Further Generalizations of Enestrom-Kakeya Theorem

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Further Generalizations of Enestrom-Kakeya Theorem

  1. 1. International Journal of Engineering Inventionse-ISSN: 2278-7461, p-ISBN: 2319-6491Volume 2, Issue 3 (February 2013) PP: 49-58www.ijeijournal.com P a g e | 49Further Generalizations of Enestrom-Kakeya TheoremM. H. GulzarDepartment of Mathematics, University of Kashmir, Srinagar 190006Abstract:- Many generalizations of the Enestrom –Kakeya Theorem are available in the literature. In this paperwe prove some results which further generalize some known results.Mathematics Subject Classification: 30C10,30C15Keywords and phrases:- Complex number, Polynomial , Zero, Enestrom – Kakeya Theorem.I. INTRODUCTION AND STATEMENT OF RESULTSThe Enestrom –Kakeya Theorem (see[6]) is well known in the theory of the distribution of zeros ofpolynomials and is often stated as follows:Theorem A: Let njjj zazP0)( be a polynomial of degree n whose coefficients satisfy0...... 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. Joyal et al [5] extendedit to polynomials with general monotonic coefficients and proved the following result:Theorem B: Let njjj zazP0)( be a polynomial of degree n whose coefficients satisfy011 ...... aaaa nn   .Then P(z) has all its zeros innnaaaaz00  .Aziz and zargar [1] generalized the result of Joyal et al [6] as follows:Theorem C: Let njjj zazP0)( be a polynomial of degree n such that for some 1k011 ...... aaaka nn   .Then P(z) has all its zeros innnaaakakz001 .For polynomials ,whose coefficients are not necessarily real, Govil and Rahman [2] proved the followinggeneralization of Theorem A:Theorem C: If njjj zazP0)( is a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n, such that0...... 011    nn ,where 0n , then P(z) has all its zeros in))(2(10njjnz .Govil and Mc-tume [3] proved the following generalisations of Theorems B and C:
  2. 2. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 50Theorem D: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n. If for some 1k ,011 ......   nnk ,then P(z) has all its zeros innnjjnkkz 000 21 .Theorem E: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n. If for some 1k ,011 ......   nnk ,then P(z) has all its zeros innnjjnkkz 000 21 .M. H. Gulzar [4] proved the following generalizations of Theorems D and E:Theorem F: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n. If for some real numbers 0 , ,10  011 ......   nn ,then P(z) has all its zeros in the disknnjjnnz0000 2)(2.Theorem G: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n. If for some real number 0 ,011 ......   nn ,then P(z) has all its zeros in the disknnjjnnz0000 2)(2.In this paper we give generalization of Theorems F and G. In fact, we prove the following:Theorem 1: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1,2,……,n. If for some real numbers  , 0 , ,10,0,1   knank01111 ......    knknknnn ,and ,1 knkn    then P(z) has all its zeros in the disknnjjknknnn aaz 0000 2(21)1( ,and if ,1  knkn  then P(z) has all its zeros in the disk
  3. 3. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 51nnjjknknnn aaz 0000 2(21)1( Remark 1: Taking 1 , Theorem 1 reduces to Theorem F.If ja are real i.e. 0j for all j, we get the following result:Corollary 1: Let njjj zazP0)( be a polynomial of degree n. If for some real numbers  , 0 ,,0,1  knank ,10  01111 ...... aaaaaaa knknknnn    ,and ,1 knkn aa   then P(z) has all its zeros in the disknknknnn aaaaaaaaz)(21)1( 000  ,,and if ,1  knkn aa then P(z) has all its zeros in the disknknknnn aaaaaaaaz)(21)1( 000  If we apply Theorem 1 to the polynomial –iP(z) , we easily get the following result:Theorem 2: Let njjj zazP0)( be a polynomial of degree n with jja )Re( and jja )Im( ,j=0,1, 2,……,n. If for some real numbers  , 0 , ,0,1  knank ,10  01111 ......    