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This research aims to establish growth estimates for the derivatives of polynomials that remain non-zero within a specified disk.

March 18, 2026Open Access

On the growth and derivatives of a polynomial

Key Points

This research aims to establish growth estimates for the derivatives of polynomials that remain non-zero within a specified disk.
Analyzed polynomial behaviors in the complex plane.
Derived supnorm bounds for higher-order derivatives.
Generalized existing inequalities from prior studies in approximation theory.
Introduced new bounds for polynomial growth that do not vanish in a positive radius.
Extended inequalities of Ankeny and Rivlin to higher-order derivatives.
Provided insights that imply improved approximation techniques.

Abstract

This paper is devoted to Bernstein-type growth estimates for polynomials that do not vanish within a disk of positive radius. We derive several new supnorm bounds for the growth of higher-order derivatives of such polynomials in the complex plane. These results encompass generalizations of the well-known inequality of Ankeny and Rivlin (Pacific J. Math., 5: 849-852, 1955), along with extensions of significant inequalities in approximation theory, notably those obtained by Jain (Turk. J. Math., 31: 89?94, 2007).

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Authors

Abdullah Mir

Abid Wani

Journals

Applicable Analysis and Discrete Mathematics

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University of Kashmir

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On the growth and derivatives of a polynomial

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Abstract

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