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Annular Bounds for the Zeros of Complex Polynomials

 

Ajeet Singh1, Neha2

Department of Mathematics Lingayas Univesity Faridabad-121002, Delhi NCR, INDIA.

* Corresponding Address:

 [email protected]

Research Article

 


Abstract: The location of zeros of complex polynomials has been investigated in frame work of Enestrom and Kakeya theorem. In this paper we extend some existing results on the zeros of complex polynomials by considering restrictions on its coefficients.

Mathematics Subjects Classification: 26C10, 30C10, 30C15

Keywords: Polynomials, Zeros, Enestrom - Kakeya theorem and the sharper bounds.


 

Introduction

The following result due to Enestrom and Kakeya [12] is well known in the theory of distribution of zeros of polynomials.

 

Theorem A (1): If P (z) = be a polynomial of degree n such that

 an ≥ an-1 ≥ an-2 ≥------≥a1 ≥a0 > 0 , a j є R                                                                                               (1)

Then P (z) does not vanish in |z|>1

This is a very elegant result but it is equally limited in scope as the hypothesis is very restrictive.

A. Joyal et al [11] extended this theorem to the polynomials whose coefficient are monotonic but not necessarily non negative and proved the following:

 

Theorem A (2): If P (z) = be a polynomial of degree n such that

 an ≥ an-1 ≥ an-2 ≥------≥a1 ≥a0 , a j є R                   

Then all the zeros of P (z) lie in              

 |z| ≤ (an – a0 + | a0|) ÷ | an|.                                                                                                                   (2)

This was further improved upon by Dewan and Govil[7].

Shah and Liman [15] relaxed the hypothesis and proved the following result.

 

Theorem B: Let P (z) = be a polynomial of degree n with complex coefficients. If

 Re (aj) =αj and Im(aj ) = βj , for j = 0,1,2-----n. such that for some λ≥ 1,

 λαn ≥ αn-1 ≥ αn-2 ≥------≥ α1 ≥ α0 ,

 βn ≥ βn-1≥ βn-2 ≥------≥β1≥ β0 > 0

Then all the zeroes of P (z) lie in

 |z + (λ -1)| ≤ [λαn - α0+ |α0| + ] ÷ | an|                                                                                            (3)

 Aziz and Zargar [1] relaxed the hypothesis of Theorem A (1) and proved the following extensions of Enestrom-Kakeya theorem.

 

Theorem C: Let P (z) = be a polynomial of degree n with complex coefficients such that for some k≥1,

 kan ≥an-1≥----- a1≥a0>0

Then all the zeros of P(z) lie in |z+k-1| ≤ k                                                                                          (4)

 

Theorem D: Let P (z) = be a polynomial of degree n with complex coefficients. If

 Re (aj) =αj and Im(aj ) = βj , for j = 0,1,2-----n. such that for some k ≥ 1,

 λαn ≤ αn-1 ≤ ----≤ ≤ ≥----≥ α1 ≥ α0

 βn ≥ βn-1≥ βn-2 ≥------≥β1≥ β0 > 0

Where 0 ≤ p ≤ n-1, then all the zeros of P(z) lie in

 |z + (λ -1)| ≤ [2αp - λαn - α0 +|α0| + ] ÷ | an|                                                                                                 (5)

Recently, Choo [5] has proved the following theorem

Theorem E: Let P (z) = be a polynomial of degree n with complex coefficients. If

 Re (aj) =αj and Im(aj ) = βj , for j = 0,1,2-----n. such that for some p and r and for some λ , μ > 0

 λαn ≤ αn-1 ≤ ----≤ ≤ ≥----≥ α1 ≥ α0

 μβn ≤ βn-1 ≤ ---- ≤ ≤ ≥----≥ β1 ≥ β0

Then P(z) has all its zeros in R1 z≤ R2 where

R1 =  and R2 =  

With

M1 = || + || +|| + 2() – (λ) - ()

And

M2 = || +|| + 2() – (λ) - () + ||

Here we notice that the annulus R1 z≤ R2 is expressed in terms of λ and μ as associated to the coefficients αn and βn in the given constraint in Theorem E.

 

Theorem 1: Let P (z) = be a polynomial of degree n with complex coefficients. If

 Re (aj) =αj and Im(aj ) = βj , for j = 0,1,2-----n. such that for some δ,η ≥ 1 and τ,σ ≤ 1

 δαn ≤ αn-1 ≤ ----≤ ≤ ≥----≥ α1 ≥ τα0

 ηβn ≤ βn-1 ≤ ---- ≤ ≤ ≥----≥ β1 ≥ σβ0                                                                                                                              (6)

where 0 ≤ p,q ≤ n-1, then all the zeros of P(z) lie in the disk

 R δη≤ |z-zδη| ≤ Rδη ,                                                                                                                               (7)

where  

zδη =  + i ,                                                                                                                  (8a)

 

 Rδη = [2()-( +)-τα0+ (1-τ)α00 + (1-σ)0+||

 (8b)

 Rδη =  

 - [+ (8c)

 

Proof: Consider the polynomial

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

 

