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International Journal of Engineering Research and Development (IJERD)
International Journal of Engineering Research and Development (IJERD)
International Journal of Engineering Research and Development (IJERD)
International Journal of Engineering Research and Development (IJERD)
International Journal of Engineering Research and Development (IJERD)
International Journal of Engineering Research and Development (IJERD)
International Journal of Engineering Research and Development (IJERD)
International Journal of Engineering Research and Development (IJERD)
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International Journal of Engineering Research and Development (IJERD)

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  • 1. International Journal of Engineering Research and Developmente-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.comVolume 6, Issue 11 (April 2013), PP. 68-7568On The Zero-Free Regions for PolynomialsM. H. GULZARDepartment of Mathematics University of Kashmir, Srinagar 190006Abstract: In this paper we find the zero-free regions for a class of polynomials whose coefficients ortheir real and imaginary parts are restricted to certain conditions.Mathematics Subject Classification: 30C10, 30C15Keywords and phrases:- Coefficient, Polynomial, Zero.I. INTRODUCTION AND STATEMENT OF RESULTSThe following result, known as the Enestrom-Kakeya Theorem [5] , is well-known in the theory of distributionof zeros of polynomials:Theorem A: Let njjj zazP0)( be a polynomial of degree n such that0...... 011   aaaa nn .Then P(z) has all its zeros in the closed unit disk 1z .In the literature there exist several generalizations and extensions of this result. Recently, Choo and Choi [1]proved the following results:Theorem B: Let njjj zazP0)( be a polynomial of degree n such thateither 132 ...... aaaa nn   and 0231 ...... aaaa nn   ,if n is odd , or022 ...... aaaa nn   and 1331 ...... aaaa nn   , if n is even. Then P(z)does not vanish in011110aaaaaaaaznnnn .Theorem C: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n, such that011 ......   nnk ,011 ......   nn .Then P(z) does not vanish in000)1(  nnnn kkaaz .Theorem D: Let njjj zazP0)( be a polynomial of degree n such that for some  ,2arg ja , j=0,1,……,n,and for some 1k ,011 ...... aaaak nn   .Then P(z) does not vanish in
  • 2. On The Zero-Free Regions for Polynomials69 11000sin2sin)(cos)(njjnnn aaakaakakaz.M. H. Gulzar [3,4] proved the following results:Theorem E: Let njjj zazP0)( be a polynomial of degree n such that for some 11 k , 12 k ,10 1  , 10 2  ,either 11321 ...... aaaak nn   and 022312 ...... aaaak nn   ,ifn is odd , or 01221 ...... aaaak nn   and 123312 ...... aaaak nn   , if n is even. Thenfor odd n, P(z) has all its zeros innnnaKaaz 11 ,where)()()(2)()( 1111011121`1 aaaaaaaakaakK nnnnnn   )( 002 aa  ,and for even n, P(z) has all its zeros innnnaKaaz 21 ,where)()()(2)()( 0011011121`2 aaaaaaaakaakK nnnnnn   )( 112 aa  .Theorem F: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n, such that011 ......   nnk .Then all the zeros of P(z) lie innnjjnnnnakakz10000 2)(2)1(.Theorem G: Let njjj zazP0)( be a polynomial of degree n such that for some  ,2arg ja , j=0,1,……,n,and for some 1k , 10 011 ...... aaaak nn   .Then P(z) has all its zeros in 11000 sin2)(2)sin(cos11njjnnaaaaakakz  .The aim of this paper is to generalize some of the above-mentioned results. More precisely, we prove thefollowing results:
  • 3. On The Zero-Free Regions for Polynomials70Theorem 1: Let njjj zazP0)( be a polynomial of degree n such that for some 11 k , 12 k ,10 1  , 10 2  ,either 11321 ...... aaaak nn   and 022312 ...... aaaak nn   ,ifn is odd , or 01221 ...... aaaak nn   and 123312 ...... aaaak nn   , if n is even. ThenP(z) does not vanish in01002111112102)()()()( aaaaaaaakaakaznnnn  , if n is odd,and in01112001112102)()()()( aaaaaaaakaakaznnnn  ,if n is even.Remark 1: Taking 11 k , 12 k , 11  , 12  , Theorem 1 reduces to Theorem B.If z is a zero of njjj zazP0)( , thennnnnnnnnaaaazaaaazz 1111  .Therefore, combining Theorem E and Theorem 1, we arrive at the following result:Corollary 1: Let njjj zazP0)( be a polynomial of degree n such that for some 11 k , 12 k ,10 1  , 10 2  ,either 11321 ...... aaaak nn   and 022312 ...... aaaak nn   ,ifn is odd , or 01221 ...... aaaak nn   and 123312 ...... aaaak nn   , if n is even. Thenfor odd n, P(z) has all its zeros in01002111112102)()()()( aaaaaaaakaakannnn   z )()()(2)()(11111011121` aaaaaaaakaakannnnnnn  1002 )(  naaa ,and for even n, P(z) has all its zeros in01112001112102)()()()( aaaaaaaakaakannnn   z )()()(2)()(10011011121` aaaaaaaakaakannnnnnn  1112 )(  naaa .Theorem 2: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n, such that011 ......   nnk .Then P(z) does not vanish in
