International Journal of Engineering Research and Development (IJERD)

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International Journal of Engineering Research and Development (IJERD)

  1. 1. International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 8, Issue 8 (September 2013), PP. 53-61 www.ijerd.com 53 | Page On The Zeros of a Polynomial in a Given Circle M. H. Gulzar Department of Mathematics University of Kashmir, Srinagar 190006 Abstract:- In this paper we discuss the problem of finding the number of zeros of a polynomial in a given circle when the coefficients of the polynomial or their real or imaginary parts are restricted to certain conditions. Our results in this direction generalize some well- known results in the theory of the distribution of zeros of polynomials. Mathematics Subject Classification: 30C10, 30C15 Keywords and phrases:- Coefficient, Polynomial, Zero. I. INTRODUCTION AND STATEMENT OF RESULTS In the literature many results have been proved on the number of zeros of a polynomial in a given circle. In this direction Q. G. Mohammad [6] has proved the following result: Theorem A: Let     0 )( j j j zazP be a polynomial o f degree n such that 0...... 011   aaaa nn . Then the number of zeros of P(z) in 2 1 z does not exceed 0 log 2log 1 1 a an  . K. K. Dewan [2] generalized Theorem A to polynomials with complex coefficients and proved the following results: Theorem B: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 0...... 011    nn . Then the number of zeros of P(z) in 2 1 z does not exceed 0 0 log 2log 1 1 a n j jn     . Theorem C: Let     0 )( j j j zazP be a polynomial o f degree n with complex coefficients such that for some real , , nja j ,......,2,1,0, 2 arg    and 011 ...... aaaa nn   . Then the number f zeros of P(z) in 2 1 z does not exceed
  2. 2. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 54 | Page 0 1 0 sin2)1sin(cos log 2log 1 a aa n j jn      . The above results were further generalized by researchers in various ways. M. H. Gulzar[4,5,6] proved the following results: Theorem D: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 0111 .......,......     knn , for .0,10,1 nk   Then the number of zeros of P(z) in 10,  z does not exceed 0 0 000 2)(2))(1( log 1 log 1 a k n j jnn      . Theorem E: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 011 ......   nn , for some ,10,0   ,then the number f zeros of P(z) in 10,  z , does not exceed 0 0 000 22)(2 log 1 log 1 a n j jnn     . Theorem F: Let     0 )( j j j zazP be a polynomial o f degree n with complex coefficients such that for some real , , nja j ,......,2,1,0, 2 arg    and 011 ...... aaaa nn    , for some ,0 ,then the number of zeros of P(z) in 10,  z , does not exceed 0 1 1 0 )1sin(cossin2)1sin)(cos( log 1 log 1 a aaa n j jn       . The aim of this paper is to find a bound for the number of zeros of P(z) in a circle of radius not necessarily less than 1. In fact , we are going to prove the following results: Theorem 1: Let     0 )( j j j zazP be a polynomial of degree n such that jja )Re( , jja )Im( and 0111 .......,......     knn ,
