[Paper Review] Inequalities for the polar derivative of a polynomial

N. A. Rather, Suhail Gulzar|arXiv (Cornell University)|Mar 1, 2013

Mathematical functions and polynomials5 references3 citations

TL;DR

This paper establishes sharp inequalities for the polar derivative of polynomials under various zero-structure constraints, generalizing classical results like Bernstein's and Turán's inequalities. It introduces a novel framework using the s-th order polar derivative and proves that for polynomials with all zeros in |z| ≤ k (k ≤ 1), the magnitude of the s-th polar derivative is bounded by a combination of the maximum and minimum modulus on |z| = k, with explicit dependence on the parameters α_j and β.

ABSTRACT

Let $ P(z) $ be a polynomial of degree $ n $ and for any real or complex number $α,$ let $D_αP(z)=nP(z)+(α-z)P^{\prime}(z)$ denote the polar derivative with respect to $α.$ In this paper, we obtain generalizations of some inequalities for the polar derivative of a polynomial.

Motivation & Objective

• To generalize classical inequalities for polynomial derivatives—such as Bernstein’s and Turán’s—by introducing the polar derivative operator.
• To extend existing results on the s-th polar derivative of polynomials that do not vanish in |z| < 1 to cases where all zeros lie in |z| ≤ k for k ≤ 1.
• To derive sharp upper bounds for the s-th polar derivative in terms of the maximum and minimum modulus of the polynomial on |z| = k.
• To unify and refine prior inequalities by incorporating complex parameters α_j and β, and to establish equality conditions.

Proposed method

• The authors define the s-th order polar derivative of a polynomial P(z) recursively via the operator D_αP(z) = nP(z) + (α - z)P'(z), generalizing the ordinary derivative.
• They introduce a comparison polynomial F(z) = z^n M / k^n, where M is the maximum modulus of P(z) on |z| = k, to construct a majorant for bounding the polar derivative.
• Using Rouche’s theorem and the method of reciprocal polynomials, they analyze the behavior of P(z) − λm (for |λ| < 1) to ensure no zeros in |z| < k when P(z) has all zeros in |z| > k.
• They apply the main inequality (Theorem 1.1) to the reciprocal polynomial g(z) = z^n ar{P}(1/ar{z}) − ar{λ}m to transfer zero distribution properties to the unit disk.
• By choosing the argument of λ to maximize the lower bound, they derive a sharp inequality combining upper and lower bounds via triangle inequality manipulation.
• The final bound is obtained by summing two inequalities—one from Theorem 1.9 and one from Theorem 1.11—resulting in a unified inequality involving both max and min modulus on |z| = k.

Experimental results

Research questions

• RQ1 How can classical inequalities for polynomial derivatives be generalized to the polar derivative operator when the polynomial has all zeros in |z| ≤ k for k ≤ 1?
• RQ2 What is the sharp upper bound for the s-th polar derivative |D_{α_s}⋯D_{α_1}P(z)| in terms of the maximum and minimum modulus of P(z) on |z| = k?
• RQ3 Can the bound be expressed in a form that unifies both the extremal cases of maximum and minimum modulus, and what role do the parameters α_j and β play?
• RQ4 Under what conditions does equality hold in the generalized polar derivative inequality, and how does this relate to known extremal polynomials like (z^n + 1)/2?

Key findings

• Theorem 1.1 provides a sharp inequality for the s-th polar derivative of a polynomial P(z) with all zeros in |z| ≤ k (k ≤ 1), bounded by a combination of the maximum modulus of P(z) on |z| = k and a comparison polynomial F(z).
• Corollary 1.2 gives a concrete upper bound for |z^s P_s(z) + β n_s Λ_s / (1+k)^s P(z)| in terms of the maximum modulus of P(z) on |z| = k, with explicit dependence on |α_j| ≥ k and |β| ≤ 1.
• Corollary 1.3 is obtained by taking |α| → ∞, yielding a bound for the s-th ordinary derivative involving the maximum modulus of P(z) on |z| = k.
• Theorem 1.11 establishes a sharp inequality combining both max and min modulus on |z| = k: the upper bound is proportional to { |z^s + βΛ_s/(1+k)^s| + |z^n α_1⋯α_s + βΛ_s/(1+k)^s|/k^n } times the max modulus.
• The equality case is achieved for the polynomial P(z) = (z^n + 1)/2 when all α_j = ∞ and β = 1, confirming sharpness.
• The proof technique, involving Rouche’s theorem and reciprocal polynomials, allows the derivation of a unified inequality that reduces to known results in limiting cases.