ON ZEROS OF POLYNOMIALS

Gulzar. World Journal of Engineering Research and Technology

ON ZEROS OF POLYNOMIALS

M. H. Gulzar*

Department of Mathematics, University of Kashmir, Srinagar, India.

Article Received on 24/01/2017 Article Revised on 14/02/2017 Article Accepted on 09/03/2017

ABSTRACT

In this paper we find ring-shaped regions containing all or a specific

number of zeros of a polynomial. Many important results follow easily

from our results.

Mathematics Subject Classification: 30C10, 30C15. KEYWORDS AND PHRASES: Polynomial, Region, Zeros.

INTRODUCTION

A classical result on the location of zeros of a polynomial is the following known as the

Enestrom-Kakeya Theorem:[2,3]

Theorem A: Let





 n

j j jz

a z

P

0 )

( be a polynomial of degree n such that

0

... 1 0

1    

a _ a a

an n .

Then all the zeros of P(z) lie in z 1.

Another classical result giving a region containing all the zeros of a polynomial is the

following known as Cauchy’s Theorem:[2,3]

Theorem B: All the zeros of the polynomial





 n

j j jz

a z

P

0 )

( of degree n lie in the circle

M

z 1 , where

n j n j

a a M max_{0  1} .

The above theorems have been generalized in various ways by the researchers.

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*Corresponding Author

M. H. Gulzar

Department of

MAIN RESULTS

In this paper we prove the following:

Theorem 1: Let





 n

j j jz

a z

P

0 )

( be a polynomial of degree n and

0 0 1 2

1

1 a a ... a a a

a a

L _n  _n  _n  _n     .

Then all the zeros of P(z) lie in

n n

n

a L z a L a R

a

   



]

[ ₀

1 0

for R1

and in

n

n a

L z a L a R

a

  

 ]

[ 0

0

for R1, provided a_n L.

Further the number of zeros of P(z) in , 1

]

[ ₀

1

0 _{  }

 

 c

c R z a L a R

a

n

n does not exceed

0

0 1

0 [ ]

log log

1

a

a L a R a c

n

n _{ }

 

for R1 and the number of zeros of P(z) in

1 , ]

[ ₀

0

 

 

 c c

R z a L a R

a

n

does not exceed

0

0

0 [ ]

log log

1

a

a L a R a c

n  



for R1.

Remark 1: If a_n a_n1...a₁ a₀ 0, then L a_n and it follows from Theorem 1 that all the zeros of P(z) lie in z 1, which is Theorem A i.e. the Enestrom-Kakeya Theorem.

If we take R=1 in Theorem 1, we get the following result:

Corollary 1: Let





 n

j j jz

a z

P

0 )

( be a polynomial of degree n and

L a_na_n1  a_n1a_n2 ... a₁a₀  a₀ .

Then the number of zeros of P(z) in , 1

0

0 _{  }



 c c

R z a L a

a

n

does not exceed

0 ) (

log log

1

a a L c

n

 .

Instead of proving Theorem 1, we prove the following more general result:

Theorem 2: Let





 n

j j jz

a z

P

0 )

( be a polynomial of degree n with

n j

a

aj) j,Im( j) j, 0,1,2,...,

. ... ... 0 0 1 2 1 1 0 0 1 2 1 1                                     n n n n n n n n M L

Then all the zeros of P(z) lie in

n n n a M L z M L a R a         ]

[ 0 0

1

0



 for R1

and in n n a M L z M L a R a        ]

[ 0 0

0



 for R1, provided an LM.

Further the number of zeros of P(z) in , 1,

]

[ _{0 0}

1

0 _{  }

     c c R z M L a R a n

n   does not

exceed

0

0 0 1

0 [ ]

log log 1 a M L a R a c n

n     

 

for R1 and the number of zeros of P(z) in

, 1 , ]

[ 0 0

0 _{  }

  

 c c

R z M L a R a

n  

does not exceed

0

0 0

0 [ ]

log log 1 a M L a R a c

n     



for R1.

Remark 2: Taking a_j real i.e. _j 0,j0,1,2,...,n, Theorem 2 reduces to Theorem 1. If we take R=1 in Theorem 2, we get the following result:

Corollary 2: Let



  n j j jz a z P 0 )

( be a polynomial of degree n with

n j

a

aj) j,Im( j) j, 0,1,2,...,

Re(    and

. ... ... 0 0 1 2 1 1 0 0 1 2 1 1                                     n n n n n n n n M L

Then the number of zeros of P(z) in 1, 1,

0 0

0 _{  }

  

L M z c c

a

a

n  

does not exceed

0 0 0 0 log log 1 a M L a a c

n     



LEMMAS

For the proof of Theorem 2, we need the following results:

Lemma 1: Let f(z) (not identically zero) be analytic for z R,f(0)0 and f(ak)0,

n

k 1,2,..., . Then









 n

j _j

i

a R f

d f

1 2

0 log (Re log (0) log

2

1  _



 .

Lemma 1 is the famous Jensen’s Theorem (see page 208 of [1]).

Lemma 2: Let f (z) be analytic for z  R,f(0)0 and f(z) M for z R. Then the number of zeros of f(z) in  ,c1

c R

z does not exceed

) 0 ( log log

1

f M

c .

