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The Number of Zeros of a Polynomial in a Disk

M. H. Gulzar

Department of Mathematics

University of Kashmir, Srinagar, 190006

Abstract: In this paper we subject the coefficients of a polynomial and their real and imaginary parts to certain restrictions and give bounds for the number of zeros in a specific region. Our results generalize many previously known results and imply a number of new results as well.

Mathematics Subject Classification: 30 C 10, 30 C 15

Keywords and Phrases: Coefficient, Polynomial, Zero

1. Introduction

A large number of research papers have been published so far on the location in the complex plane of some or all of the zeros of a polynomial in terms of the coefficients of the polynomial or their real and imaginary parts. The famous Enestrom-Kakeya Theorem states [6] that if the coefficients of the polynomial

j n

j jz

a z

P

= =

0

)

( satisfy

0≤a0 ≤a1 ≤...≤an−1 ≤an,

then all the zeros of P(z) lie in the closed disk z ≤1. By putting a restriction on

the coefficients of a polynomial similar to that of the Enestrom-Kakeya Theorem, Mohammad [7] proved the following result:

Theorem A: Let j

n

j jz

a z

P

= =

0

)

( be a polynomial such that

0<a0 ≤a1 ≤...≤an−1 ≤an. Then the number of zeros of P(z) in

2 1

(2)

0

log 2 log

1 1

a an

+ .

For polynomials with complex coefficients, Dewan [1] proved the following results:

Theorem B: Let j

n

j jz

a z

P

= =

0

)

( be a polynomial such that

, 0,1,2,..., ,

2

argaj − ≤ ≤ j = n

π α β

for some real αand β and

0< a0 ≤ a1 ≤...≤ an−1 ≤ an . Then the number of zeros of P(z) in

2 1

z does not exceed

0

1

0

sin 2 ) 1 sin (cos

log 2 log

1

a

a a

n

j j

n

− = +

+

+ α α

α

.

Theorem C: Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that

0<α0 ≤α1 ≤...≤αn−1 ≤αn. Then the number of zeros of P(z) in

2 1

z does not exceed

0 0

log 2 log

1 1

a

n

j j

n

= + +

β α

.

Regarding the number of zeros of P(z) in z ≤δ,0<δ <1, Gulzar [4,5] proved the

following generalizations of the above results:

Theorem D: Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that for some 0<k ≤1,0<τ ≤1 and 0≤ln,

kαn ≤αn−1 ≤...≤αl+1 ≤αl ≥αl−1 ≥...≥α1 ≥τα0.

(3)

0

log 1 log

1

a M

δ

,

where

M

= + + −

+ − +

= n

j j n

n

l k

0 0

0

0 ( ) 2

2 ) (

2α α α α τ α α β .

Theorem E: : Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that for some 0<k1,k2 ≤1,0<τ1,τ2 ≤1 ,0≤ln

and 0≤mn,

kn ≤αn−1 ≤...≤αl+1 ≤αl ≥αl−1 ≥...≥α1 ≥τ1α0

kn ≤βn−1 ≤...≤βm+1 ≤βm ≥βm−1 ≥...≥β1 ≥τ2β0.

Then for 0<δ <1 the number of zeros of P(z) in z ≤δ does not exceed

0

log 1 log

1

a M

δ

,

where

M′ = an +(1−k1n +(1−k2n +2(αlm)−(k1αn +k2βn)+(1−τ10

−τ1α0 +(1−τ2)β0 −τ2β0.

Recently, Gardner and Sheilds [2] proved the following results:

Theorem F: Let j

n

j jz

a z

P

= =

0

)

( be a polynomial of degree n such that for some t>0,

n l≤ ≤

0 and for some real numbers α,β ,

, 0,1,2,..., ,

2

argaj −β ≤α ≤π j= n

and

tn antn−1an1 ≤...≤tl+1al+1tl altl−1al1 ≥...≥ta1a0 >0.

