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
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 n1 ...... 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 n1 ...... 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 n1 ...... 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 n1 ...... 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 an1 z n1 ...... a1 z a0 ) an z n1 (an an1 ) 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 n1 ) z n ...... ( 1 0 ) z 0 } For
z 1 , we have, by using the hypothesis
F ( z) an (k 1) n k n n1 n1 n2 ...... 1 (1 ) 1 ...... 1 0 0 n n 1 n 1 n 2 . ....... 1 0 0
an (k 1) n k n n1 n1 n2 ...... 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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