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  • 1. International Journal of Computational Engineering Research||Vol, 03||Issue, 4||www.ijceronline.com ||April||2013|| Page 172On The Zeros of Polynomials and Analytic FunctionsM. H. GulzarDepartment of Mathematics University of Kashmir, Srinagar 190006I. INTRODUCTION AND STATEMENT OF RESULTSRegarding the zeros of a polynomial , Jain [2] proved the following results:Theorem A: LetnnppppzazazazaazP ............)(1110be a polynomial of degreen such that ppaa 1for some  np ,......,2,1 ,,11 nnpjjjpaaaMM   nnaMnp  ),1(,111pjjjpaamm .0),2( 1 mnpThen P(z) has at least p zeros in1)( 1paaMpzpp,provided11)( 1 paaMp ppand11110paaMppaaMpmappppp.Theorem B: LetnnppppzazazazaazP ............)(1110be a polynomial of degree nsuch that ppaa 1for some  np ,......,2,1 ,,,.......,1,0,2arg nja jfor some real  and  and011...... aaaa nn .Then P(z) has at least p zeros in11paaLpzpp,wherenpjjjpnnpaaaaaLL11sin)(cos)( andAbstractIn this paper we obtain some results on the zeros of polynomialsand related analytic functions, which generalize and improve upon the earlier well-known results.Mathematics Subject Classification: 30C10, 30C15Key-words and phrases: Polynomial, Analytic Function , Zero.
  • 2. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 17311101sin)(cos)(pjjjppaaaall  0),12( 1 lnp ,provided111011pppppppaaLppaaLpla .In this paper, we prove the following results:Theorem 1: LetnnppppzazazazaazP ............)(1110be a polynomial of degree nsuch that ppaa 1for some  1,......,2,1  np and ,00111............ aaaaaa ppnn  .Then P(z) has at least p zeros in1)()2(100paaMpKzaKaaKa ppnnn,wherepnnaaaM  2 ,provided 11011011ppppppppaaMpaapaaMpaand K<1.Remark 1: Taking 0 in Theorem 1, we get the following result:Corollary 1: LetnnppppzazazazaazP ............)(1110be a polynomial of degree nsuch that ppaa 1for some  1,......,2,1  np0111............ aaaaaa ppnn .Then P(z) has at least p zeros in1)(2111100paaMpKzaKaaa ppnnn,wherepnnaaaM 1,provided 1110111011ppppppppaaMpaapaaMpaand 11K .This result was earlier proved by Roshan Lal et al [4] .If the coefficients are positive in Theorem 1, we have the following result:Corollary 2: LetnnppppzazazazaazP ............)(1110be a polynomial of degree nsuch that ppaa 1for some  1,......,2,1  np and ,0.0............ 0111 aaaaaa ppnnThen P(z) has at least p zeros in1)()})1(2{1222020paaMpKzaKaKa ppnn,wherepnaaM  )(22 ,provided
  • 3. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 174 1120112011ppppppppaaMpaapaaMpaand 12K .If the coefficients of the polynomial P(z) are complex, we prove the following result:Theorem 2: LetnnppppzazazazaazP ............)(1110be a polynomial of degree nsuch that ppaa 1for some  1,......,2,1  np and ,00111............ aaaaaa ppnn and for some real  and  ,.,......,1,0,2arg nja jThen P(z) has at least p zeros in1133paaMpKzpp,where1113sin)(cos)1sin)(cos(npjjjpnaaaaM  ,provided11313011pppppppaaMpmpaaMpaRemark 2: Taking 0 in Theorem 2, it reduces to Theorem B.Remark 3: It is easy to see that 3K <1.Next, we prove the following result on the zeros of analytic functions :Theorem 3: Let 0)(0 jjjzazf be analytic in 4Kz  and for some natural number p withpaapp 121,............ 11210  pppaaaaaa ,for some 0 ,and 111010121ppppppppppaaapaapaaapa  .Then f(z) has at least p zeros in pppaaappKz141.Remark 4: Taking 0 , Theorem 3 reduces to the following result:Corollary 3: Let 0)(0 jjjzazf be analytic in 4Kz  and for some natural number p withpaapp 121,
  • 4. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 175............ 11210  pppaaaaaa ,for some 0 ,and 11101011ppppppppppaaapaapaaapa .Then f(z) has at least p zeros in pppaaappKz141.Cor.3 was earlier proved by Roshan Lal et al [4].II. LEMMAFor the proofs of the above results , we need the following lemma due to Govil and Rahman [1]:Lemma : If 1a and 2a are complex numbers such that,2,1,2arg  ja j for some real numbers  and  ,then sin)(cos)( 212121aaaaaa  .III. PROOFS OF THE THEOREMS3.1 Proof of Theorem 1: Consider the polynomial)()1()( zPzzF jnpjjjppppjjjjnnppppzaazaazaaazazazazaaz11111101110)()()()............)(1(1nnza)()( zz   ,where)( z 1110)(pjjjjzaaaand1111)()()( nnnpjjjpppzaaazaaz .For Kz  (<1), we have, by using the hypothesis, 1111)(nnpjnjjjpppKaKaaKaaz 11)1(11111)(npjpnjjpnnnpnnppppKaaKaaKaKKaa pnnnnppppaaaaaKKaa   1111)(  pnnnnppppaaaaaKKaa   1111)(  pnnppppaaaKKaa 2)(11
