DOI: 10.15415/mjis.2016.42015
A Ring-shaped Region for the Zeros of a Polynomial M. H. GULZAR1, A. W. MANZOOR2 Department of Mathematics, University of Kashmir, Srinagar Email: gulzarmh@gmail.com Received:July 19, 2015| Revised: December 12, 2015| Accepted: February 11, 2016 Published online: March 30, 2016 The Author(s) 2016. This article is published with open access at www.chitkara.edu.in/publications
Abstract: Let P ( z ) =
n
∑ a j z j be a polynomial of degree n .Then according j =0
to Cauchy’s classical result all the zeros of P(z) lie in z ≤ 1 + A , where
A = max0≤ j≤n−1
aj an
. The purpose of this paper is to present a ring-shaped
region containing all or a specific number of zeros of P(z). Mathematics Subject Classification: 30C10, 30C15. Keywords: Polynomial, Region, Zeros. 1. INTRODUCTION The following result known as Cauchy’s Theorem [2] is well known in the theory of distribution of zeros of polynomials: n Theorem A. All the zeros of the polynomial P ( z ) = ∑ a j z j of degree n lie j =0
in the circle z ≤ 1 + M , where M = max0≤ j≤n−1
aj an
.
Several generalizations and improvements of this result are available in the literature. Mohammad [3] used Schwarz Lemma and proved the following result: n
Theorem
B.
All
the
zeros
of
the
polynomial
P( z) = ∑ a j z j
j =0 M z≤ an ≤ M , where of degree n lie in if an M = max z =1 an z n−1 + ...... + a0 = max z =1 a0 z n−1 + ...... + an−1 .
In this paper, we find a ring-shaped region containing all the zeros of P(z). We also obtain a result of similar nature for the number of zeros of the polynomial. In fact we prove the following results:
Mathematical Journal of Interdisciplinary Sciences Vol. 4, No. 2, March 2016 pp. 177–182
Gulzar, MH Manzoor, AW
Theorem 1.All the zeros of the polynomial P ( z ) = the ring-shaped region r1 ≤ z ≤ r2 , where 4
r1 =
2
2
2
[ R a1 ( M − a0 ) + 4 a1 R 2M
2
1 3 2 M ]
n
∑ ajz j
of degree n lie in
j =0
− a1 R 2 ( M − a0 )
2
,
2 M12
r2 =
2
[ R 4 an−1 ( M1 − an )2 + 4 an
1 R 2 M 3 ]2
,
− an−1 R 2 ( M1 − an )
M = max z = R an z n + ...... + a1z , M1 = max z = R a0 z n + ...... + an−1z ,
R being any positive number.
Theorem 2. The number of zeros of the polynomial P ( z ) = n in the ring-shaped region r1 ≤ z ≤
n
∑ a j z j of degree
j =0 R ,1 < c ≤ R , does not exceed c
1 M log(1 + ), log c a0 where M is as in Theorem 1. 2. LEMMAS For the proofs of the above theorems we make use of the following results: Lemma1. Let f(z) be analytic in z ≤1 ,f(0)=a, where a <1 , f ′(0) = b and f ( z ) ≤1 for z = 1 . Then, for z ≤1 , 2
f (z) ≤
(1 − a ) z + b z + a (1 − a ) 2
a (1 − a ) z + b z + (1 − a )
The example
b z − z2 1+ a f ( z) = b 1− z − az 2 1+ a a+
shows that the estimate is sharp. Lemma 1 is due to Govil, Rahman and Schmeisser [1].
178
.
Lemma 2. Let f(z) be analytic for z ≤ R ,f(0)=0, f ′(0) = b and f ( z ) ≤ M for z = R . Then, for z ≤ R ,
M z M z + R2 b . f (z) ≤ 2 . M+ b z R Lemma 2 is a simple deduction from Lemma 1 . Lemma 3.If f(z) is analytic for z ≤ R , f (0) ≠ 0
and
f ( z ) ≤ M for
R z = R , then the number of zeros of f(z)in z ≤ ,1 < c < R is less than or c 1 M . equal to log log c f (0 ) Lemma 3 is a simple deduction from Jensen’s Theorem (see [4]). Proof of Theorem 1. Let Q(z)= an z n + an−1z n−1 + ...... + a1z . Then Q(z) is analytic for z ≤ R ,Q(0)=0, Q ′(0) = a1 and for z ≤ R
Q( z ) = an z n + an−1z n−1 + ...... + a1z ≤ M Hence, by Lemma 2, for z ≤ R ,
Q( z ) ≤
M z M z + a1 R 2 . . M + a1 z R2
Therefore, for z ≤ R ,
P( z ) = Q( z ) + a0 ≥ a0 − Q( z ) M z M z + a1 R 2 ≥ a0 − 2 . M + a1 z R >0 if
a0 R 2 ( M + a1 z ) − M z ( M z + a1 R 2 ) > 0 i.e. if
2
M 2 z + a1 R 2 ( M − a0 ) z − a0 R 2 M < 0
which is true if
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A Ring-shaped Region for the Zeros of a Polynomial
Gulzar, MH Manzoor, AW
2
4
z<
2
[ R a1 ( M − a0 ) + 4 a0 R
2
1 3 2 M ]
− a1 R 2 ( M − a0 )
2M 2
.
