•  
  •  
 

Turkish Journal of Mathematics

Author ORCID Identifier

NIRMAL SINGHA: 0000-0003-3060-0165

FAHREDDİN G. ABDULLAYEV: 0000-0002-9711-0796

BARCHAD CHANAM: 0000-0001-6397-9066

Abstract

A well-known theorem due to Ankeny and Rivlin states that if p(z) is a polynomial of degree n such that p(z) has no zero in |z| < 1, then
max|z|=R≥1 |p(z)| ≤ (Rn + 1 / 2) max|z|=1 |p(z)|.
This research examines the polynomial p(z), ensuring that it has no zero in the disk |z| < k, where k ≥ 1. At the same time, we investigate the sth derivative of this polynomial, where 0 ≤ s < n. In our effort to establish integral formulations of the inequalities related to the derivatives of this class of polynomials, we have successfully extended and generalized Ankeny and Rivlin’s inequality to integral settings. Additionally, part of our findings provides integral analogs of results by Mir [J. Anal., 27 (2019), 851−857]. Moreover, another aspect of our work leads to an improvement in the result of Jain [Turk. J. Math., 31 (2007), 89−94], which we have also verified using an example. We have also compared our results with a previously known result using this numerical example, where the bounds that are in terms of integral means are estimated numerically by numerical integration using Simpson’s 1/3rd rule and illustrate graphically the obtained inequalities as regards sharpness.

DOI

10.55730/1300-0098.3603

Keywords

Polynomial, zero, maximum modulus, sth derivative, integral inequality, Simpson's 1/3 rd rule

First Page

491

Last Page

517

Publisher

The Scientific and Technological Research Council of Türkiye (TÜBİTAK)

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Download

Included in

Mathematics Commons

Share

COinS