International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 13, Issue 3 (March 2017), PP.01-08
A Ring-Shaped Region Containing All or A Specific Number of The Zeros of A Polynomial M. H. Gulzar 1 , A. W. Manzoor 2 Department Of Mathematics, University Of Kashmir, Srinagar
ABSTRACT: According to a Cauchy’s classical result all the zeros of a polynomial P( z )
n
a j 0
degree n lie in
z 1 A , where A max 0 j n 1
aj
j
z j of
. In this paper we find a ring-shaped region containing
an
all or a specific number of zeros of P(z). Mathematics Subject Classification: 30C10, 30C15. Keywords and Phrases: Polynomial, Region, Zeros.
I.
INTRODUCTION
A classical result giving a region containing all the zeros of a polynomial is the following known as Cauchy’s Theorem [4]: n
Theorem A. All the zeros of the polynomial
P( z ) a j z j of degree n lie in the circle j 0
z 1 M , where M max 0 j n 1
aj an
.
The above result has been generalized and improved in various ways. Mohammad [5] used Schwarz Lemma and proved the following result: n
Theorem B. All the zeros of the polynomial
P( z ) a j z j of degree n lie in z j 0
where
M if a n M , an
M max z 1 a n z n 1 ...... a0 max z 1 a0 z n 1 ...... a n1 .
Recently Gulzar [3] proved the following result : n
P( z ) a j z j of degree n lie in the ring-shaped region
Theorem C.All the zeros of the polynomial
j 0
r1 z r2 , where r1
1 3 2
[ R a1 (M a 0 ) 4 a1 R M ] a1 R 2 ( M a 0 ) 4
2
2
2
2
2M 2 2 2M 1
r2
1 3 2
,
,
[ R a n 1 ( M 1 a n ) 4 a n R M ] a n 1 R ( M 1 a n ) 4
2
2
2
M max z R a n z n ...... a1 z , M 1 max z R a0 z n ...... a n 1 z , 1
2
A Ring-Shaped Region Containing All Or A Specific Number Of The Zeros Of A Polynomial R being any positive number. n
P( z ) a j z j of degree n in the ring-shaped region
Theorem C.The number of zeros of the polynomial
j 0
R ,1 c R ,does not exceed c 1 M log(1 ), log c a0
r1 z
where M is as in Theorem 1.
II.
MAIN RESULTS
In this paper we prove the following results: n
n
n 1
j 0
j 1
j 0
P( z ) a j z j be a polynomial of degree n and let L a j , L a j . Then all the
Theorem 1. Let
zeros of P(z) lie in
r1 z r2 , where for R 1 , 1
[ R 4 a1 ( R n L a 0 ) 2 4 a 0 R 3n 2 L3 ] 2 a1 R 2 ( R n L a 0 ) 2
r1
2 R 2 n L3 a 0
,
2 R 2 n L 3 a 0
r2
1
4
[ R a1 and for R 1
2
( R L a 0 ) 2 4 a 0 R 3n 2 L 3 ] 2 a1 R 2 ( R n L a 0 ) n
1
[ a1 ( RL a0 ) 2 4 a0 RL3 ] 2 a1 ( RL a0 ) 2
r1
2 RL3 a 0
,
2 RL 3 a 0
r2
,
1 3 2
[ a1 ( RL a 0 ) 4 a 0 RL ] a1 ( RL a 0 ) 2
2
R being any positive number. n
Theorem2. Let
n
P( z ) a j z j be a polynomial of degree n and let L a j . Then the number of zeros j 0
j 1
R L a0 R 1 1 Rn L ,1 c does not exceed = log log(1 ) for R 1 c log c a0 log c a0 n
of P(z) in r1 z
and
RL a0 1 RL 1 log(1 ) for R 1 . = log log c a0 log c a0
For different values of the parameters in Theorems 1 and 2, we get many interesting results. For example for R=1, we get the following results: Corollary 1. Let
n
n
n 1
j 0
j 1
j 0
P( z ) a j z j be a polynomial of degree n and let L a j , L a j . Then all
the zeros of P(z) lie in
r1 z r2 , where,
2
A Ring-Shaped Region Containing All Or A Specific Number Of The Zeros Of A Polynomial
r1
1 3 2
[ a1 ( L a 0 ) 4 a 0 L ] a1 ( L a 0 ) 2
2
2 L3 a 0
,
2 L 3 a 0
r2
.
1 3 2
[ a1 ( L a 0 ) 4 a 0 L ] a1 ( L a 0 ) 2
Corollary 2. Let
2
n
n
j 0
j 1
P( z ) a j z j be a polynomial of degree n and let L a j . Then the number of zeros
of P(z) in r1 z
L a0 1 L R 1 ,1 c does not exceed log(1 ). = log c log c a0 log c a0 III.
