, 21 −−++"  ( ) ( ) ( );,;0,;, frSfrNfcrN m +′−−−"  (ii) ( ) ( ) ( ) ( ) ( )gbrNgbrNgrNGrTGrT ;,;,;0,,, 21 +++′≤  ( ) ( ) ( )gcrNgcrNgbrN m ;,;,;, 21 −−++"  ( ) ( ) ( ).,;0,;, grSgrNgcrN m +′−−−"  Proof. By the Nevanlinna’s first fundamental theorem and Lemma 2.2, we get ( ) ( )11,, OFrTFrT +⎟⎠⎞⎜⎝⎛=  ( ) ( )11,;0, OFrmFrN +⎟⎠⎞⎜⎝⎛+=  ( ) ( ) ( )1;0,,;0, OFrmFFrmFrN +′+⎟⎠⎞⎜⎝⎛ ′+≤  ( ) ( ) ( ) ( )FrSFrNFrNFrT ,;0,;0,, +′−+′=  ( ) ( ) ( ) ( )fbrNfbrNfrNFrT ;,;,;0,, 21 +++′≤  ( ) ( ) ( )fcrNfcrNfbrN m ;,;,;, 21 −−++"  ( ) ( ) ( ).,;0,;, frSfrNfcrN m +′−−−"  Similarly, we get ( )., GrT  This proves the lemma. Lemma 2.8. Let f and g be two nonconstant meromorphic functions. Then ( ) ( ) ,111 ≡/′−′− gggfff mnmn  where n is a positive integer. Proof. If possible let ( ) ( ) .111 ≡′−′− gggfff mnmn  Let 0z  be a 1-point Harina P. Waghamore and S. Rajeshwari 8 of f with multiplicity p. Then 0z  is a pole of g with multiplicity q, say, such that ( ) ( ) ,21111 ++≥+++=−− mnqmnpm  i.e., .13−++≥ mmnp  Hence ( ) .311;1 ++−−>Θ mnmf  Similarly, we can now show that ( ) 311; ++−−≥ωΘ mnmf  and ( ) ,311;2 ++−−≥ωΘ mnmf  where ω is the imaginary cube root of unity. Therefore ( ) ( ) ( ) ( ) ,23133;;;1 2 >++−−≥ωΘ+ωΘ+Θ mnmfff  a contradiction. This proves the lemma. Lemma 2.9. Let ,11111 ⎥⎦⎤⎢⎣⎡+−++=+nmnffFmn  ,11111 ⎥⎦⎤⎢⎣⎡+−++=+nmnggGmn  where 2≥n  is an integer. If ,11 GF ≡  then .gf ≡  Proof. Let .fgh =  If possible, suppose that h is nonconstant. Since ,11 GF ≡  it follows that .111111−−⋅+++= +++mnnmhhnmnf  Generalization of Nonlinear Differential … 9 Since mf  has no simple pole, it follows that 0=− kuh  has no simple root for ,3...,,2,1 += nk  where .12 ⎟⎠⎞⎜⎝⎛++π= mnikk eu  Hence ( ) 21; >Θ huk  for ,3...,,2,1 += nk  which is impossible. Therefore h is a constant. If ,1≠h  it follows that f is a constant, which is not the case. So 1=h  and hence .gf ≡  This proves the lemma. Lemma 2.10. If 1F  and 1G  be defined as in Lemma 2.9. Then  (i) ( ) ( ) ( ) ⎟⎠⎞⎜⎝⎛+++++′≤ mfnmnrNfrNFrTFrT ;11,;0,,, 11  ( ) ( ) ( ),,;0,;1, frSfrNfrN m +′−−  (ii) ( ) ( ) ( ) ⎟⎠⎞⎜⎝⎛+++++′≤ mgnmnrNgrNGrTGrT ;11,;0,,, 11  ( ) ( ) ( ).,;0,;1, grSgrNgrN m +′−−  The lemma can be proved in the line of the proof of Lemma 2.7. Lemma 2.11. Let 1F  and 1G  be defined as in Lemma 2.9, where ( )2+≥ mn  is an integer. Then 11 GF ′≡′  implies .11 GF ≡  The proof is similar to that of Lemma 2.3. 