On the Number of Zeros of a Polynomial in a Given Domain

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On the Number of Zeros of a Polynomial in a Given

Domain

M. H. Gulzar

Department of Mathematics University of Kashmir, Srinagar 190006

Email: [email protected]

Abstract: In this paper we obtain bounds for the number of zeros of a polynomial in a given domain when the coefficients of the polynomial or their real or imaginary parts are restricted to certain conditions. Our results generalize some known results in the theory of the distribution of zeros of polynomials.

Mathematics Subject Classification: 30C10, 30C15

Keywords and phrases: Coefficient, Polynomial, Zero.

1. Introduction and Statement of Results In the literature we find a large number of published research papers

(e.g.[2,4,5,7,8,9,10,11,12,13,16,17,19,20]) concerning the number of zeros of a polynomial in a circle.

S. K. Singh [18] proved the following result on the number of zeros of a polynomial:

Theorem A: Let





 

0

) (

j j jz a z

P be a polynomial o f degree n such that min_0jn a_j 1

andmax0jn aj  an , then the number of zeros of P(z) in

k R

z  does not exceed

k R a

n _n n

log

} )

1 log{(

2 

where

max{ 1 , 2 ,. 3 ,...}

n n

n n

n n

a a a a a a

R    .

For the class of polynomials with real coefficients, Q. G. Mohammad [14] proved the following result:

Theorem B: Let





 

0

) (

j j jz a z

P be a polynomial o f degree n such that

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Then the number of zeros of P(z) in

2 1



z does not exceed

0 log 2 log

1 1

a a_n

 .

Bidkham an d Dewan [3] generalized Theorem B in the following way:

Theorem C: Let





 

0

) (

j j jz a z

P be a polynomial o f degree n such that

a_n a_n1...a_k1a_k a_k1...a₁a₀ 0,

for some k,0k n.Then the number of zeros of P(z) in

2 1



z does not exceed

    

 

    

0 0

0 2

log 2 log

1

a

a a a a

an n k

.

Theorem D: Let





 

0

) (

j j jz a z

P be a polynomial o f degree n with complex coefficients such that

for some real , , ,0 ,

2

arga_j     jn and for some 0t 1,0k n,

n

n k

k k k

a t a

t a t a

t

a  ...  1 _1 ...

1

0 .

Then the number of zeros of P(z) in

2 1



z does not exceed

     

    

 _{ }  _{ }







0 1

0 1

) 1 sin (cos

sin 2 cos 2

log 2 log

1

a t

a t a t a

t n

n n

j j j k

k    

.

Ebadian et al [6] generalized the above results by proving the following

results:

Theorem E: Let





 

0

) (

j j jz a z

P be a polynomial o f degree n such that

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for some k, 0k n.Then the number of zeros of P(z) in

2

R z  ,

R>0, does not exceed

    

 

      

0 0 0

1

) (

) (

log 2 log

1

a

a a R a a R a R

a_n n k _k n _{k n}

for R1

and

    

 

      

0 0 0

1

) (

) (

log 2 log

1

a

a a R a a R a R

a_n n _k n _{k n}

for R1 .

Theorem F: Let









0 ) (

j j jz

a z

P be a polynomial o f degree n with complex coefficients such that

for some real , , ,0 ,

2

argaj     jn

 

 and for some R0,0kn,

n

n k

k k k

a R a

R a R a

R

a₀  ₁ ...  1 _1 ... .

Then the number of zeros of P(z) in

2

R

z  does not exceed

     

    

 _{ }  _{ }







0

1

0 1

) 1 sin (cos

sin 2 cos 2

log 2 log

1

a R

a R a R R

a

R n

n n

j

j j k

k    

.

In this paper we give generalizations of Theorems E and F. More precisely we prove the following results:

Theorem 1: Let





 

0

) (

j j jz a z

P be a polynomial o f degree n with Re(aj)j, Im(aj)j such

that for some k,,,0k1,0 1, 0n,

kn n1...1 1...0.

Then the number of zeros of P(z) in  (R0,c1)

c R

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    

     

 

    

     

 

     



     

 







 



 

0

1

1

1

1 0

0 0 0 0

1

] 2

) (

[

] 2

) (

[

log log

1

a k

R

R a R a

c

n

j j n

n n n

n

j j n

n

 



  



 

    

 

     

for R1

and

    

     

 

    

     

 

     



     

 







 



 

0

1

1

1

1 0

0 0 0 0

1

] 2

) (

[

] 2

) (

[

log log

1

a k

R

R a R

a

c

n

j j n

n n n

n

j j n

n

 



  

 

    

 

     

for R1.

Remark 1: Taking k1, 1, c=2,Theorem 1 reduces to Theorem E if aj are real i.e. j 0,j.