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) 1for z 1 . Then, for
z 1, (1 a ) z b z a (1 a ) 2
f ( z)
a (1 a ) z b z (1 a ) 2
.
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 [2]. Lemma 2.Let f(z) be analytic for Then, for
z R ,f(0)=0, f (0) b and f ( z) M for z R .
z R, M z M z R2 b . f ( z) 2 . Mbz R
Lemma 2 is a simple deduction from Lemma 1 .
z R , f (0) 0 and f ( z) M for z R , then the number of zeros of 1 M R log f(z)in z ,1 c R is less than or equal to . c log c f (0) Lemma 3.If f(z) is analytic for
Lemma 3 is a simple deduction from Jensen’s Theorem (see [1]).
IV.
PROOFS OF THEOREMS
Proof of Theorem 1. Let G(z)= a n z a n 1 z n
Then G(z) is analytic for
n 1
...... a1 z .
z R ,G(0)=0, G (0) a1 and for z R
G( z ) an z n an1 z n1 ...... a1 z
3
A Ring-Shaped Region Containing All Or A Specific Number Of The Zeros Of A Polynomial
a n z a n 1 z n
n 1
...... a1 z
a n R n a n 1 R n 1 ...... a1 R R n [ a n a n 1 ...... a1 ] RnL for R 1 and for R 1 ,
G( z) RL .
z R,
Hence, by Lemma 2, for
G( z )
R n L z R n L z a1 R 2 , . n R2 R L a1 z
for R 1 and for R 1 ,
G( z ) Therefore, for
RL z RL z a1 R 2 . . RL a1 z R2
z R , R 1,
P( z) G( z) a0 a0 G( z ) R n L z R n L z a1 R 2 a0 . R2 R n L a1 z 0 if
a0 R 2 (R n L a1 z ) R n L z (R n L z a1 R 2 ) 0 i.e. if
R 2 n L2 z a1 R 2 ( R n L a0 ) z a0 R n 2 L 0 2
which is true if 1
[ R 4 a1 ( R n L a 0 ) 2 4 a 0 R 3n 2 L3 ] 2 a1 R 2 ( R n L a 0 ) 2
z
2 R 2 n L3 a 0
z R , R 1, P( z) 0 if
Similarly, for
a0 R 2 (RL a1 z ) RL z (RL z a1 R 2 ) 0 i.e. if
L2 z a1 ( RL a0 ) z a0 RL 0 2
which is true if 1
[ a1 ( RL a 0 ) 2 4 a 0 RL3 ] 2 a1 ( RL a 0 ) 2
z
2 RL3 a 0
This shows that P(z) does not vanish in
4
.
.
A Ring-Shaped Region Containing All Or A Specific Number Of The Zeros Of A Polynomial
[ R a1 ( R L a 0 ) 4 a 0 R 2
4
z for
n
2
3n 2
1 3 2
L ] a1 R 2 ( R n L a 0 )
2 R 2 n L3 a 0
z R, R 1
and P(z) does not vanish in 1
[ a1 ( RL a 0 ) 2 4 a 0 RL3 ] 2 a1 ( RL a 0 ) 2
z for
2 RL3 a 0
z R , R 1.
In other words, all the zeros of P(z) lie in 1
[ R 4 a1 ( R n L a 0 ) 2 4 a 0 R 3n 2 L3 ] 2 a1 R 2 ( R n L a 0 ) 2
z i.e.
2 R 2 n L3 a 0
z r1 for z R , R 1
and in
z i.e.
1 3 2
[ a1 ( RL a 0 ) 4 a 0 RL ] a1 ( RL a 0 ) 2
2
2 RL3 a 0
z r1 for z R , R 1 .
On the other hand, let
1 Q( z ) z n P ( ) a 0 z n a1 z n 1 ...... a n 1 z a n z H ( z) an , where
H ( z ) a 0 z n a1 z n 1 ...... a n 1 z . Then H(z) is analytic and
H ( z) R n L for z R , H(0)=0, H (0) a n 1 . Hence, by Lemma 2, for
z R, H ( z)
R n L z R n L z a1 R 2 , . n R2 R L a1 z
for R 1 and for R 1 ,
RL z RL z a1 R 2 .. H ( z) . RL a1 z R2 Therefore, for
z R , R 1,
Q( z) H ( z) a0 a0 H ( z ) R n L z R n L z a1 R 2 a0 . n R2 R L a1 z 0 if
5
A Ring-Shaped Region Containing All Or A Specific Number Of The Zeros Of A Polynomial
a0 R 2 ( R n L a1 z ) R n L z ( R n L z a1 R 2 ) 0 i.e. if
R 2n L2 z a1 R 2 ( R n L a0 ) z a0 R n 2 L 0 2
which is true if
[ R a1 ( R L a 0 ) 4 a 0 R 2
4
z
n
2
3n 2
1 3 2
L ] a1 R 2 ( R n L a 0 )
2 R 2 n L 3 a 0
.