Then the number of zeros of P(z) in z ≤δt,0<δ <1 is less than

0 *

log 1 log

1

a M

δ

, where

− =

+ +

+ + + +

+ −

= 1

1 1 1

1 0

*

sin 2 ) cos sin

1 ( cos

2 ) sin cos

1 (

n

j j j n

n l

l t a t a t

a t

a

(4)

Theorem G: Let j n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that for some t>0, and some 0≤ln,

tnαntn−1αn1 ≤...≤tl+1αl+1tlαltl−1αl1 ≥...≥tα1 ≥α0 ≠0.

Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than

0 * *

log 1 log

1

a M

δ

,

where

* *

M

=

+ +

+ + +

+ −

= n

j

j j n

n n l

lt t t

t

0

1 1

1 0

0 ) 2 ( ) 2

(α α α α α β .

Theorem H: Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that for some t>0, ,0≤lnand 0≤mn,

tnαntn−1αn1 ≤...≤tl+1αl+1tlαltl−1αl1 ≥...≥tα1 ≥α0 ≠0

1 1 0

1 1

1 1

1

...

... β β β β β

β

β ≤ ≤ ≤ ≤ ≥ − ≥ ≥ ≥

+ + −

t t

t t

t

t m m

m m l m n

n n

n .

Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than

0 * * *

log 1 log

1

a M

δ

,

where

* * *

M =(α0 −α0)t+2αltl+1+1n −αn)tn+1 +(β0 −β0)t+2βmtm+1 +(βn −βn)tn+1.

In this paper, we prove the following results:

Theorem 1: Let j

n

j jz

a z

P

= =

0

)

( be a polynomial of degree n such that for some t>0,

n l≤ ≤

0 ,0<k ≤1,0<τ ≤1 and for some real numbers α,β ,

, ,..., 2 , 1 , 0 , 2

argaj −β ≤α ≤π j= n

and

ktn antn−1an1 ≤...≤tl+1al+1tl altl−1al1 ≥...≥ta1 ≥τ a0 >0.

Then the number of zeros of P(z) in z ≤δt,0<δ <1 is less than

0 1

log 1 log

1

a M

δ

, where

(5)

− =

+ +

+ 1

1 1

sin 2 )

1 (

n

j j j n

nt a t

a

k α .

Remark 1: For different values of the parameters, we get many interesting results.

For example for k=1,τ =1, Theorem 1 reduces to Theorem F . For t=1, it reduces

to Theorem D .

Theorem 2: Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that for some t>0,0<k ≤1,0<τ ≤1 and 0≤ln,

ktnαntn−1αn1 ≤...≤tl+1αl+1tlαltl−1αl1 ≥...≥tα1 ≥τα0 ≠0.

Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than

0 2

log 1 log

1

a M

δ

,

where

2

M

=

+ +

+

+ + + +

+ + −

= n

j

j j n

n n n

n l

lt t k t t

t t

0

1 1

1 1

0 0

0 ( ) 2 2 ( ) 2

2α τ α α α α α α β .

Remark 2: For different values of the parameters, we get many interesting results.

For example for k=1,τ =1, Theorem 2 reduces to Theorem G . For t=1, it reduces

to the following result:

Corollary 1: Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that for some 0<k ≤1,0<τ ≤1 and 0≤ln,

kαn ≤αn−1 ≤...≤αl+1 ≤αl ≥αl−1 ≥...≥α1 ≥τα0 ≠0.

Then for 0<δ <1 the number of zeros of P(z) in z ≤δ does not exceed

0 2

* log 1 log

1

a M

δ

,

where

* 2

M

= + + −

+ + + −

= n

j j n

n n

l k

0 0

0

0 ( ) 2 2 ( ) 2

2α τ α α α α α α β .

Applying Theorem 2 to the polynomial –iP(z) , we get the following result:

Corollary 2: Let j

n

j jz

a z

P

= =

0

)

(6)

n j

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

Im( =β = such that for some t>0,0<k ≤1,0<τ ≤1 and 0≤ln,

1 1 0

1 1

1 1

1

...