Lemma 2 is a simple deduction from Lemma 2.

PROOFS OF THEOREMS

Proof of Theorem 2: Consider the polynomial

) ( ) 1 ( )

(z z P z

F  

   



 a z a a z

a z

a a z

a

a z a z

a z a z

n n n n

n

n n n n

) (

) (

... )

(

) ...

)( 1 (

1 1

1 1

1

0 1 1

1

 

 



 

  

  

 



   

 

0 0

1 )

(

... a a za



... )

{( )

( ... )

( 1 1 0 0 1

1        



  _{ } n

n n n

n n n

nz z z i z

a       

} )

(₁₀ ₀

 z

For z 1so that j n

z j

,..., 2 , 1 , 1

1 _{  }

, we have, by using the hypothesis

... ...

{ )

( 1 1 0 0 1

1

 

  

  



  _{ } n

n n n

n n n

n z z z z

a z

F       

} 0 0

1  

  

 z

1 0

1 0 1 2

1

1 . ...

{

[   _       _    _

 n n _{n n n n}

n n n

n

z z

z z

a

z         

}]

... 1 ₁0 0

1

n n

n n

z z

z

   



    

  _

1 0

0 1 2

1

1 . ...

{

[   _  _  _       _

 n n n n n n n

n z a

z         

}]

...   



     

0

)] (

[ 

 

 zn a_n z L M

if

n

a M L z  

provided a_n LM.

This shows that those zeros of F(z) whos modulus is greater than 1 lie in

n

a M L z   .

Since the zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above

inequality, it follows that all the zeros of F(z) and hence all the zeros of P(z) lie in

n

a M L z   .

On the other hand, we have

... )

{( ) (

... )

( )

( 0 1 1 0 1

1        



  _{ } n

n n n

n n n

nz a z z i z

a z

F      

} ) (₁ ₀ z



) ( 0 G z

a 



Where G(z)a_nzn1(_n _n1)zn...(₁₀)zi{(_n_n1)zn ...

} ) (₁₀ z

 .

For z R, we have, by using the hypothesis

z z

z z

z a z

G n n n

n n n n

n 1 1 0 1 1 ₀

1

... ...

)

(     _       _    

] [

] ...

... [

... ...

0 0 1

0 1 1

0 1 1

1

0 1 1

0 1 1

1

 

  

   



  

 

 



    

          

   

 

  

 



 



 



M L a R

a R

R R

R R

R a

n n

n n n

n n n

n n n n

n n n n

for R1. For R1,

] [

)

(z R a LM ₀  ₀

G _n .

Since G(z) is analytic for z R,G(0)0, it follows by Schwarz Lemma that in z R,

z M

L a R z

G n

n

] [

)

( 0 0

1     

 

for R1 and

z M

L a R z

Hence for z R,

) ( )

(z a0 G z

F  

z M

L a R a

z G a

n n

] [

) (

0 0 1

0 0

      

  



for R1 and

F(z)  a₀ R[a_n LM₀  ₀ ]z

for R1.

Thus for R1, F(z) 0if

]

[ 0 0

1

0



 

    _

M L a R

a z

n

n

and for R1, F(z) 0if

]

[ 0 0

0



 

   

M L a R

a z

n

.

In other words, all the zeros of F(z) lie in

]

[ _{0 0}

1

0



 

    _

M L a R

a z

n

n for R1 and in

]

[ _{0 0}

0



 

   

M L a R

a z

n

for R1.

Since the zeros of F(z) are also the zeros of P(z), it follows that all the zeros of P(z) lie in

]

[ _{0 0}

1

0



 

    _

M L a R

a z

n

n for R1 and in

]

[ _{0 0}

0



 

   

M L a R

a z

n

for R1.

Again , for z R, we have, by using the hypothesis

F(z)  a_n zn1  a₀ _n _n1 zn ...₁ ₀ z  _n _n1 zn ...  ₁₀ z

0 1 ... 1 0 1 ...

1        

  _{ } n

n n n

n n n

nR a R R R

a      

 ₁₀ R

1[ ₁ ... _{1 0 1} ...

0         

  _{ }

n n n

n n

n _a

R

a      

 ₁₀]

 a₀ Rn1[a_n LM ₀  ₀]

for R1 ,

F(z)  a₀ R[a_n LM₀  ₀].

Hence, by using Lemma 2 and the above observations, it follows that the number of zeros of

F(z) and therefore P(z) in , 1,

]

[ 0 0

1

0 _{  }

   

 c

c R z M

L a R

a

n

n   does not exceed

0

0 0 1

0 [ ]

log log

1

a M L a R a c

n

n     

 

for R1 and the number of zeros of P(z) in

, 1 , ]

[ _{0 0}

0 _{  }

  

 c c

R z M

L a R

a

n  

does not exceed

0

0 0

0 [ ]

log log

1

a M L a R a c

n     



for R1.

That completes the proof of Theorem 2.

REFERENCES

1. Ahlfors L. V. Complex Analysis, 3rd edition, Mc-Grawhill.

2. Marden M, Geometry of Polynomials, Mathematical Surveys Number 3, Amer. Math.

Soc. Providence, RI, 1966.

3. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University