z R , R 1, Q( z) 0 if
Similarly, for
a0 R 2 ( RL a1 z ) RL z ( RL z a1 R 2 ) 0 i.e. if
L2 z a1 ( RL a0 ) z a0 RL 0 2
which is true if
z
1 3 2
[ a1 ( RL a 0 ) 4 a 0 RL ] a1 ( RL a 0 ) . 2 RL 3 a 0 2
2
This shows that Q(z) does not vanish in 1
[ R 4 a1 ( R n L a 0 ) 2 4 a 0 R 3n 2 L 3 ] 2 a1 R 2 ( R n L a 0 ) z 2 R 2 n L 3 a 0 2
for
z R, R 1
and Q(z) does not vanish in 1
[ a1 ( RL a 0 ) 2 4 a 0 RL 3 ] 2 a1 ( RL a 0 ) z 2 RL 3 a 0 2
for
z R , R 1. 1 z
In other words, all the zeros of Q ( z ) z P ( ) lie in n
1
4
z for
[ R a1
2
( R L a 0 ) 2 4 a 0 R 3n 2 L 3 ] 2 a1 R 2 ( R n L a 0 ) 2 R 2 n L 3 a 0 n
z R, R 1
and in 1
[ a1 ( RL a 0 ) 2 4 a 0 RL3 ] 2 a1 ( RL a 0 ) 2
z for
2 RL3 a 0
z R , R 1. 1 z
Hence all the zeros of P ( ) lie in
[ R a1 ( R L a 0 ) 4 a 0 R 4
z
2
n
2
3n 2
L ] a1 R 2 ( R n L a 0 )
2 R 2 n L 3 a 0
6
1 3 2
A Ring-Shaped Region Containing All Or A Specific Number Of The Zeros Of A Polynomial for
z R, R 1
and in 1
z for
[ a1
2
( RL a 0 ) 2 4 a 0 RL 3 ] 2 a1 ( RL a 0 ) 2 RL 3 a 0
z R , R 1.
Therefore , all the zeros of P(z) lie in
2 R 2 n L 3 a 0
z
1
4
[ R a1 i.e.
2
( R L a 0 ) 2 4 a0 R 3n 2 L 3 ] 2 a1 R 2 ( R n L a0 ) n
z r2 for z R , R 1 and in 2 RL 3 a 0
z
1
[ a1 ( RL a 0 ) 2 4 a 0 RL 3 ] 2 a1 ( RL a 0 ) 2
i.e.
z r2 for z R , R 1 .
Thus we have proved that all the zeros of P(z) lie in
r1 z r2 . That completes the proof of Theorem 1.
Proof of Theorem 2. To prove Theorem 2, we need to show only that the number of zeros of P(z) in
z
1 RL R 1 Rn L ,1 c does not exceed log(1 ) for R 1 . log(1 ) for R 1 and c log c a0 log c a0
Since P(z) is analytic for
z R , P(0)= a 0 and for z R ,
P( z ) a n z n a n 1 z n 1 ....... a1 z a0 an z an1 z n
n 1
....... a1 z a0
a n R n a n 1 R n 1 ....... a1 R a0 R n L a0 for R 1 and
P( z) RL a0 for R 1 , it follows , by Lemma 3, that the number of zeros of P(z) in r1 z
R ,1 c does not exceed c
R n L a0 1 1 Rn L = log log(1 ) for R 1 log c a0 log c a0 and
RL a0 1 RL 1 log(1 ) for R 1 and Theorem 2 follows. = log log c a0 log c a0 REFERENCES
[1]. [2]. [3].
Ahlfors L. V. ,Complex Analysis, 3rd edition, Mc-Grawhill. Govil N. K., Rahman Q. I and Schmeisser G.,On the derivative of a polynomial, Illinois Math. Jour. 23, 319-329(1979). Gulzar M. H. ,Manzoor A. W. ,A Ring-shaped Region for the Zeros of a Polynomial,
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A Ring-Shaped Region Containing All Or A Specific Number Of The Zeros Of A Polynomial [4]. [5].
Mathematical Journal of Interdisciplinary Sciences, Vol.4, No.2(March 2016), 177-182. Marden M., Geometry of Polynomials, Mathematical Surveys Number 3, Amer. Math.. Soc. Providence, RI, (1966). Mohammad Q. G., On the Zeros of a Polynomial, Amer. Math. Monthly, 72, 631-633 (1966).
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