On the Zeros of A Polynomial Inside the Unit Disc

Page 1

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


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