... β β β β τβ

β

β ≤ ≤ ≤ ≤ ≥ − ≥ ≥ ≥

+ + −

t t

t t

t

kt l

l l l l l n

n n n

.

Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than

0 * * 2

log 1 log

1

a M

δ

,

where

* * 2

M

=

+ +

+

+ + + +

+ + −

= n

j

j j n

n n n

n l

lt t k t t

t t

0

1 1

1 1

0 0

0 ( ) 2 2 ( ) 2

2β τ β β β β β β α .

If aj is real i.e. βj =0,∀j, Theorem 2 gives the following result:

Corollary 3: Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n such that for some

t>0,0<k ≤1,0<τ ≤1 and 0≤ln,

... 1 1 ... 1 0 0

1 1 1

1 ≤ ≤ ≤ ≥ ≥ ≥ ≥ ≠

≤ −

+ + −

a ta a

t a t a t a

t a

kt l l

l l l l n

n n

n τ .

Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than

0 * * * 2

log 1 log

1

a M

δ

,

where

* * * 2

M =2a0 t−τ(a0 +a0)t+2altl+1+2antn+1−k(an +an)tn+1.

Theorem 3: Let j

n

j jz

a z

P

= =

0

)

( b e a polynomial of degree n with Re(aj)=αj,

n j

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

Im( =β = such that for some t>0,0<k1,k2 ≤1,0<τ1,τ2 ≤1 ,0≤ln

and 0≤mn,

k1tnαntn−1αn1 ≤...≤tl+1αl+1tlαltl−1αl1 ≥...≥tα1 ≥τ1α0 ≠0

1 1 2 0

1 1

1 1

1

2 β ≤ β ≤...≤ β ≤ β ≥ β − ≥...≥ β ≥τ β

− +

+ −

t t

t t

t t

k n n n n l l l m l m .

Then for 0<δ <1 the number of zeros of P(z) in z ≤δt is less than

0 3

log 1 log

1

a M

δ

,

where

3

M 1 1

1 1

0 0 1

0 ( ) 2 ( )

2 − + + + + + − + +

= n

n n n

n l

lt t k t

t

t τ α α α α α α

α

1

2 1 1

0 0 1

0 ( ) 2 ( )

2 − + + + + + − + +

+ n

n n n

n m

mt t k t

t

t τ β β β β β β

(7)

Remark 3: For different values of the parameters, we get many interesting results.

For example for k=1,τ =1, Theorem 3 reduces to Theorem H . For t=1, it reduces

to Theorem E .

2. Lemmas

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

Lemma 1: Let z1,z2Cwith z1 ≥ z2 and , 1,2

2

argzj −β ≤α ≤π j = for some real

numbers α and β. Then

z1z2 ≤(z1z2)cosα +(z1 + z2)sinα .

The above lemma is due to Govil and Rahman [3].

Lemma 2: Let F(z) be analytic in zR, F(z) ≤Mfor zRand F(0)≠0. Then

for 0<δ <1 the number of zeros of F(z) in the disk z ≤δt is less than

0

log 1 log

1

a M

δ

.

For the proof of this lemma see [8, P.171].

3. Proofs of Theorems Proof of Theorem 1: Consider the polynomial

n

n n n l

l l l l

l z a z a z a z a z

a z

a z a a z t z P z t z

F( )=( − ) ( )=( − )( 0 + 1 )+ 2 2 +...+ 1 −1 + + +1 +1+... 1 −1+

1

1 1

2 1 2 0

0 0

1 0

1 1

1 1

1 0

1 0

) (

) (

... ) (

)] (

) [(

) (

... )

( ) (

... ) (

+ +

+ −

+ +

− +

− + + −

+ −

+ − + =

− −

+ + −

+ −

+ + − + =

l l l l l l

n n n n n l

l l l l l

z a t a z a t a z

a t a z a a a

t a t a

z a z a t a z

a t a z a t a z

a t a t a

τ

τ

1

1) ( )]

[(

...+ − − − − +

+ n

n n n n n

nt a ka t a t z a z

ka

For z =t, we have by using the hypothesis and Lemma 1,

1 1

1 2

1 2 0 1 0 0

0 ...

