Well-known Polynomial Inequalities Research Articles

Extension of Some Polynomial Inequalities to the Polar Derivative and the Generalized Polar Derivative

In this paper certain polynomial inequalities for the polar derivative and the generalized polar derivative with restricted zeros are given, which generalize and refine some well-known polynomial inequalities due to Lax, Tura ́n, AlSaeedi, Rather, Ali, Shafi, and Dar and others.

SOME POLYNOMIAL INEQUALITIES IN THE COMPLEX DOMAIN WITH PRESCRIBED ZEROS

In this paper certain polynomial inequalities with restricted zeros are given, which generalize and refine some well-known polynomial inequalities due to Ankeny and Rivlin, Dewan, Singh and Mir, Mir, Wani and Hussain and others.

On the derivative of a polynomial

In this paper, we establish some inequalities for the complex polynomial. Our results constitute generalizations and refinements of some well-known polynomial inequalities.

Inequalities for the polar derivative of a polynomial with restricted zeros

For a polynomial p(z) of degree n, we consider an operator Dα which map a polynomial p(z) into Dαp(z) := (α-z)p'(z)+ np(z) with respect to α. It was proved by Liman et al. [A. Liman, R. N. Mohapatra and W. M. Shah, Inequalities for the Polar Derivative of a Polynomial, Complex Analysis and Operator Theory, 2010] that if p(z) has no zeros in |z| < 1 then for all α, β _ C with |α| ≥ 1, |β| ≤ 1 and |z| = 1, |zDαp(z) + nβ |α| 1 2 p(z) |≤ n 2 {[| α+β |α| 1 2 |] + max | p (z)|z|-1 |} - [|α + β |α| - 1 2 |-| z + β |α| - 1 2|] min | p(z) |z|=1|}. In this paper we extend above inequality for the polynomials having no zeros in |z| < 1, except s-fold zeros at the origin. Our result generalize certain well-known polynomial inequalities.

Обобщение некоторых известных неравенств для производноИ многочленов

For a polynomial P(z) of degree n which has no zeros in z < 1, Dewan et al. [4] established the inequality $$\left| {zP'(z) + \frac{{\pi \beta }}{2}P(z)} \right| \leqslant \frac{n}{2}\left\{ {\left( {\left| {\frac{\beta }{2}} \right| + \left| {1 + \frac{\beta }{2}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right| - \left( {\left| {1 + \frac{\beta }{2}} \right| - \left| {\frac{\beta }{2}} \right|} \right)\mathop {\min }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|} \right\},$$ for any β ≤ 1 and z = 1. In this paper we improve the above inequality for the sth derivative of a polynomial which has no zeros in z < k, k ≤ 1. Our results generalize certain well-known polynomial inequalities.

On some inequalities concerning the polar derivative of a polynomial

In this paper we establish some inequalities concerning to polar derivative of polynomial having all its zeros inside or outside a unit circle and thereby present some compact generalizations of certain well-known polynomial inequalities.

Some Results on the Polar Derivative of a Polynomial

Let P(z) be a polynomial of degree n and for any complex number α, let Dα P(z) = n P(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect to α. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.

AN $L^p$ INEQUALITY FOR POLYNOMIALS NOT VANISHING IN A DISK

For the class of polynomials P(z) = a0 + P n=µ az � , 1 ≤ µ ≤ n, of degree n not vanishing in the disk |z| r ≥ 1, p > 0 and present compact generalizations of certain well-known polynomial inequalities.

Inequalities for a polynomial and its derivative

For a polynomial p(z) of degree n which has no zeros in |z|<1, Liman et al. (Appl. Math. Comput. 218:949-955, 2011) established for all α,β∈C with |α|≤1, |β|≤1, R>r≥1 and |z|=1. In this paper, we extend the above inequality for the polynomials having no zeros in |z|<k, k≤1. Our result generalizes certain well-known polynomial inequalities.MSC:30A10, 30C10, 30D15.

Some Inequalities for the Polar Derivative of Polynomials in Complex Domain

Let p(z) be a polynomial of degree n. In this paper we prove results con- cerning maximum modulus of the polar derivative of p(z) with restricted zeros. Our results refine and generalize certain well-known polynomial inequalities.

On the maximum modulus of a polynomial and its polar derivative

For a polynomial p(z) of degree n, having all zeros in |z| ≤ 1, Jain is shown that In this paper, the above inequality is extended for the polynomials having all zeros in |z| ≤ k, where k ≤ 1. Our result generalizes certain well-known polynomial inequalities. (2010) Mathematics Subject Classification. Primary 30A10; Secondary 30C10, 30D15.

Inequalities for the Polar Derivative of a Polynomial

Let P(z) be a polynomial of degree at most n. We consider an operator Dα which maps a polynomial P(z) into DαP(z) := nP(z) + (α − z)P′(z) and prove results concerning the estimates of |DαP(z)| on the disk |z| = R ≥ 1, and thereby obtain extensions and generalizations of a number of well-known polynomial inequalities.

Generalization of certain well-known polynomial inequalities

If P ( z ) = ∑ ν = 0 n c ν z ν is a polynomial of degree n having no zeros in | z | < 1 , then for | β | ⩽ 1 , it was proved by Jain [V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Mat. 32 (52) (1997) 45–51] that | z P ′ ( z ) + n β 2 P ( z ) | ⩽ n 2 { | 1 + β 2 | + | β 2 | } max | z | = 1 | P ( z ) | , | z | = 1 . In this paper, we shall first obtain a result concerning minimum modulus of polynomials and next we improve upon the above inequality for the polynomials with restricted zeros. Our results refine and generalize certain well-known polynomial inequalities including some results of Bernstein, Lax, Malik and Vong, and Aziz and Dawood.

Inequalities for the Polar Derivative of a Polynomial

Let be a polynomial of degree and for any real or complex number , and let denote the polar derivative of the polynomial with respect to . In this paper, we obtain new results concerning the maximum modulus of a polar derivative of a polynomial with restricted zeros. Our results generalize as well as improve upon some well-known polynomial inequalities.

Extensions of some polynomial inequalities to the polar derivative

Let p ( z ) be a polynomial of degree n and for any real or complex number α, let D α p ( z ) = n p ( z ) + ( α − z ) p ′ ( z ) denote the polar derivative of the polynomial p ( z ) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros inside or outside a circle. Our results shall generalize and sharpen some well-known polynomial inequalities.