For different values of the parameters k,,etc. we get many interesting results e.g. if we take

1



 , we get the following result:

Corollary 1: Let





 

0

) (

j j jz a z

P be a polynomial o f degree n with Re(aj)j, Im(aj)j such

that for some k,0k 1,

kn n1...1 1...0,

n

 

0 . Then the number of zeros of P(z) in  (R0,c1)

c R

www.ijsrp.org                                       





      0 1 1 1 1 0 0 0 1 ] 2 ) ( [ ] 2 [ log log 1 a k R R a R a c n j j n n n n n j j n n                   

for R1

and                                       





      0 1 1 1 1 0 0 0 1 ] 2 ) ( [ ] 2 [ log log 1 a k R R a R a c n j j n n n n n j j n n                  

for R1.

Taking k1, n in Theorem 1, we get the following result:

Corollary 2: Let



   0 ) ( j j jz a z

P be a polynomial o f degree n with Re(aj)j, Im(aj)j such

that for some ,0 1,

n n1 ...0.

Then the number of zeros of P(z) in  (R0,c1)

c R

z does not exceed

            _{      }

_

  0 1 0 0 0 0 1

0 [ ( ) 2 ]

log log 1 a R a c n j j n n

n        

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and

     

     

    









0

1 0

0 0 0

0 [ ( ) 2 ]

log log

1

a R

a

c

n

j j n

n       



for R1.

Taking  1, 0in Theorem 1, we get the following result:

Corollary 3: Let









0 ) (

j j jz

a z

P be a polynomial o f degree n with Re(aj)j, Im(aj)j such

that for some k,0k1,

kn n1 ...a1 0,

then the number of zeros of P(z) in  (R0,c1)

c R

z does not exceed

     

    

 _{     }

_

 

0

1 0

0 1

0 [2 ( ) 2 ]

log log

1

a k R

a

c

n

j j n

n n

n      

for R1

and

     

    

 _{     }

_



0

1 0

0

0 [2 ( ) 2 ]

log log

1

a k

R a

c

n

j j n

n

n     



for R1.

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Theorem 2: Let



   0 ) ( j j jz a z

P be a polynomial o f degree n with Re(aj)j, Im(aj)j such

that for some k,0k 1,0 1,

kn n1...1 1...0,

n

 

0 , then the number of zeros of P(z) in  (R0,c1)

c R

z does not exceed

                                        





      0 1 1 1 1 0 0 0 0 0 1 ] 2 ) ( [ ] 2 ) ( [ log log 1 a k R R a R a c n j j n n n n n j j n n                      

for R1

and                                         





      0 1 1 1 1 0 0 0 0 0 1 ] 2 ) ( [ ] 2 ) ( [ log log 1 a k R R a R a c n j j n n n n n j j n n                     

for R1.

Theorem 3: Let



   0 ) ( j j jz a z

P be a polynomial o f degree n with complex coefficients such that

for some real , , ,0 ,

2

arga_j     jn and for some R0,0n,0k 1,0 1,

n n a kR a R a R a R

a  ...   _1 ...

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Then the number of zeros of P(z) in  (R0,c1)

c R

z does not exceed

n n n

n n

j

j j

a R R

a k a R R

a R a R c

1 1

1

1 1

0

2 ) 1 sin (cos

sin 2 cos 2

{ 1 log[ log

1   



 _{ }



_{   }

 

Ra₀ (cossin1)2Ra₀}].

Remark 2: For different values of the parameters k,,etc., we get many interesting results from Theorem 3 as in case of Theorem 1. For k1, 1, c=2,Theorem 3 reduces to Theorem F.

2.Lemmas

For the proofs of the above results we need the following results:

Lemma 1: If f(z) is analytic in z R,but not identically zero, f(0)0 and

n k

a

f( _k)0, 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: If f(z) is analytic and f(z) M(r)in z r, then the number of zeros of f(z) in

c r z 

(r>0,c>1)does not exceed

) 0 (

) ( log log

1

f r M c .

Lemma 2 is a simple deduction from Lemma 1.

Lemma 3: Let









0 ) (

j j jz

a z

P be a polynomial o f degree n with complex coefficients such that

for some real , , ,0 ,

2

arga_j    jn and

, 0

,

1 j n

a

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ta_j a_j1 (ta_j  a_j1)cos(ta_j  a_j1)sin.

Lemma 3 is due to Govil and Rahman [7].