  • 4. On The Zero-Free Regions for Polynomials71 11000002)()(njjnnnn kaaz.Remark 2: If in Theorem 2, we have, in addition,011 ......   nn ,then 01  nnjjj , and we have the following result:Theorem 3: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n, such that011 ......   nnk ,011 ......   nn .Then P(z) does not vanish in00000)()(  nnnn kaaz .Remark 3: Taking 1 , Theorem 3 reduces to Theorem C.Combining Theorem 2 and Theorem F, we get the following result:Corollary 2: : Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n, such that011 ......   nnk .Then P(z) has all its zeros in11000002)()(njjnnnn kaaz])1(2)(2[1110000   nnjjnnnakkaTheorem 4: Let njjj zazP0)( be a polynomial of degree n such that for some  ,2arg ja , j=0,1,……,n,and for some 1k , 10 011 ...... aaaak nn   .Then P(z) does not vanish in 11000sin2)1sin(cos)1sin(cosnjjn aaaakaz.Remark 4: Taking 1 , Theorem 4 reduces to Theorem D.Combining Theorem 4 and Theorem G, we get the following result:
  • 5. On The Zero-Free Regions for Polynomials72Corollary 3 : Let njjj zazP0)( be a polynomial of degree n such that for some  ,2arg ja , j=0,1,……,n,and for some 1k , 10 011 ...... aaaak nn   .Then P(z) has all its zeros in11000sin2)1sin(cos)1sin(cosnjjn aaaakaz nnjjnnakaaaaaka)1(sin2)(2)sin(cos1 11000  .II. LEMMAFor the proofs of the above results, we need the following lemma:Lemma: For any two complex numbers 0b and 1b such that 10 bb  and1,0,2arg  jbj for some real  , sin)(cos)( 101010 bbbbbb  .The above lemma is due to Govil and Rahman [2].III. PROOFS OF THE THEOREMSProof of Theorem 1: Let n be odd. Consider the polynomial)()1()( 2zPzzF ......)()()......)(1(131211201112nnnnnnnnnnnnnnzaazaazazaazazazaz01202313 )()( azazaazaa )(0 zqa  ,where......)()()( 1312112  nnnnnnnnnn zaazaazazazqzazaazaa 1202313 )()( 1312121112)()1()(  nnnnnnnnnnnn zaakzakzaakzaza20223113113112 )()1()(......)1( zaazazaazak nn   zaza 1202 )1(  For 1z ,......)1()1()( 123121111   nnnnnnnn akaakakaakaazq10202211113 )1()1( aaaaaaa  010021111121 2)()()()( aaaaaaaakaak nnnn   1M .
  • 6. On The Zero-Free Regions for Polynomials73Since q(z) is analytic for 1z and q(0)=0, it follows , by Rouche’s theorem, thatzMzq 1)(  for 1z .Hence, it follows that, for 1z ,)()( 0 zqazF zMazqa100 )(0if10Maz  .It is easy to see that 01 aM  and the proof is complete if n is odd.The proof for even n is similar.Proof of Theorem 2: Consider the polynomial)()1()( zPzzF ......)()()......)(1(121110111nnnnnnnnnnnnzaazaazaazazazaz001212 )()( azaazaa 21212111).....()()( zzzza nnnnnnnn   njjjj ziaz11001 )()( )(0 zqa  ,where.)(.....)()()( 21212111zzzzazq nnnnnnnn   njjjj ziz1101 )()( 12111)()1()(  nnnnnnnnnn zzkzkza njjjj zizzz11001212 )()1()()(...... For 1z ,12211 ......)1()(    nnnnnn kkazq)()1(11001 njjj 110000 2)()(njjnnnnn ka 2MSince q(z) is analytic for 1z and q(0)=0, it follows , by Rouche’s theorem, thatzMzq 2)(  for 1z .Hence, it follows that, for 1z ,
  • 7. On The Zero-Free Regions for Polynomials74)()( 0 zqazF zMazqa200 )(0if20Maz  .It is easy to see that 02 aM  and the proof is completeProof of Theorem 4: Consider the polynomial)()1()( zPzzF ......)()()......)(1(121110111nnnnnnnnnnnnzaazaazaazazazaz001212 )()( azaazaa )(0 zqa  ,where21212111)(.....)()()( zaazaazaazazq nnnnnnnn  zaa )( 01 12111)()1()(  nnnnnnnnnn zaazakzakazazazaazaa 001212 )1()()(......   .For 1z , we have , by using the lemma,12211 ......)1()( aaaaakakaazq nnnnnn  001 )1( aaa  nnnnnn akaakaaka )1(sin)(cos)( 11   0010112122121)1(sin)(cos)(sin)(cos)(......sin)(cos)(aaaaaaaaaaaaa nnnn 1100 sin2)1sin(cos)1sin(cosnjjn aaaak 3MSince q(z) is analytic for 1z and q(0)=0, it follows , by Rouche’s theorem, thatzMzq 3)(  for 1z .Hence, it follows that, for 1z ,)()( 0 zqazF zMazqa300 )(0if
  • 8. On The Zero-Free Regions for Polynomials7530Maz  .It is easy to see that 03 aM  and the proof is completeREFERENCES[1]. Y. Choo and G. K. Choi, On the Zero-free Regions of Polynomials, Int.Journal of Math. Analysis,Vol.5,2011,no. 20, 975-981.[2]. N. K. Govil and Q. I. Rahman, On the Enestrom-Kakeya Theorem,Tohoku Math.J. 20(1968), 126-136.[3]. M. H. Gulzar, Distribution of the Zeros of a Polynomial, Int. Journal Of Innovative Research andDevelopment, Vol.2 Issue 3, March 2013,416-427.[4]. M. H. Gulzar Some Refinements of Enestrom-Kakeya Theorem, , Int.Journal of Mathematical Archive-2(9), September 2011, 1512-1519.[5]. M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer.Math. Soc. (RI Providence) , 1966.

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