![ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (143-145), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 143
Paper Publications
On The Zeros of Polar Derivatives
1
P.Ramulu, 2
G.L.Reddy
1
Department of Mathematics, Govt.Degree College, Wanaparthy, Mahabubnagar, India-509103
2
Schools of Mathematics and Statistics, University of Hyderabad, India-500046
Abstract: We extend some existing results on the zeros of polar derivatives of polynomials by considering more
general coefficient conditions. As special cases the extended results yield much simpler expressions for the upper
bounds of zeros than those of the existing results.
Mathematics Subject Classification: 30C10, 30C15.
Keywords: Zeros of polynomial, Eneström-Kakeya theorem, Polar derivatives.
1. INTRODUCTION
Let denote ( ) ( ) ( ) ( ) is a polar derivative of P(z) with respect to real number . The polynomial
( ) is of degree at most n-1 and it generalizes the ordinary derivative in the sense that
( )
( ). Many
results on the location of zeros of polynomials are available in the literature. In literature [3-5] attempts have been made to
extend and generalize the Enestrom-Kakeya theorem. Existing results in the literature also show that there is a need to
find bounds for special polynomials, for example, for those having restrictions on the coefficient, there is always need for
refinement of results in this subject to find location of zeros of polar derivatives of polynomials. Among them the
Eneström-Kakeya theorem [1-2] given below is well known in the theory of zero distribution of polynomials.
Theorem (Enestrom-Kakeya theorem): Let P(z)= ∑ be a polynomial of degree n such that
then all the zeros of P(z) lie in |z| 1.
Here we establish zeros of polar derivatives of polynomial by using Eneström-Kakeya Theorem.
Theorem 1. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar
derivative of P(z) with respect to real number such that
( ) ( ) then all the zeros of polar derivative ( )
lie in , -.
Corollary 1. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar
derivative of P(z) with respect to real number such that
( ) ( ) then all the zeros of polar derivative
( ) lie in .
Remark 1.
By taking in theorem 1, then it reduces to Corollary 2.
Theorem 2. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar
derivative of P(z) with respect to real number such that
( ) ( ) then all the zeros of polar derivative ( )
lie in , -.](https://image.slidesharecdn.com/onthezerosofpolarderivatives-261-170203085417/85/On-the-Zeros-of-Polar-Derivatives-1-320.jpg)
![ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (143-145), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 144
Paper Publications
Corollary 2. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar
derivative of P(z) with respect to real number such that
( ) ( ) then all the zeros of polar derivative
( ) lie in , -.
Remark 2.
By taking in theorem 2, then it reduces to Corollary 2.
2. PROOFS OF THE THEOREMS
Proof of the Theorem 1.
Let ( ) be a polynomial of degree n
Let denote ( ) ( ) ( ) ( ) is polar derivative of P(z) with respect to real number of degree n-1.
If then ( ) ( ) ( )
( ) ( ) ( )
Let us consider the polynomial ( ) ( ) ( ) so that
Q(z)=(1-z)( ( ) ( ) )
( ) ( ) ,( ) ( ) - ,( )
( ) - ,( ) -
Also if then
Now ( ) * ( ) (
) ( ) ( ) ( ) +
, *
( ) ( ) ( ) ( ) ( )
+ ]
, * ( )
( ) ( ) ( ) ( ) +-
, * , - , - ,( )
( ) - ,( ) ( ) - ,( ) - +-
, * +-
if , -
This shows that if Q(z) 0 provided , -
Hence all the zeros of Q(z) with |z| lie in , -.
But those zeros of Q(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Since all the zeros of
polar derivative ( ) are also the zeros of Q(z) lie in the circle defined by the above inequality and this completes the
proof of the Theorem 1.
Proof of the Theorem 2.
