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 n1 ...... 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