)

( ≤ + − + − + − + + − + l+l l+

l l

lt a t a t a t

a t

a t a t a t a t a a t a z

F τ τ

1

1

...+ − + − + +

+ n

n n n n n n

nt a t ka t a t a t

ka

1

1 1

2 1 2 0 1 0

0 (1 ) ...

+ +

− + −

− + + − + − + −

+

= l

l l l l

lt a t a t a t

a t

a t a t a t a t a t

a τ

1 1 (1 )

...+ − + − + +

+ n

n n n n

n

nt a t k a t a t

(8)

a0 t+(1−τ)a0t+[(a1t−τa0)cosα+(a1ta0 )sinα]t 1 1 1 1 1 1 1 1 2 1 2 1 2 ) 1 ( ] sin ) ( cos ) [( ... ] sin ) cos ) [( ] sin ) ( cos ) [( ... ] sin ) ( cos ) [( + − − + + + − − + − + + + − + + + + − + − + − + + + + − + n n n n n n n n n l l l l l l l l l l t a t a k t t a k a t a k a t t a a t a a t a t a a t a t a t a a t a α α α α α α α α

2a0t τ a0 (1 cosα sinα) 2a t 1cosα a t 1(1 ksinα kcosα) n

n l

l + + −

+ − + − = + +

− = + + − + 1 1 1 sin 2 ) 1 ( n j j j n

nt a t

a

k α

=M1.

Since F(z) is analytic for zt and F(z) ≤M1 for z =t, it follows by Lemma 2 and

Maximum Modulus Theorem that the number of zeros of F(z) and hence of P(z) in

t

z ≤δ is less than or equal to

0 1 log 1 log 1 a M δ

and the theorem follows.

Proof of Theorem 2: Consider the polynomial

n n n n l l l l l

l z a z a z a z a z

a z a z a a z t z P z t z

F = − = − + + + + − + + + + + 1 −1+

1 1 1 1 2 2 1

0 ) ... ...

)( ( ) ( ) ( ) ( 1 1 1 2 1 2 0 0 0 1 0 1 1 1 1 1 0 1 0 ) ( ) ( ... ) ( )] ( ) [( ) ( ... ) ( ) ( ... ) ( + + − + − + + − − + − + + − + − + − + = − − + + − + − + + − + = l l l l l l n n n n n l l l l l l z a t a z a t a z a t a z a a a t a t a z a z a t a z a t a z a t a z a t a t a τ τ 1

1) ( )]

[( ...+ − − − − + + n n n n n n

nt a ka t a t z a z

ka

=(α0 +iβ0)t+[{(α1 +iβ1)t−τ(α0 +iβ0)}+{τ(α0 +iβ0)−(α0 +iβ0)}]z

1 1 1 1 1 1 1 1 ) ( )}] ( ) ( { )} ( ) ( [{ ... )] ( ) [( )] ( ) [( ... + − − + + + − − + − + − + − + − + + + + − + + + − + + + n n n n n n n n n n n n l l l l l l l l l l z i z i t i k i t i k z i t i z i t i β α β α β α β α β α β α β α β α β α 1 1 1 0 0 0 1

0 [( ) ( )] ... ( ) ( )

+ + − + − − + + − + − + = l l l l l

lt z t z

z t

t α τα τα α α α α α

α ]. ) ( ... ) ( [ )] ( ) [( ... 1 1 0 1 0 1 1 + − + − − − + + − + + − − − − + + n n n n n n n n n n n n z z t z t t i z z t t k t k β β β β β β α α α α α

(9)

1 1 1 2 1 2 0 1 0 0 0 ... )