Thus P(z) does not vanish in 4
z<
2
2
[ R a1 ( M − a0 ) + 4 a0 R
2
1 3 2 M ]
2M 2
− a1 R 2 ( M − a0 ) .
On the other hand, let
1 R( z ) = z n P( ) = a0 z n + a1z n−1 + ...... + an−1z + an z = H ( z ) + an ,
where
H ( z ) = a0 z n + a1z n−1 + ...... + an−1z . Then H(z) is analytic and H ( z ) ≤ M1 for z ≤ R , H(0)=0, H ′(0) = an−1 . Hence, by Lemma 2, for z ≤ R ,
M1 z M1 z + an−1 R 2 . H (z) ≤ 2 . M1 + an−1 z R Therefore, for z ≤ R ,
R ( z ) = an + H ( z ) ≥ an − H ( z ) M z M z + an−1 R 2 ≥ a n − 12 . 1 M1 + an−1 z R >0 if
an R 2 ( M1 + an−1 z ) − M1 z ( M1 z + an−1 R 2 ) > 0 i.e. if
2
M12 z + an−1 R 2 ( M1 − an ) z − an R 2 M1 < 0
which is true if
180
4
z<
2
2
[ R an−1 ( M1 − an ) + 4 an R
2
1 3 2 M1 ]
− an−1 R 2 ( M1 − an )
1 3 2 M1 ]
− an−1 R 2 ( M1 − an ) .
.
2 M12
Thus P(z) does not vanish in 4
z<
2
2
[ R an−1 ( M1 − an ) + 4 an R
2
2 M12
Hence all the zeros of R(z) lie in 1
2
z≥
[ R 4 an−1 ( M1 − an )2 + 4 an R 2 M13 ]2 − an−1 R 2 ( M1 − an )
.
2 M12 n
1 z
Thus all the zeros of z P ( ) lie in 4
z≥
2
2
[ R an−1 ( M1 − an ) + 4 an R
2
1 3 2 M1 ]
− an−1 R 2 ( M1 − an )
.
2 M12
1 z
Hence all the zeros of P ( ) lie in 4
z≥
2
2
[ R an−1 ( M1 − an ) + 4 an R
2
1 3 2 M1 ]
2 M12
− an−1 R 2 ( M1 − an ) .
This implies that all the zeros of P(z) lie in
2 M12
z≤ 2
[ R 4 an−1 ( M1 − an )2 + 4
1 2 3 2 an R M1 ]
.
− an−1 R 2 ( M1 − an )
Consequently, it follows that all the zeros of P(z) lie in the ring-shaped region r1 ≤ z ≤ r2 and the proof of Theorem 1 is complete. Proof of Theorem 2. To prove Theorem 2, we need to show only that the number
R 1 M ,1 < c < R does not exceed log(1 + ). c log c a0 Since P(z) is analytic for z ≤ R , P(0)= a0 and for z ≤ R , of zeros of P(z) in z ≤
P( z ) = an z n + an−1z n−1 + ....... + a1z + a0
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A Ring-shaped Region for the Zeros of a Polynomial
Gulzar, MH Manzoor, AW
≤ an z n + an−1z n−1 + ....... + a1z + a0 ≤ M + a0
,
it follows , by Lemma 3, that the number of zeros of P(z) in z ≤
1
does not exceed log log c follows.
M + a0 a0
=
R ,1 < c ≤ R c
1 M log(1 + ) and Theorem 2 log c a0
ACKNOWLEDGEMENT The authors are thankful to the referee for his value-able suggestions. REFERENCES [1] Govil N. K., Rahman Q. I and Schmeisser G.,On the derivative of a polynomial, Illinois Math. Jour. 23, 319-329(1979). [2] Marden M,, Geometry of Polynomials, Mathematical Surveys Number 3, Amer. Math.. Soc. Providence, RI, (1966). [3] Mohammad Q. G., On the Zeros of a Polynomial, Amer. Math. Monthly, 72, 631-633 (1966). http://dx.doi.org/10.2307/2313853 [4] Titchmarsh E.C., Theory of Functions, Oxford University Press, London, (1949).
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