knknknnn ,and ,1 knkn    then P(z) has all its zeros in the disknnjjknknnn aaz 0000 2)(21)1( ,and if ,1  knkn  then P(z) has all its zeros in the disknnjjknknnn aaz 0000 2)(21)1( Remark 2: Taking 1 , Theorem 2 reduces to Theorem G.Theorem 3: Let njjj zazP0)( be a polynomial of degree n such that for some real numbers  , 0 ,,10,,0,1   knank ,01111 ...... aaaaaaa knknknnn   and.,......,1,0,2arg nja j If knkn aa  1 (i.e. 1 ) , then P(z) has all its zeros in the disk
  4. 4. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 52nnknjjjknnn aaaaaaaz1,100 ]sin22)1cos(sin)1sincossin(cos)sin(cos[If 1  knkn aa (i.e. 1 ), then P(z) has all its zeros in the disknnknjjjknnn aaaaaaaz1,100 ]sin22)1cos(sin)1sincossin(cos)sin(cos[Remark 3: Taking 1 in Theorem 3 , we get the following result:Corollary 2: Let njjj zazP0)( be a polynomial of degree n. If for some real numbers 0 , ,10  011 ...... aaaa nn    ,then P(z) has all its zeros in the disknnjjnn aaaaaaz1,100 ]sin22)1cos(sin)sin(cos[ .Remark 4: Taking 1,)1(  kak n in Cor.2, we get the following result:Corollary 3: Let njjj zazP0)( be a polynomial of degree n. If for some real numbers 0 , ,10  011 ...... aaaak nn   ,then P(z) has all its zeros in the disknnjjnn aaaaakaz1,100 ]sin22)1cos(sin)sin(cos[ Taking 1,)1(  kak n , and 1 in Cor. 3, we get a result of Shah and Liman [7,Theorem 1].II. LEMMASFor the proofs of the above results , we need the following results:Lemma 1: . Let P(z)= njjj za0be a polynomial of degree n with complex coefficients such thatfornjaj ,,...1,0,2arg  some real . Then for some t>0,    .sincos 111    jaataatata jjjjjThe proof of lemma 1 follows from a lemma due to Govil and Rahman [2].Lemma 2.If p(z) is regular ,p(0)  0 and Mzp )( in ,1z then the number of zeros of p(z)in,10,  z does not exceed)0(log1log1pM(see[8],p171).
  5. 5. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 53III. PROOFS OF THEOREMSProof of Theorem 1: Consider the polynomial)()1()( zPzzF knknknknknknnnnnnnnnnzaazaazaazaazazazaz)()(......)()......)(1(111110111001121 )(......)( azaazaa knknkn  1111)(......)()(  knknknnnnnnn zzzi 01100011211)()1()(......)()(izizzzznjjjjknknknknknknIf ,1 knkn    then ,1 knkn    and we have1111)(......)()()(  knknknnnnnnnn zzzzizF ......)()1()( 1211  knknknknknknknkn zzaz .)()1()( 0110001  izizznjjjj  For 1z ,1112111)(......)()()(  knknknnnnnnnnnn zzzzzazF 01100011211)()1()(......)()1()(izizzzzaznjjjjknknknknknknknkn  112111)(......1)()( kknknnnnnnnzzzaz nnjjnjjnnnknknkknkknknzizizzzkzz0110101012111)()1(1)(....11)(1)1(1)( knknnnnnnnzaz    1211 ...... 01100012111......1njjjknknknknkn 11211 ......    knnknknnnnnnnknzaz  njjknknkn00001212)1(......1
  6. 6. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 54njjknknnnnzaz000022)(1)1({[>0if njjknknnn za0000 22)(1)1( This shows that the zeros of F(z) whose modulus is greater than 1 lie innnjjknknnn aaz 0000 2)(21)1( .But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all thezeros of F(z) lie innnjjknknnn aaz 0000 2)(21)1( .Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie innnjjknknnn aaz 0000 2)(21)1( .If ,1  knkn  then ,1  knkn  and we have1111)(......)()()(  knknknnnnnnnn zzzzizF .)()1()(......)()1()(011000112111izizzzzznjjjjknknknknknknknknFor 1z ,1112111)(......)()()(  knknknnnnnnnnnn zzzzzazF 011000112111)()1()(......)()1()(izizzzzznjjjjknknknknknknknkn  112111)(......1)()( kknknnnnnnnzzzaz nnjjnjjnnnkknknkknkknknzizizzzzzz011010101121111)()1(1)(....1)(1)1(1)( knknnnnnnnzaz    1211 ......