Let ( ) be a polynomial of degree n
Let denote ( ) ( ) ( ) ( ) is polar derivative of P(z) with respect to real number of degree n-1.](https://image.slidesharecdn.com/onthezerosofpolarderivatives-261-170203085417/85/On-the-Zeros-of-Polar-Derivatives-2-320.jpg)
![ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Vol. 2, Issue 1, pp: (143-145), Month: April 2015 – September 2015, Available at: www.paperpublications.org
Page | 145
Paper Publications
If then ( ) ( ) ( )
( ) ( ) ( )
Let us consider the polynomial ( ) ( ) ( ) so that
Q(z)=(1-z)( ( ) ( ) )
( ) ( ) ,( ) ( ) - ,( )
( ) - ,( ) -
Also if then
Now ( ) * ( ) (
) ( ) ( ) ( ) +
, *
( ) ( ) ( ) ( ) ( )
+ ]
, * ( )
( ) ( ) ( ) ( ) +-
, * , - , - ,( )
( ) - ,( ) ( ) - , ( ) - +-
, * +-
if , -
This shows that if Q(z) 0 provided , -
Hence all the zeros of Q(z) with |z| lie in , -.
But those zeros of Q(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Since all the zeros of
polar derivative ( ) are also the zeros of Q(z) lie in the circle defined by the above inequality and this completes the
proof of the Theorem 2..
REFERENCES
[1] 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.
[2] S.KAKEYA, On the limits of the roots of an alegebraic equation with positive coefficient, Tôhoku Math.J 2 (1912-
1913),140-142.
[3] 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).
[4] Gulshan Singh, Wali Mohammad Shah, Yash Paul, Inequalities for the Polar Derivative of a Polynomial, Advances
in Pure Mathematics, 2011, 1, 23-27.
[5] P.Ramulu, Some Generalization of Eneström-Kakeya Theorem, International Journal of Mathematics and Statistics
Invention (IJMSI), Volume 3 Issue 2 || February. 2015 || PP-52-59.](https://image.slidesharecdn.com/onthezerosofpolarderivatives-261-170203085417/85/On-the-Zeros-of-Polar-Derivatives-3-320.jpg)
On the Zeros of Polar Derivatives
- 1. ISSN 2350-1022 International Journal of Recent Research in Mathematics Computer Science and Information Technology Vol. 2, Issue 1, pp: (143-145), Month: April 2015 – September 2015, Available at: www.paperpublications.org Page | 143 Paper Publications On The Zeros of Polar Derivatives 1 P.Ramulu, 2 G.L.Reddy 1 Department of Mathematics, Govt.Degree College, Wanaparthy, Mahabubnagar, India-509103 2 Schools of Mathematics and Statistics, University of Hyderabad, India-500046 Abstract: We extend some existing results on the zeros of polar derivatives of polynomials by considering more general coefficient conditions. As special cases the extended results yield much simpler expressions for the upper bounds of zeros than those of the existing results. Mathematics Subject Classification: 30C10, 30C15. Keywords: Zeros of polynomial, Eneström-Kakeya theorem, Polar derivatives. 1. INTRODUCTION Let denote ( ) ( ) ( ) ( ) is a polar derivative of P(z) with respect to real number . The polynomial ( ) is of degree at most n-1 and it generalizes the ordinary derivative in the sense that ( ) ( ). Many results on the location of zeros of polynomials are available in the literature. In literature [3-5] attempts have been made to extend and generalize the Enestrom-Kakeya theorem. Existing results in the literature also show that there is a need to find bounds for special polynomials, for example, for those having restrictions on the coefficient, there is always need for refinement of results in this subject to find location of zeros of polar derivatives of polynomials. Among them the Eneström-Kakeya theorem [1-2] given below is well known in the theory of zero distribution of polynomials. Theorem (Enestrom-Kakeya theorem): Let P(z)= ∑ be a polynomial of degree n such that then all the zeros of P(z) lie in |z| 1. Here we establish zeros of polar derivatives of polynomial by using Eneström-Kakeya Theorem. Theorem 1. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar derivative of P(z) with respect to real number such that ( ) ( ) then all the zeros of polar derivative ( ) lie in , -. Corollary 1. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar derivative of P(z) with respect to real number such that ( ) ( ) then all the zeros of polar derivative ( ) lie in . Remark 1. By taking in theorem 1, then it reduces to Corollary 2. Theorem 2. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar derivative of P(z) with respect to real number such that ( ) ( ) then all the zeros of polar derivative ( ) lie in , -.
