Extensions of some polynomial inequalities to the polar derivative

Abstract

Let p(z) be a polynomial of degree n and for any real or complex number α, let D α p(z) = np(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.

Some Bounds for the Polar Derivative of a Polynomial

“…Theorem 4 (see [8]). If ] , 1 ≤ ≤ , 0 ̸ = 0, be a polynomial of degree having all its zeros in | | ≤ , ≥ 1.…”

Inequalities for the polar derivative of a polynomial with restricted zeros

“…Recently Dewan et al [9] generalized the inequality (1.6) to the polynomial of the form p(z) = a 0 + n ν=t a ν z ν , 1 ≤ t ≤ n, and proved if p(z) = a 0 + n ν=t a ν z ν , 1 ≤ t ≤ n, is a polynomial of degree n having no zeros in |z| < k, k ≥ 1 then for |α| ≥ 1,…”

Some integral mean estimates for polynomials

“…where m = Min |z|=k |P (z)| and q(z) = z n P ( 1 z ). The above lemma is due to Dewan, Singh and Mir [5]. Lemma 3.…”