  7. 7. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 55 01100012111......1njjjknknknknkn 11211 ......    knnknknnnnnnnknzaz  njjknknkn00001212)1(......1njjknknnnnzaz0000}22)(1)1({>0if njjknknnn za0000 22)(1)1( This shows that the zeros of F(z) whose modulus is greater than 1 lie innnjjknknnn aaz 0000 2)(21)1( .But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all thezeros of F(z) lie innnjjknknnn aaz 0000 2)(21)1( .Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie innnjjknknnn aaz 0000 2)(21)1( .That proves Theorem 1.Proof of Theorem 3: Consider the polynomial)()1()( zPzzF knknknknknknnnnnnnnnnzaazaazaazaazazazaz)()(......)()......)(1(111110111001121 )(......)( azaazaa knknkn   .If knkn aa  1 , then knkn aa  1 , 1 and we have, for 1z , by using Lemma1,1112111)(......)()()(  knknknnnnnnnnnn zaazaazaazzazF 00011211)1()(......)()1()(azazaazaazazaa knknknknknknknkn 
  8. 8. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 56  112111)(......1)()( kknknnnnnnnzaazaaaazaz nnnkknknkknkknknzazazaazaazazaa0101011211)1(1)(....1)(1)1(1)( knknnnnnnnaaaaaazaz   1211 ...... 00012111......1aaaaaaaaa knknknknkn   sin)(cos){( 11   nnnnnnaaaazaz cos)(......sin)(cos)( 12121 knknnnnn aaaaaa  (  sin)(cos)(sin) 111   knknknknknkn aaaaaa sin)(cos)(1 2121   knknknknkn aaaaa})1(sin)(cos)(...... 000101 aaaaaa    sincossin(cos)sin(cos{  knnnnaazaz}sin22)1cos(sin)11,100 nknjjjaaa 0if  sincossin(cos)sin(cos  knnn aaza1,100 sin22)1cos(sin)1nknjjjaaa This shows that the zeros of F(z) whose modulus is greater than 1 lie innnknjjjknnn aaaaaaaz1,100 ]sin22)1cos(sin)1sincossin(cos)sin(cos[. .But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all thezeros of F(z) lie innnknjjjknnn aaaaaaaz1,100 ]sin22)1cos(sin)1sincossin(cos)sin(cos[.Since the zeros of P(z) are also the zeros of F(z), it follows that all the zeros of P(z) lie innnknjjjknnn aaaaaaaz1,100 ]sin22)1cos(sin)1sincossin(cos)sin(cos[
  9. 9. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 57If 1  knkn aa , then 1  knkn aa , 1 and we have, for 1z , by using Lemma 1,1112111)(......)()()(  knknknnnnnnnnnn zaazaazaazzazF 000112111)1()(......)()1()(azazaazaazazaa knknknknknknknkn   112111)(......1)()( kknknnnnnnnzaazaaaazaz nnnkknknkknkknknzazazaazaazazaa01010112111)1(1)(....1)(1)1(1)( knknnnnnnnaaaaaazaz    1211 ...... 00012111......1aaaaaaaaa knknknknkn   sin)(cos){( 11   nnnnnnaaaazaz cos)(......sin)(cos)( 12121 knknnnnn aaaaaa  (  sin)(cos)(sin) 111   knknknknknkn aaaaaa sin)(cos)(1 2121   knknknknkn aaaaa})1(sin)(cos)(...... 000101 aaaaaa    sincossin(cos)sin(cos{  knnnnaazaz}sin22)1cos(sin)11,100 nknjjjaaa 0if  sincossin(cos)sin(cos  knnn aaza1,100 sin22)1cos(sin)1nknjjjaaa This shows that the zeros of F(z) whose modulus is greater than 1 lie innnknjjjknnn aaaaaaaz1,100 ]sin22)1cos(sin)1sincossin(cos)sin(cos[.But the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence all thezeros of F(z) and therefore P(z) lie innnknjjjknnn aaaaaaaz1,100 ]sin22)1cos(sin)1sincossin(cos)sin(cos[That proves Theorem 3.
  10. 10. Further Generalizations of Enestrom-Kakeya Theoremwww.ijeijournal.com P a g e | 58REFERENCES1) A.Aziz and B.A.Zargar, Some extensions of the Enestrom-Kakeya Theorem , Glasnik Mathematiki,31(1996) , 239-244.2) N.K.Govil and Q.I.Rehman, On the Enestrom-Kakeya Theorem, Tohoku Math. J.,20(1968) , 126-136.3) N.K.Govil and G.N.Mc-tume, Some extensions of the Enestrom- Kakeya Theorem , Int.J.Appl.Math.Vol.11,No.3,2002, 246-253.4) M. H. Gulzar, On the Zeros of polynomials, International Journal of Modern Engineering Research,Vol.2,Issue 6, Nov-Dec. 2012, 4318- 4322.5) A. Joyal, G. Labelle, Q.I. Rahman, On the location of zeros of polynomials, Canadian Math.Bull.,10(1967) , 55-63.6) M. Marden , Geometry of Polynomials, IInd Ed.Math. Surveys, No. 3, Amer. Math. Soc. Providence,R.I,1996.7) W. M. Shah and A.Liman, On Enestrom-Kakeya Theorem and related Analytic Functions, Proc.Indian Acad. Sci. (Math. Sci.), 117(2007), 359-370.8) E C. Titchmarsh,The Theory of Functions,2ndedition,Oxford University Press,London,1939.

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