  3. 3. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 55 | Page for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR c R z does not exceed 0 0 000 1 ]2)(2))(1([ log log 1 a kR c n j jnn n      for 1R and 0 1 00000 ]2)())(1([ log log 1 a kRa c n j jnn     for 1R . Remark 1: Taking R=1 and  1 c in Theorem 1, it reduces to Theorem D . If the coefficients ja are real i.e. jj  ,0 , then we get the following result from Theorem 1: Corollary 1: Let     0 )( j j j zazP be a polynomial o f degree n such that 0111 .......,...... aaakaaaa nn    , for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR c R z does not exceed 0 000 1 )](2))(1([ log log 1 a aaaaakaaR c nn n   for 1R and 0 0000 )]())(1([ log log 1 a aaaaakaaRa c nn   for 1R . Applying Theorem 1 to the polynomial –iP(z) , we get the following result: Theorem 2: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 0111 .......,......     knn , for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR c R z does not exceed 0 0 000 1 ]2)(2))(1([ log log 1 a kR c n j jnn n      for 1R and 0 1 00000 ]2)())(1([ log log 1 a kRa c n j jnn    
  4. 4. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 56 | Page for 1R . Taking n in Theorem 1, we get the following result : Corollary 2: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 011 .......   nnk , for .0,10,1 nk   Then the number of zeros of P(z) in )1,0(  cR c R z does not exceed 0 0 000 1 ]2)(2)([ log log 1 a kR c n j jnn n     for 1R and 0 0 0000 ]2)()([ log log 1 a kRa c n j jnn    for 1R . Theorem 3: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 011 ......   nn , for some ,10,  ,then the number of zeros of P(z) in )1,0(  cR c R z , does not exceed 0 0 000 1 ]22)([ log log 1 a R c n j jnn n     for 1R and 0 0 0000 ]2)([ log log 1 a Ra c n j jnn    for 1R . Remark 2: Taking R=1 and  1 c in Theorem 3, it reduces to Theorem E . If the coefficients ja are real i.e. jj  ,0 , then we get the following result from Theorem 3: Corollary 3: Let     0 )( j j j zazP be a polynomial o f degree n such that 011 ...... aaaa nn    , for some .10,  Then the number of zeros of P(z) in )1,0(  cR c R z , does not exceed 0 000 1 ]2)([ log log 1 a aaaaaR c nn n   for 1R
  5. 5. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 57 | Page and 0 0000 ])([ log log 1 a aaaaaRa c nn   Applying Theorem 3 to the polynomial –iP(z) , we get the following result: Theorem 4: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 011 ......   nn , for some .10,0   Then the number of zeros of P(z) in )1,0(  cR c R z , does not exceed 0 0 000 1 ]22)([ log log 1 a R c n j jnn n     for 1R and 0 0 0000 ]2)([ log log 1 a Ra c n j jnn    for 1R . Taking 1,)1(  kk n in Corollary 3 , we get the following result: Corollary 4: Let     0 )( j j j zazP be a polynomial o f degree n such that jja )Re( , jja )Im( and 011 ......   nnk , for some .10,  Then the number of zeros of P(z) in )1,0(  cR c R z , does not exceed 0 0 000 1 ]22)()([ log log 1 a kR c n j jnn n     for 1R and 0 0 0000 ]2)()([ log log 1 a kRa c n j jnn    for 1R . Theorem 5: Let     0 )( j j j zazP be a polynomial o f degree n such that 011 ...... aaaa nn    for some .10,  Then the number of zeros of P(z) in )1,0(  cR c R z , does not exceed }]2)1sin(cos)1sin)(cos{([ 1 log log 1 00 1 0   n n aR ac
  6. 6. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 58 | Page for 1R and }])1sin(cos)1sin)(cos{([ 1 log log 1 000 0 aaaRa ac n   for 1R . For different values of the parameters ,,k in the above results, we get many other interesting results. II. LEMMAS For the proofs of the above results we need the following results: Lemma 1: If f(z) is analytic in Rz  ,but not identically zero, f(0)  0 and nkaf k ,......,2,1,0)(  , then    n j j i a R fdf 1 2 0 log)0(log(Relog 2 1     . Lemma 1 is the famous Jensen’s theorem (see page 208 of [1]). Lemma 2: If f(z) is analytic and )()( rMzf  in rz  , then the number of zeros of f(z) in 1,  c c r z does not exceed )0( )( log log 1 f rM c . Lemma 2 is a simple deduction from Lemma 1. Lemma 3: Let     0 )( j j j zazP be a polynomial o f degree n with complex coefficients such that for some real , , ,0, 2 arg nja j    and ,0,1 njaa jj   then any t>0,  sin)(cos)( 111   jjjjjj aataatata . Lemma 3 is due to Govil and Rahman [4]. III. PROOFS OF THEOREMS Proof of Theorem 1: Consider the polynomial F(z) =(1-z)P(z) )......)(1( 01 1 1 azazazaz n n n n    0011 1 )(......)( azaazaaza n nn n n    1 110 1 )(......)(        zzaza n nn n n       n j j jj ziz zzkk 0 10001 1 211 )()]()[( ......)(])1()[(       For Rz  , we have by using the hypothesis 0 1 )( aRazF n n   +      RkRRn nn 1 1 11 ......     RRRRk 001 1 21 )1(......)1(           n j jj 1 1 )( 