( ≤ + − + − + − + + − + l+l l+

l l

lt t t t

t t t t t t z

F α τα α α τα α α α α α α

1 1 0 1 0 1 1 ) ( ... ) ( ... + − + − + + + + + + + + − + − + + n n n n n n n n n n n n n t t t t t t t t t t k t t k β β β β β β α α α α α l l

lt t

t t t t t

t (1 ) 0 ( 1 0) ( 2 1) 2 ... ( 1)

0 + − + − + − + + − +

≤ α τ α α α α α α α

= + + + − + + + + − + − + + − + n j j j n n n n n n n l l

l t t k t t k t t t

0 1 1 1 1 1

1 ) ... ( ) (1 ) 2

(α α α α α α β

= + + + + + + + + − − = n j j j n n n n n l

lt t k t t

t t 0 1 1 1 1 0 0

0 ) 2 2 ( ) 2

2

( α α τα α α α α β

=M2.

The theorem now follows as in the proof of Theorem 1.

Proof of Theorem 3: As in the proof of Theorem 2,

1 1 1 0 0 1 0 1 1

0 [( ) ( )] ... ( ) ( )

)

(z = t+ t− + − z+ + ltl zl + l+tl zl+

F α α τ α τ α α α α α α

]. )} ( ) {( ... ) ( ) ( ... )} ( ) {( [ )] ( ) [( ... 1 2 1 2 1 1 1 0 0 2 0 2 1 0 1 1 1 1 + − + + − + − − − − − + + − + − + + − + − + + − − − − + + n n n n n n n m m m m m m n n n n n n n z z t t k t k z t z t z t t i z z t t k t k β β β β β β β β β β β τ β τ β β α α α α α

For z =t, we have by using the hypothesis,

1 1 1 2 1 2 0 1 0 0 0 ... )

( ≤ + − + − + − + + − + l+l l+

l l

lt t t t

t t t t t t z

F α τα α α τα α α α α α α

1 2 1 2 1 1 1 0 0 2 0 2 1 0 1 1 1 1 ... ... ... + − + + − + − + − + − + + − + − + + − + − + + + − + − + + n n n n n n n n m m m m m m n n n n n n n n t t t t k t t k t t t t t t t t t t t t k t t k β β β β β β β β β β β τ β τ β β α α α α α l l

lt t

t t t t

t

t (1 1) 0 ( 1 1 0) ( 2 1) 2 ... ( 1)

0 + − + − + − + + − −

(10)

1 1

1 1

0 0 1

0 ( ) 2 ( )

2 − + + + + + − + +

= n

n n n

n l

lt t k t

t

t τ α α α α α α

α

1

2 1 1

0 0 1

0 ( ) 2 ( )

2 − + + + + + − + +

+ n

n n n

n m

mt t k t

t

t τ β β β β β β

β .

=M3.

The result now follows as in the proof of Theorem 1.

References

[1] K.K. Dewan, Extremal Properties and Coefficient Estimates for Polynomials with Restricted Coefficients and on Location of Zeros of Polynomials,Ph.D Thesis, IIT Delhi, 1980.

[2] R. Gardner and B. Sheilds, The Number of Zeros of a Polynomial in a Disk, Journal of Classical Analysis, Vol.3, No.2(2013), 167-176.

[3] N. K. Govil and Q. I. Rahman, On the Enestrom-Kakeya Theorem, Tohoku Mathematics Journal, 20(1968), 126-136.

[4] M. H. Gulzar, On the Zeros of Polynomials with Restricted Coefficients ISST Journal of Mathematics &Computing System, Vol.3 No.2 (July- December 2012), 33-40.

[5] M. H. Gulzar, On the Zeros of a Polynomial, Int. Journal of Scientific and Research Publications, Vol. 2 Issue 7,July 2012 , 1-7.

[6] M. Marden, , Geometry of Polynomials,Math. Surveys No. 3, Amer. Math

Society(R.I. Providence) (1966).

[7] Q. G. Mohammad, On the Zeros of Polynomials, American Mathematical Monthly, 72 (6) (1965), 631-633.

[8] E. C. Titchmarsh, The Theory of Functions, 2nd Edition, Oxford University Press

London, 1939.

References

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