Certain Inequalities for the generalized polar derivative of a polynomial

TL;DR

This work extends Bernstein‑type and polar‑derivative inequalities to the generalized polar derivative $D_{\alpha}^{\gamma}[P](z)$ for polynomials with zeros in $|z|\le k$ ($k\ge1$). By defining the generalized derivative $P^{\gamma}(z)$ and its polar form, the authors derive sharp lower bounds on $\max_{|z|=1}|P^{\gamma}(z)|$ and $\max_{|z|=1}|D_{\alpha}^{\gamma}[P](z)|$ in terms of $\max_{|z|=1}|P(z)|$, the coefficient data $a_0,a_n$, and the parameters $\wedge$ and $\gamma_m$. The bounds are shown to be sharp for extremal polynomials such as $P(z)=(z+k)^n$ and extend prior results to the regime $k\ge1$, unifying and broadening earlier findings in the literature on polar derivatives. These results enhance the understanding of zero-placement effects on derivative inequalities and have potential applications in stability and zero localization analyses on the unit circle.

Abstract

Recently Rather et al. \cite{NT} considered the generalized derivative and the generalized polar derivative and studied the relative position of zeros of generalized derivative and generalized polar derivative with respect to the zeros of polynomial.\\ \indent In this paper, we establish some inequalities that estimate the maximum modulus of generalized derivative and the generalized polar derivative of the polynomial $P(z)$, which is also the extension of recently known results.

Theorems & Definitions (18)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem F
• Theorem G
• Theorem H
• Theorem I
• Theorem 1