  7. 7. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 59 | Page   )1(......[ 1110 1    kkRaRa nn nn n ]2)1(...... 1 1 000121      n j jn   nn nn n kRaRa     00000 1 )())(1([ ]2 1 1     n j j 000 1 2)())(1([     kR nn n ]2 0   n j j for 1R and   n j jnn kRazF 1 00000 ]2)())(1([)(   for 1R . Therefore , by Lemma 3, it follows that the number of zeros of F(z) and hence P(z) in )0,0(  cR c R z does not exceed 0 0 000 1 2)(2))(1([ log log 1 a kR c n j jnn n      for 1R and 0 1 00000 2)())(1([ log log 1 a kRa c n j jnn     for 1R . That proves Theorem 1. Proof of Theorem 3: Consider the polynomial F(z) =(1-z)P(z) )......)(1( 01 1 1 azazazaz n n n n    0011 1 )(......)( azaazaaza n nn n n    2 1210 1 )(......)( zzzaza n nn nn n       n j j jj ziz 0 10001 )()]()[(  . For Rz  , we have by using the hypothesis 0 1 )( aRazF n n   + RRRR n nn n 01 2 121 ......    + j n j jj RR   0 10 )()1(  011210 1 ......[     nnn n R   n j j 0 0 2)1(  ]
  8. 8. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 60 | Page ]22)([ 0 000 1    n j jnn n R  for 1R and   n j jnnRazF 0 0000 ]2)([)(  for 1R . Therefore , by Lemma 2, it follows that the number of zeros of F(z) and hence P(z) in )0,0(  cR c R z does not exceed 0 0 000 1 ]22)([ log log 1 a R c n j jnn n     for 1R and 0 0 0000 ]2)([ log log 1 a Ra c n j jnn    for 1R . That proves Theorem 3. Proof of Theorem 5: Consider the polynomial F(z) =(1-z)P(z) )......)(1( 01 1 1 azazazaz n n n n    0011 1 )(......)( azaazaaza n nn n n    2 1210 1 )(......)( zaazaazaza n nn nn n     zaaaa )]()[( 0001   . For Rz  , we have by using the hypothesis and Lemma 3 0 1 )( aRazF n n   + RaaRaaRaaR n nn n 01 2 121 ......    + R0)1(  n nnnn nn n RaaaaRaRa ]sin)(cos)[( 110 1     Raaaa RaRaaaa ]sin)(cos)[( )1(]sin)(cos)[(...... 0101 2 0 2 1212     ]2)1sin(cos)1sin)(cos[( 00 1 aaR n n    for 1R and ])1sin(cos)1sin)(cos[( 000 aaaRa n   for 1R . Hence , by Lemma 2, it follows that the number of zeros of F(z) and therefore P(z) in )0,0(  cR c R z does not exceed }]2)1sin(cos)1sin)(cos{([ 1 log log 1 00 1 0   n n aR ac
  9. 9. On The Zeros of a Polynomial in a Given Circle www.ijerd.com 61 | Page for 1R and }])1sin(cos)1sin)(cos{([ 1 log log 1 000 0 aaaRa ac n   for 1R . That completes the proof of Theorem 5. REFERENCES [1]. L.V.Ahlfors, Complex Analysis, 3rd edition, Mc-Grawhill [2]. K. K. Dewan, Extremal Properties and Coefficient estimates for Polynomials with restricted Zeros and on location of Zeros of Polynomials, Ph.D. Thesis, IIT Delhi, 1980. [3]. N. K. Govil and Q. I. Rahman, On the Enestrom- Kakeya Theorem, Tohoku Math. J. 20(1968),126- 136. [4]. M. H. Gulzar, On the Zeros of a Polynomial inside the Unit Disk, IOSR Journal of Engineering, Vol.3, Issue 1(Jan.2013) IIV3II, 50-59. [5]. M. H. Gulzar, On the Number of Zeros of a Polynomial in a Prescribed Region, Research Journal of Pure Algebra-2(2), 2012, 35-49. [6]. M. H. Gulzar, On the Zeros of a Polynomial, International Journal of Scientific and Research Publications, Vol. 2, Issue 7,July 2012,1-7. [7]. Q. G. Mohammad, On the Zeros of Polynomials, Amer. Math. Monthly 72(1965), 631-633.

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