Some Generalization of Eneström-Kakeya Theorem

International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 3 Issue 2 || February. 2015 || PP-52-59

Some Generalization of Eneström-Kakeya Theorem P.Ramulu Department of Mathematics, Govt.Degree College, Wanaparthy, Mahabubnagar, Telangana,India 509103.

Abstract: In this paper we prove some extension of the Eneström-Kakeya theorem by relaxing the hypothesis of this result in several ways and obtain zero-free regions for polynomials with restricted coefficients and there by present some interesting generalizations and extensions of the Enestrom-Kakeya Theorem.

Mathematics Subject Classification: 30C10, 30C15.

Keywards: Zeros of polynomial, Eneström-Kakeya theorem.

1. INTRODUCTION The well known Results Eneström-Kakeya theorem [1,2] in theory of the distribution of zeros of polynomials is the following.

Theorem ( ).): Let P(z)= be a polynomial of degree n such that then all the zeros of P(z) lie in |z| 1.

Applying the above result to the polynomial

Theorem (

). If P(z) =

P(

we get the following result:

be a polynomial of degree n such that

then P(z) does not vanish in |z|<1 In the literature [3-9], there exist several extensions and generalizations of the Enestrom-Kakeya Theorem.

Recently B. A. Zargar [9] proved the following results:

Theorem (

). If P(z) =

be a polynomial of degree n such that for some k then P(z) does not vanish in the disk |z|<

Theorem (

). If P (z) =

0 ≤ ρ<an,

Theorem (

Theorem (

1,

.

be a polynomial of degree n such that for some real number then P(z) does not vanish in the disk |z|<

). If P (z) =

). If P (z) =

.

be a polynomial of degree n such that for some real number k 1, then P(z) does not vanish in the disk |z|<

be a polynomial of degree n such that for some real number

0

then P(z) does not vanish in the disk |z|< In this paper we give generalizations of the above mentioned results. In fact, we prove the following results:

www.ijmsi.org

52 | P a g e

Some Generalization Of EnestrĂśmâ&#x20AC;Ś

Theorem 1. Let P(z) =

be a polynomial of degree n with real coefficients such that for some k

1

0, then all the zeros of P(z) does not vanish in the disk

Corollary 1.. Let P(z) =

be a polynomial of degree n with real coefficients such that for some

then all the zeros of P(z) does not vanish in the disk

Corollary 2. Let P(z) = some k

1

be a polynomial of degree n with positive real coefficients such that for

0,

then all the zeros of P(z) does not vanish in the disk

Remark 1. (i) By taking

and k = 1 in theorem 1, then it reduces to Corollary 1.

(ii) By taking

theorem 1, then it reduces to Corollary 2.

Theorem 2. Let P(z) =

be a polynomial of degree n with real coefficients such that for some

, then all the zeros of P(z) does not vanish in the disk

Corollary 3. Let P(z) =

be a polynomial of degree n with real coefficients such that for some

, then all the zeros of P(z) does not vanish in the disk

Corollary 4. Let P(z) =

be a polynomial of degree n with positive real coefficients such that for

some , then all the zeros of P(z) does not vanish in the disk

www.ijmsi.org

53 | P a g e

Some Generalization Of EnestrĂśmâ&#x20AC;Ś

Remark 2. (i) By taking (ii) By taking

in theorem 2 then it reduces to Corollary 3. in theorem 2, then it reduces to Corollary 4.

Theorem 3. Let P(z) =

be a polynomial of degree n with real coefficients such that for some

, 0, then all the zeros of P(z) does not vanish in the disk

Corollary 5. Let P(z) =

.

be a polynomial of degree n with real coefficients such that for some ,

then all the zeros of P(z) does not vanish in the disk

Corollary 6. Let P(z) =

.

be a polynomial of degree n with Positive real coefficients such that for

some ,

0,

then all the zeros of P(z) does not vanish in the disk

.

Remark 3. (i) By taking (ii) By taking

and r = 1 in theorem 3, then it reduces to Corollary 5. in theorem 3, then it reduces to Corollary 6..

Theorem 4. Let P(z) =

be a polynomial of degree n with real coefficients such that for some

, 0, then all the zeros of P(z) does not vanish in the disk

Corollary 7. Let P(z) =

.

be a polynomial of degree n with real coefficients such that for some

,

then all the zeros of P(z) does not vanish in the disk

Corollary 8. Let P(z) =

.

be a polynomial of degree n with positive real coefficients such that for

some ,

0,

www.ijmsi.org

54 | P a g e

Some Generalization Of EnestrĂśmâ&#x20AC;Ś then all the zeros of P(z) does not vanish in the disk

.

Remark 4. (i) By taking (ii) By taking

in theorem 4, then it reduces to Corollary 7. in theorem 4, then it reduces to Corollary 8.

2. Proofs of the Theorems Proof of the Theorem 1.

Let P(z)= Let Consider the polynomial J(z)=

P( )

And R(z)= (z-1)J(z) so that Then R(z)=(z-1)( =

)

{ }

Also if

then

Now

[|z| -

]

[|z| ] [|z| ] [|z| if |z|

] [

]

This shows that all the zeros of R(z) whose modulus is greater than 1 lie in the closed disk |z| [ ] But those zeros of R(z) whose modulus is less than or equal to 1 already lie in the above disk.