- 2. ISSN 2350-1022 International Journal of Recent Research in Mathematics Computer Science and Information Technology Vol. 2, Issue 1, pp: (143-145), Month: April 2015 – September 2015, Available at: www.paperpublications.org Page | 144 Paper Publications Corollary 2. Let P(z)= ∑ be a polynomial of degree n and let denote ( ) ( ) ( ) ( ) is a polar derivative of P(z) with respect to real number such that ( ) ( ) then all the zeros of polar derivative ( ) lie in , -. Remark 2. By taking in theorem 2, then it reduces to Corollary 2. 2. PROOFS OF THE THEOREMS Proof of the Theorem 1. Let ( ) be a polynomial of degree n Let denote ( ) ( ) ( ) ( ) is polar derivative of P(z) with respect to real number of degree n-1. If then ( ) ( ) ( ) ( ) ( ) ( ) Let us consider the polynomial ( ) ( ) ( ) so that Q(z)=(1-z)( ( ) ( ) ) ( ) ( ) ,( ) ( ) - ,( ) ( ) - ,( ) - Also if then Now ( ) * ( ) ( ) ( ) ( ) ( ) + , * ( ) ( ) ( ) ( ) ( ) + ] , * ( ) ( ) ( ) ( ) ( ) +- , * , - , - ,( ) ( ) - ,( ) ( ) - ,( ) - +- , * +- if , - This shows that if Q(z) 0 provided , - Hence all the zeros of Q(z) with |z| lie in , -. But those zeros of Q(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Since all the zeros of polar derivative ( ) are also the zeros of Q(z) lie in the circle defined by the above inequality and this completes the proof of the Theorem 1. Proof of the Theorem 2. Let ( ) be a polynomial of degree n Let denote ( ) ( ) ( ) ( ) is polar derivative of P(z) with respect to real number of degree n-1.
- 3. ISSN 2350-1022 International Journal of Recent Research in Mathematics Computer Science and Information Technology Vol. 2, Issue 1, pp: (143-145), Month: April 2015 – September 2015, Available at: www.paperpublications.org Page | 145 Paper Publications If then ( ) ( ) ( ) ( ) ( ) ( ) Let us consider the polynomial ( ) ( ) ( ) so that Q(z)=(1-z)( ( ) ( ) ) ( ) ( ) ,( ) ( ) - ,( ) ( ) - ,( ) - Also if then Now ( ) * ( ) ( ) ( ) ( ) ( ) + , * ( ) ( ) ( ) ( ) ( ) + ] , * ( ) ( ) ( ) ( ) ( ) +- , * , - , - ,( ) ( ) - ,( ) ( ) - , ( ) - +- , * +- if , - This shows that if Q(z) 0 provided , - Hence all the zeros of Q(z) with |z| lie in , -. But those zeros of Q(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Since all the zeros of polar derivative ( ) are also the zeros of Q(z) lie in the circle defined by the above inequality and this completes the proof of the Theorem 2.. REFERENCES [1] 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. [2] S.KAKEYA, On the limits of the roots of an alegebraic equation with positive coefficient, Tôhoku Math.J 2 (1912- 1913),140-142. [3] 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). [4] Gulshan Singh, Wali Mohammad Shah, Yash Paul, Inequalities for the Polar Derivative of a Polynomial, Advances in Pure Mathematics, 2011, 1, 23-27. [5] P.Ramulu, Some Generalization of Eneström-Kakeya Theorem, International Journal of Mathematics and Statistics Invention (IJMSI), Volume 3 Issue 2 || February. 2015 || PP-52-59.