 = -zn{( α n + n)z+(δ-1)+i(η-1)+[( δαn αn-1)zn +(αn-1- αn-2)zn-1 +--------------------- + {(α1-τα0)+( τα0 –α0)z+ α0]+i[( ηzn + ( βn-1- β n-2 )z n-1+-----+{(β1-σβ0)+( σβ0 –β0 )}z+β0]

 

Now if |z|>1,  < 1, j= 0, 1, 2---n-1

 Therefore,

 |F (z)| ≥ |z|n [|anz+( δ-1)+i(η-1) –{

 }]

  >0, if

  | z +  + i | >

 {2(αp + βq)-(δαn +) - }

 

This shows that the zeros of F (z) having modulus greater than 1 lie in

 

{| z +  + i |

 {2(αp + βq)-(δαn +) - }                                         (9)

 

Since all the zeros of P(z) with modulus greater than 1 lie in the disc given by eq(9), it can be shown that Rδη ≥ 1. Consequently the zeros of P (z) with modulus less than or equal to one are already contained in the disk |z-zδη| ≤ Rδη .                                                                                                                                                                         (10)

In order to prove the lower bound Rδη ≤ |z-zδη| we first prove the following lemma.

 

Lemma: Let P (z) = be a polynomial of degree n with complex coefficients. Then for |z|<1, we show that

 

|z| ≤  =  

Proof: Let |z|<1.

 Consider F (z) = (1-z) P (z)

= χ (z) + a0,                                                                                                                               (11)

Where

χ(z)= {( α n +iβn)z+(δ-1)+i(η-1)+[( δαn - αn-1)zn +( αn-1- αn-2)z n-1 +------------ + {(α1-τα0)+( τα0–α0)z +       i[( η - zn + ( βn-1- β n-2 )z n-1+-----+{(β1-σβ0)+( σβ0- β0 )}z]

   | χ(z)| =|{( α n+iβn)z+(δ-1)+i(η-1)+[( δαn - αn-1)zn +( αn-1- αn-2)z n-1 +--------- +

 {(α1-τα0)+( τα0- α0)z +i [( η - zn + ( βn-1- β n-2 )z n-1+-----+{(β1-σβ0)+( σβ0- β0)}z] |

 ≤ | anz|+|( δ-1)+|( η-1)| +[M1]

Where

 M1 =2(αp + βq)-(δαn +)-                                                                                                                                                                                                                               (12)

Since χ (0)=0,it follows by Schwarz lemma that

 | χ (z)|≤ M1|z| for |z|<1

 

Therefore for |z|<1,

 | F (z)| = | χ(z) + a0 | ≥ |a0|-| χ(z) | > 0 , if

  |a0| > |z|[M2] ,

 

where M2 = |an|++ M1                                                                                             

                    = |an|+2(αp + βq)-(δαn +)-

                                                                                                                                                             (13)

Thus, |z| ≤  

             =                                           (14)

Hence P(z) does not vanish in |z| <  . It can be shown that M2 ≤ |a0| so that |z|≤1. Hence P (z) has all its zeros in  |z|.                                                                                                                                                        (15)

                                                                                                                                   

Now we prove the second part of the main theorem (1)

 Since |z - zδη | ≥ |z |- |zδη | , (16)

 then using eq(15) of above lemma in eq(16), we have

 |z - zδη | ≥ |z |- |zδη |  - |zδη |

 This implies  - |zδη | ≤ |z - zδη |

  -  + i ≤ |z - zδη |                                                                                                     (17)

 

 From eq(17) we obtain Rδη ≤ |z-zδη| ,                                                                                                    (18)

 where Rδη is given in eq 8( c )

 On combining eq(10) and eq(18) the above theorem is completely proved.

 

Conclusion

We get (i) if τ = 1, σ1, then all the zeros of P(z) lie in the disk

 R 22≤ |z-zδη| ≤ R11, (19)

where,

R11 = [2()-( +)-α00 + (1-σ)0+|| (19a)

 

 R22=  

  - [+ (19b)

and

 

(ii) if σ = 1, τ, then all the zeros of P(z) lie in the disk

 R 44≤ |z-zδη| ≤ R33 , (20)

                                                 

where ,

 R33 = [2()-( +)-α00 + (1-σ)0+|| (20a)

 

 R44 =  

 - [+ (20b)

and zδη is given by eq(8a)

(iii) Further we note with regard to the upper bound of the Theorem 1 given as |z-zδη| ≤ Rδη ,

 where

 zδη =  + i = A+iB where A=  and B= 1

 and

 Rδη = [2()-( +)-τα0+ (1-τ) α00 + (1-σ)0+|| ]

 and that if we transfer the centre of the above disc at the origin so that equation (9) can be written as

 |z| = |  + zδη | ≤ |z-zδη| +|zδη|

 ≤ Rδη+ |zδη|

 ≤ {2(αp + βq)-(δαn +)- } +  (21)

Comparing this bound with upper bound of Theorem E given by:

|z | ≤ R11 =  

 ≤  || +|| + 2() – (λ) - () + ||}

 ≤ 2(αp + βq)-( λ+ μ)-( ++|| ] +  (20)

We here find that the present bound given by (21) corresponding to τ = 1=σ is sharper than eq (20) of Choo [5], in view of  < A+B.

References

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