  • 5. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 176ppnnppppaaaaapaa2)()(11 pnnppnnppaaaaaaaap22)(111112pppppnnpaaaaap111pppppaaMp(1)Also for Kz  (<1),1110)(pjjjjKaaaz1110)(pjjjaaKa)( 010aaKa p 12)( 0110paaaaaaapappnnpp 01101aapaaMpa ppp (2)Since, by hypothesis 11011011ppppppppaaMpaapaaMpa ,it follows from (1) and (2) that)()( zz   for Kz  .Hence, by Rouche’s theorem, )( z and )()( zz   i.e. F(z) have the same number of zeros in Kz Since the zeros of P(z) are also the zeros of F(z) and since )( z has at least p zeros in Kz  ,it followsthat P(z) has at least p zeros in Kz  .That prove the first part of Theorem 1.To prove the second part, we show that P(z) has no zero innnnKaaaaz002 .Let )()1()( zPzzF 11101110)()............)(1(nnjnjjjnnppppzazaaazazazazaaz)(0zga  ,where
  • 6. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 1771111)()( nnnjjjjzazaazg .For Kz  (<1), we have, by using the hypothesis,111)( nnjnjjjzazaazg111 nnjnjjjKaKaa nnnjjjnnKaaaaaK1111)( nnnnnKaaaaaK   011 nnnnnKaaaaaK   011 nnnKaaaK  02  .Since g(z) is analytic for Kz  , g(0)=0, we have, by Schwarz’s lemma, zKaaaKzgnnn 02)(  for Kz  .Hence, for Kz  ,)()( 0zgazF  zKaaaKazgannn0002)(0if)2( 00nnnKaaaKaz.This shows that F(z) and therefore P(z) has no zero in)2( 00nnnKaaaKaz.That proves Theorem 1 completely.Proof of Theorem 2: Consider the polynomial)()1()( zPzzF jnpjjjppppjjjjnnppppzaazaazaaazazazazaaz11111101110)()()()............)(1(1nnza)()( zz   ,where)( z 1110)(pjjjjzaaaand
  • 7. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 1781111)()()( nnnpjjjpppzaaazaaz ..For 3Kz  (<1), we have, by using the hypothesis, 1313131)(nnpjnjjjpppKaKaaKaaz 11)1(3113131331npjpnjjpnnnpnnppppKaaKaaKaKKaa 11111331npjjjnnnppppaaaaaKKaa  11111331npjjjnnnppppaaaaaKKaa    cos}){( 11331 nnnppppaaaKKaasin)(cos)(......sin)(cos)(sin}){(2121111nnnnppppnnaaaaaaaaaa  cos)1sin)(cos(1331 pnppppaaKKaa 111sin)(npjjjaa 31331MKKaapppp311313111MpaaMppaaMpaapppppppp1131pppppaaMpmpaaMpapp1130mKa  30(3)Also , for 3Kz  , we have, by using the lemma and the hypothesis,jpjjjzaaaz 1110)(1113011310pjjjpjjjjaaKaKaaa
  • 8. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 179 .......sin)(cos)( 010130  aaaaKa     sin)(cos)( 2121  ppppaaaa    sin)(cos)( 10130 jjpaaaaKamKa  30(4)Thus, for 3Kz  , we have from (3) and (4), )()( zz   .Since )( z and )( z are analytic for 3Kz  , it follows by Rouche’s theorem that )( z and)()( zz   i.e. F(z) have the same number of zeros in 3Kz  . But the zeros of P(z) are also the zeros ofF(z). Therefore, we conclude that P(z) has at least p zeros in Kz  , as the same is true of )( z . That provesTheorem 2.Proof of Theorem 3: Consider the function)()1()( zfzzF 11111102210)()()(......))(1(pjjjjppppjjjjzaazaazaaazazaaz),()( zz  where1110)()(pjjjjzaaaz ,111)()()(pjjjjpppzaazaaz .For 4Kz  ( 4K <1, by hypothesis forpaapp 121), we have )1(4111441)(pjpjjjppppKaaKKaaz 111441pjjjppppaaKKaapppppaKKaa1441pppppppppaaaappaaappaa  ))(1())(1(111111ppppppaaap. (5)and11410)(pjjjjKaaaz  41221100...... Kaaaaaaa pp 
  • 9. On The Zeros Of Polynomials...www.ijceronline.com ||April||2013|| Page 180  41221100..... Kaaaaaaa pp      410041221100412211002..........KaaaKaaaaaaaKaaaaaaappppp 101021 ppppaapaaapa  (6)Since, by hypothesis, 111010121ppppppppppaaapaapaaapa  ,it follows from (5) and (6) that )()( zz   for 4Kz  .Since )( z and )( z are analytic for 3Kz  , it follows, by Rouche’s theorem, that )( z and)()( zz   i.e. F(z) have the same number of zeros in 3Kz  . But the zeros of P(z) are also the zeros ofF(z). Therefore, we conclude that P(z) has at least p zeros in Kz  , as the same is true of )( z . That provesTheorem 3.REFERENCES[1] N. K. Govil and Q. I. Rahman, On the Enestrom-Kakeya Theorem, Tohoku Math. J.20 (1968), 126-136.[2] V.K. Jain, On the zeros of a polynomial, Proc.Indian Acad. Sci. Math. Sci.119 (1) ,2009, 37-43.[3] M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math. Soc. Providence, RI, 1966.[4] Roshan Lal, Susheel Kumar and Sunil Hans, On the zeros of polynomials and analytic functions,Annales Universitatis Mariae Curie –Sklodowska Lublin Polonia, Vol.LXV, No. 1, 2011, Sectio A,97-108.