A Ring-Shaped Region Containing All or A Specific Number of The Zeros of A Polynomial

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 n1 .

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 .

Mbz 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  an1 z n1  ...... 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 L2 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

L2 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  an1 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,

7

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).

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