3. Proof of the Theorems In this section, we present the proofs of the main results. Proof of Theorem 1.1. Let F and G be defined as in Lemma 2.3. If possible, suppose that ( ) ( ) { ( ) ( ) ( )FrNGrNFrNGrTFrT ′∞+′+′≤′+′ ;,;0,;0,2,, 222  ( )} ( ) ( ).,,;,2 GrSFrSGrN ′+′+′∞+  Harina P. Waghamore and S. Rajeshwari 10 Then by Lemmas 2.2, 2.6 and 2.7, we get ( ) ( )GrTFrT ,, +  ( ) ( ) ( ) ( )fbrNfbrNfrNFrT ;,;,;0,, 21 +++′≤  ( ) ( ) ( )fcrNfcrNfbrN m ;,;,;, 21 −−++"  ( ) ( ) ( )GrTfrNfcrN m ′+′−−− ,;0,;,"  ( ) ( ) ( ) ( )gbrNgbrNgbrNgrN m ;,;,;,;0, 21 +++++ "  ( ) ( ) ( ) ( )grNgcrNgcrNgcrN m ′−−−−− ;0,;,;,;, 11 "  ( ) ( ) ( ) ( ){ }GrNFrNGrNFrN ′∞+′∞+′+′≤ ;,;,;0,;0,2 2222  ( ) ( ) ( ) ( )fbrNfbrNfbrNfrN m ;,;,;,;0, 21 +++++ "  ( ) ( ) ( ) ( )frNfcrNfcrNfcrN m ′−−−−− ;0,;,;,;, 21 "  ( ) ( ) ( ) ( )gbrNgbrNgbrNgrN m ;,;,;,;0, 21 +++++ "  ( ) ( ) ( ) ( )grNgcrNgcrNgcrN m ′−−−−− ;0,;,;,;, 21 "  ( ) ( )grSfrS ,, ++  ( ) ( ( ) ) ( ) ( )grNfrNfrNfrN m ;0,4;0,21;0,2;0,4 2 +′+−+≤  ( ( ) ) ( ) ( )frNgrNgrN m ;,4;0,21;0,2 2 ∞+′+−+  ( ) ( ) ( )fbrNfrNgrN ;,;0,;,4 1++∞+  ( ) ( ) ( ) ( )fcrNfcrNfbrNfbrN m ;,;,;,;, 212 −−+++ "  ( ) ( ) ( ) ( )gbrNgrNfrNfcrN m ;,;0,;0,;, 1++′−−−"  ( ) ( ) ( ) ( )gcrNgcrNgbrNgbrN m ;,;,;,;, 212 −−+++ "  ( ) ( ) ( ) ( ),,,;0,;, grSfrSgrNgcrN m ++′−−−"  Generalization of Nonlinear Differential … 11 ( ) ( )frTmn ,1++  ( ) ( ) ( ) ( ) ( ) ( )grSfrSgrmTgrTfrmTfrT ,,,2,11,2,11 +++++≤  ( ) ( ) ( ) ( ) ( ) ( ).,,,211,211 grSfrSgrTmfrTm +++++≤  So by Lemma 2.2, we get ( ) ( ) ( ){ } ( ) ( ),,,,,10 grSfrSgrTfrTmn +≤+−−  which is a contradiction. Hence by Lemma 2.5 either GF ′≡′  or .1≡′′GF  Since by Lemma 2.1 ,1≡/′′GF  it follows by Lemma 2.3 and Lemma 2.4 .gf ≡  This proves the theorem. Proof of Theorem 1.2. Let 1F  and 1G  be defined as in Lemma 2.9. If possible suppose that ( ) ( ) { ( ) ( ) ( )12121211 ;,;0,;0,2,, FrNGrNFrNGrTFrT ′∞+′+′≤′+′  ( )} ( ) ( ).,,;, 1112 GrSFrSGrN ′+′+′∞+  Then by Lemmas 2.2, 2.6 and 2.10, we get ( ) ( )11 ,, GrTFrT +  ( ) ( ) ( ) ( ){ }12121212 ;,;,;0,;0,2 GrNFrNGrNFrN ′∞+′∞+′+′≤  ( ) ( )frNfnmnrNfrN m ′−⎟⎠⎞⎜⎝⎛+++++ ;0,;11,;0,  ( ) ( ) ⎟⎠⎞⎜⎝⎛+++++− mm gnmnrNgrNfrN ;11,;0,;1,  ( ) ( ) ( ) ( )grSfrSgrNgrN m ,,;1,;0, ++−′−  ( ) ( ) ( )frNfrNfrN m ′++≤ ;0,2;1,2;0,4 22  ( ) ( ) ( )grNgrNgrN m ′+++ ;0,2;1,2;0,4 22  Harina P. Waghamore and S. Rajeshwari 12 ( ) ( ) ( )frNgrNfrN ;0,;,4;,4 +∞+∞+  ( ) ( )frNfrNfnmnrN mm ′−−⎟⎠⎞⎜⎝⎛++++ ;0,;1,;11,  ( ) ( )mm grNgnmnrNgrN ;1,;11,;0, −⎟⎠⎞⎜⎝⎛+++++  ( ) ( ) ( )grSfrSgrN ,,;0, ++′−  ( ) ( ) ( ) ( ) ( ) ( )grSfrSgrTmfrTm ,,,211,211 +++++≤  and so by Lemma 2.2, we get ( ) ( ) ( ){ } ( ) ( ),,,,,10 grSfrSgrTfrTmn +≤+−−  which is a contradiction. Hence by Lemma 2.5 either 11 GF ′≡′  or .111 ≡′′GF  Since by Lemma 2.8 ,111 ≡/′′GF  it follows by Lemmas 2.9 and 2.11 that .gf ≡  This proves the theorem. Acknowledgement The authors thank the anonymous referees for their valuable suggestions which led to the improvement of the manuscript. References  [1] M. L. Fang and W. Hong, A unicity theorem for entire functions concerning differential polynomials, Indian J. Pure Appl. Math. 32(9) (2001), 1343-1348.  [2] C. Y. Fang and M. L. Fang, Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl. 44(5-6) (2002), 607-617.  [3] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, The Clarendon Press, Oxford, 1964.  [4] I. Lahiri, Uniqueness of meromorphic functions when two linear differential polynomials share the same 1-points, Ann. Polon. Math. 71(2) (1999), 113-128. Generalization of Nonlinear Differential … 13  [5] I. Lahiri and A. Sarkar, Uniqueness of meromorphic function and its derivative, JIPAM J. Inequal. Pure Appl. Math. 5(1) (2005), Article 20 (electronic). http://jipam.vu.edu.au/.  [6] I. Lahiri and P. Sahoo, Uniqueness of non-linear differential polynomials sharing 1-points, Georgian Math. J. 12(1) (2005), 131-138.  [7] I. Lahiri and Rupa Pal, Non-linear differential polynomials sharing 1-points, Bull. Korean Math. Soc. 43(1) (2006), 161-168.  [8] W. C. Lin, Uniqueness of differential polynomials and a problem of Lahiri, Pure Appl. Math. 17(2) (2001), 104-110 (in Chinese).  [9] W. C. Lin and H. X. Yi, Uniqueness theorems for meromorphic function, Indian J. Pure Appl. Math. 35(2) (2004), 121-132.  [10] C. C. Yang, On deficiencies of differential polynomials II, Math. Z. 125 (1972), 107-112. 

Generalization of nonlinear differential polynomials sharing 1-points

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Generalization of nonlinear differential polynomials sharing 1-points

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