Therefore, it follows that all the zeros of R(z) and hence J(z) lie in |z| [ ] Since P(z) =

J(

it follows, by replacing z by

,

Then all the zeros of P(z) lie in

|z| Hence P(z) does not vanish in the disk

www.ijmsi.org

55 | P a g e

Some Generalization Of Eneström…

|z| This completes the proof of the Theorem 1.

Proof of the Theorem 2.

Let P(z)= Let Consider the polynomial J(z)=

P( )

And R(z)= (z-1)J(z) so that Then R(z)=(z-1)( =

)

{ } then

Also if Now

[|z| -

]

[|z|

+1+…+|

−1−

|+|

| }]

[|z| ] [|z| if |z|

{4

|

|+2

+

}]

[

]

This shows that all the zeros of R(z) whose modulus is greater than 1 lie in the closed disk |z| [ ] But those zeros of R(z) whose modulus is less than or equal to 1 already lie in the above disk.

Therefore, it follows that all the zeros of R(z) and hence J(z) lie in |z| [ ] Since P(z) = J( it follows, by replacing z by Then all the zeros of P(z) lie in

,

|z| Hence P(z) does not vanish in the disk

|z| This completes the proof of the Theorem 2.

www.ijmsi.org

56 | P a g e

Some Generalization Of Eneström…

Proof of the Theorem 3.

Let P(z)= Let Consider the polynomial J(z)=

P( )

And R(z)= (z-1)J(z) so that Then R(z)=(z-1)( =

)

{ }

Also if

then

Now

[|z| -

]

[|z|

+1|+…+|

−1+ −

− |+|

| }]

[|z| ] [|z| if |z|

{ |

|+2

+

| )+

+2 } ]

[

]

This shows that all the zeros of R(z) whose modulus is greater than 1 lie in the closed disk |z| [ ] But those zeros of R(z) whose modulus is less than or equal to 1 already lie in the above disk.

Therefore, it follows that all the zeros of R(z) and hence J(z) lie in |z| [| |+2 + | )+ +2 ] Since P(z) = J( it follows, by replacing z by Then all the zeros of P(z) lie in

,

|z| Hence P(z) does not vanish in the disk

|z| This completes the proof of the Theorem 3.

Proof of the Theorem 4.

Let P(z)= Let Consider the polynomial J(z)=

P( )

And R(z)= (z-1)J(z) so that

www.ijmsi.org

57 | P a g e

Some Generalization Of Eneström…

Then R(z)=(z-1)( =

)

{ }

Also if

then

Now

[|z|

]

[|z|

+1+…+|

−1−

|+|

| }]

[|z| ] [|z| if |z|

{

2

[

2

+ 4 }] + 4 ]

This shows that all the zeros of R(z) whose modulus is greater than 1 lie in the closed disk |z| [ 2 + 4 ] But those zeros of R(z) whose modulus is less than or equal to 1 already lie in the above disk.

Therefore, it follows that all the zeros of R(z) and hence J(z) lie in |z| 2 + 4 ] Since P(z) = J( it follows, by replacing z by Then all the zeros of P(z) lie in

,

|z| Hence P(z) does not vanish in the disk |z| This completes the proof of the Theorem 4.

REFERENCES [1].

[2].

[3].

[4].

[5].

G.Eneström,Remarquee sur un théorème relatif aux racines de l’equation +...+ =0 oü tous les coefficient sont et positifs, Tôhoku Math.J 18 (1920),34-36.

S.KAKEYA, On the limits of the roots of an alegebraic equation with positive coefficient, Tôhoku Math.J 2 (1912-1913),140-142.

Dewan, K. K., and Bidkham, M., On the Enesrtrom-Kakeya Theorem I., J. Math. Anal. Appl., 180 (1993) , 29-36.

Kurl Dilcher, A generalization of Enesrtrom-Kakeya Theorem, J. Math. Anal. Appl., 116 (1986) , 473488.

Joyal, A., Labelle, G. and Rahman, Q. I., On the Location of zeros of polynomials, Canad. Math. Bull., 10 (1967),53-63.

www.ijmsi.org

58 | P a g e

Some Generalization Of EnestrĂśmâ&#x20AC;Ś [6].

[7].

[8].

[9].

[10]

Marden, M., Geometry of Polynomials, IInd. Edition, Surveys 3, Amer. Math. Soc., Providence, (1966) R.I. Milovanoic, G. V., Mitrinovic , D. S., Rassiass Th. M., Topics in Polynomials, Extremal problems, Inequalities,Zeros, World Scientific , Singapore, 1994.

Rahman, Q. I., and Schmeisser, G., Analytic Theory of Polynomials, 2002, Clarendon Press, Oxford.

Zargar, B. A., Zero-free regions for polynomials with restricted coefficients, International Journal of Mathematical Sciences and Engineering Applications, Vol. 6 No. IV (July 2012), 33-42.

P.Ramulu, G.L .Reddy ,On the Enestrom-Kakeya theorem. International Journal of Pure and Applied Mathematics, Vol. 102 No.4, 2015.(Up Coming issue)

www.ijmsi.org

59 | P a g e