On the Zeros of A Polynomial Inside the Unit Disc

International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 13, Issue 2 (February 2017), PP.67-70

On the Zeros of A Polynomial Inside the Unit Disc M.H. Gulzar Department Of Mathematics, University Of Kashmir, Srinagar

ABSTRACT: In this paper we find the number of zeros of a polynomial inside the unit disc under certain conditions on the coefficients of the polynomial.

Mathematics Subject Classification: 30C10, 30C15.

Keywords and Phrases: Coefficients, Polynomial, Zeros.

I.

INTRODUCTION

In the context of the Enestrom-Kakeya Theorem [4] which states that all the zeros of a polynomial n

P( z )   a j z j with a n  a n 1  ...... a1  a0  0 lie in z  1 , Q. G. Mohammad [ 5] proved the j 0

following result giving a bound for the number of zeros of P(z) in z 

1 : 2

n

Theorem A: Let

P( z )   a j z j be a polynomial of degree n such that j 0

a n  a n 1  ...... a1  a0  0 , Then the number of zeros of P(z) in z 

a 1 1 log n . does not exceed 1  2 log 2 a0

Various bounds for the number of zeros of a polynomial with certain conditions on the coefficients were afterwards given by researchers in the field (e.g. see [1],[2],[3]).

II.

MAIN RESULTS

In this paper we find a bound for the number of zeros of a polynomial in a closed disc of radius less than 1 and prove n

Theorem 1: Let

P( z )   a j z j be a polynomial of degree n with Re( a j )   j , Im(a j )   j , j 0

j  0,1,2,......,n such that for some  ,0    n  1 and for some k  1, o    1 , k n   n1  ......   1    and

L       1    1    2  ...... 1   0   0 .

Then the number of zeros of P(z) in

z   ,0    1 does not exceed n

a n  (k  1)  n  k n     L  (1   )    2  j

1 log

1



Taking a j real i.e.

j 0

log

a0

.

 j  0, j  0,1,2,.....,n , Theorem 1 reduces to the following result: n

j Corollary 1: Let P( z )   a j z be a polynomial of degree n such that for some  ,0    n  1 and for j 0

some k  1, o    1 ,

kan  a n 1  ...... a 1  a 67

On The Zeros Of A Polynomial Inside The Unit Disc and

L  a  a 1  a 1  a 2  ...... a1  a0  a0 Then the number f zeros of P(z) in

1 1

log

Taking

log

z   ,0    1 does not exceed

k ( a n  a n )  a  L  (1   ) a a0

.

   1 in Cor. 1 , we get the following result: n

P( z )   a j z j be a polynomial of degree n such that for some ,0    n  1 and for

Corollary 2: Let

j 0

some k  1,

kan  a n1  ...... a 1  a and

L  a  a 1  a 1  a 2  ...... a1  a0  a0 Then the number f zeros of P(z) in

1 1

log

log

z   ,0    1 does not exceed

k ( a n  a n )  a  L a0



.

Taking k  1 in Cor. 1 , we get the following result: n

P( z )   a j z j be a polynomial of degree n such that for some ,0    n  1 and for

Corollary 3: Let

j 0

some o    1 ,

a n  a n1  ...... a 1  a

and

L  a  a 1  a 1  a 2  ...... a1  a0  a0 Then the number f zeros of P(z) in

1 1

log

log

z   ,0    1 does not exceed

a n  a n  a  L  (1   ) a a0

.

   1 Taking in Theorem 1 , we get the following result: n

Corollary 4: Let

P( z )   a j z j be a polynomial of degree n with Re( a j )   j , Im(a j )   j , j 0

j  0,1,2,......,n such that for some  ,0    n  1 and for some k  1, , k n   n1  ......   1    and

L       1    1    2  ...... 1   0   0 .

Then the number f zeros of P(z) in

z   ,0    1 does not exceed n

a n  (k  1)  n  k n     L  2  j

1 log

1



log

j 0

a0

Taking k  1 in Theorem 1 , we get the following result:

68

.

On The Zeros Of A Polynomial Inside The Unit Disc n

Corollary 5: Let

P( z )   a j z j be a polynomial of degree n with Re( a j )   j , Im(a j )   j , j 0

j  0,1,2,......,n such that for some  ,0    n  1 and for some o    1 ,  n   n1  ......   1    and

L       1    1    2  ...... 1   0   0 .

Then the number f zeros of P(z) in

z   ,0    1 does not exceed n

a n   n     L  (1   )    2  j

1 log

1

j 0

log

.

a0



Similarly for other different values of the parameters, we get many other interesting results.

III.

LEMMA

For the proof of Theorem 1, we need the following result:

z  1, f (0)  0 and f ( z)  M for z  1. Then the number of zeros of 1 M log f(z) in z   ,0    1 does not exceed .

1 f ( 0) log  Lemma: Let f (z) be analytic for

(for reference see [6] ).

IV.

PROOF OF THEOREM 1

Consider the polynomial

F ( z)  (1  z) P( z)

 (1  z )(an z n  an1 z n1  ...... a1 z  a0 )  an z n1  (an  an1 ) z n  ...... (a 1  a ) z  1  (a  a 1 ) z   ...... (a1  a0 ) z  a0  a n z n 1  (k  1) n z n  (k n   n 1 ) z n  ( n 1   n  2 ) z n 1  ...... (  1    ) z  1

 (  1)  z  1  (     1 ) z   ...... (1   0 ) z   0  i{( n   n1 ) z n  ...... ( 1   0 ) z   0 } For

z  1 , we have, by using the hypothesis

F ( z)  an  (k  1)  n  k n   n1   n1   n2  ......   1     (1   )         1  ......  1   0   0   n   n 1   n 1   n  2 .  .......  1   0   0

 an  (k  1)  n  k n   n1   n1   n2  ......   1     (1   )         1  ......  1   0   0   n   n 1   n 1 .   n  2  .......  1   0   0 n

 a n  (k  1)  n  k n     L   (      )  2  j j 0

Since F(z) is analytic for

z  1 , F (0)  a0  0 , it follows by the Lemma that the number of zeros 69

On The Zeros Of A Polynomial Inside The Unit Disc

z   ,0    1 des not exceed

of F(z) in

n

a n  (k  1)  n  k n     L   (      )  2  j

1 log

1

j 0

log

a0



.

Since the zeros of P(z) are also the zeros of F(z) , it follows that the number of zeros of P(z) in

z   ,0    1 des not exceed n

a n  (k  1)  n  k n     L   (      )  2  j

1 log

1

log



j 0

a0

.

That completes the proof of Theorem 1.

REFERENCES [1].

[2].

[3].

[4].

[5].

[6].

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.

K.K.Dewan, Theory of Polynomials and Applications, Deep & Deep Publications, 2007, Chapter 17.

M. H. Gulzar, On the Number of Zeros of a Polynomial in a Prescribed Region, Reseach Journal of Pure Algebra, Vol.2(2) , 2012, ,35-46.

M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math.Soc.(1966).

Q. G Mohammad, On the Zeros of Polynomials, Amer. Math. Monthly, Vol.72 , 1965, 631-633 E. C. Titchmarsh, Theory of Functions, 2nd Edition, Oxford University Press, 1949.

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