3.Proofs of Theorems

Proof of Theorem 1: Consider the polynomial

F(z) =(1-z)P(z)

(1z)(a_nzn a_n1zn1 ...a₁za₀)

1 1 0 0

1

) (

... )

(a a z a a z a

z

a n

n n n

n      



  _







 



 _{      }



 1

1

1 1

0 1

) (

] ) 1 ( ) [(

n

j

j j j n

n n

n n

nz a ka a k a z a a z

a



a a z a a a z

j

j j

j ) [( ) ( 1) ]

( _{1 0 0}

2

1    

 



 

 



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







 



 _{      }

 1

1 1 1

0 1

) (

] ) 1 ( ) [(

) (

n

j

j j j n

n n

n n

n R a k k R R

a z F



  

 





 

 

 

  

 

 n

j

j j j j

j j

j R R R

1

1 0

0 1 1

1) [( ) (1 ) ] ( )

(       



.

Which gives

( ) [ ( ) 2 ]

1

1 0

1





 

 _{       }

 n

j j n

n n n

n n

n R a R k

a z F

 

   

   

] 2

) (

[

1

1 0

0 0

0





      

  _{      }  _

j j

R

for R1

and

( ) [ ( ) 2 ]

1

1 0

1





 

 _{       }

 n

j j n

n n n

n n

n R a R k

a z F

 

   

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[ ( ) 2 ]

1

1 0

0 0

0





      

 _      _  

j j R

for R1.

Thus

1 [ { ( ) 2 }

) 0 (

)

( 1

1 0

1

0





 

 _{       }

 n

j j n

n n n

n n

n R a R k

a a F

z F

 

   

   

[ ( ) 2 ]

1

1 0

0 0

0





      

  _{      }  _

j j

R

for R1

and

1 [ { ( ) 2 }

) 0 (

)

( 1

1 0

1

0





 

 _{       }

 n

j j n

n n n

n n

n R a R k

a a F

z F

 

   

   

[ ( ) 2 ]

1

1 0

0 0

0





      

 _      _  

j j

R

for R1.

Hence , by Lemma 2,it follows that the number of zeros of F(z) in  (R0,c1)

c R

z does not

exceed

    

     

 

    

     

 

     



     

 







 



 

0

1

1

1

1 0

0 0 0 0

1

] 2

) (

[

] 2

) (

[

log log

1

a k

R

R a R

a

c

n

j j n

n n n

n

j j n

n

 



  



 

    

 

     

for R1

www.ijsrp.org                                         





      0 1 1 1 1 0 0 0 0 0 1 ] 2 ) ( [ ] 2 ) ( [ log log 1 a k R R a R a c n j j n n n n n j j n n                     

for R1.

As the number of zeros of P(z) in  (R0,c1)

c R

z does not exceed the

number of zeros of F(z) in  (R0,c1)

c R

z , the theorem follows.

Proof of Theorem 3: Consider the polynomial

F(z) =(R-z)P(z)

(Rz)(a_nzn a_n1zn1 ...a₁za₀)

a z Ra Ra a z Ran an zn Ra a z

n n n n

n ( ) ( ) ... ( 1 0)

1 2 1 1

0

1       

       1 2 1 1 0 1 ) ( )] ( ) [( _{  }   _{      }   n n n n n n n n n

nz Ra kRa a kRa Ra z Ra a z

a z a a a Ra z a Ra z a Ra z a Ra )] ( ) [( ) ( ... ) ( ) ( ... 0 0 0 1 2 1 2 1 1 1              _         

For z R, we have

1 2 1 1 1 0 1 1 )

(z  a_n Rn Ra  k  Rn a_n  kRa_n a_n Rn  Ra_n a_n Rn

F , ) 1 ( ... ... 0 0 1 2 1 2 1 1 1 R a R a Ra R a Ra R a Ra R a Ra              _         

which gives ,by using Lemma 3and the hypothesis

n n n n n n n n

n R Ra k R a a kRa a kRa R

a z

www.ijsrp.org

R a R

a a R

a a R R

a a R a

a R

R a

a R a

a R

R a

R a a

R a

R a

R a

a R

an n n n n

0 0

1

0 1 2

1 2 1

2

1 1

1 1

1

1 1

2 1

2

) 1 ( ] sin ) (

cos ) [(

] sin ) (

cos ) [(

... ]

sin ) (

cos ) [(

] sin ) (

cos ) [(

... ]

sin ) (

cos ) [(

 



 

 

 

 

 

 

 



 

 



  



 

 

 

 

 



  



 

 



 

 



 

 



n

n n

n R R a

a k a

R 1 cos 1(cos sin 1) 2 1

2       

  _   





 

  

 1

1 0

0(cos sin 1) 2 2 sin

n

j

j j

a R R

a R a

R   



_n n n _n

n

j

j j

a R R

a k a R R

a

R 1 1

1

1 1

2 ) 1 sin (cos

sin 2 cos

2  





 _{    }

  _  



 

Ra0 (cos sin 1)2Ra0 .

The rest of the proof can be completed on the same lines as in the proof of Theorem 1.

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Polynomials.Critical points, Zeros and Extremal Properties, Clarendon

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Morav.,2 (1998), 91-96.

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