Akademik Veri Yönetim Sistemi
Turkish Journal of Mathematics, cilt.49, sa.4, ss.491-517, 2025 (SCI-Expanded, Scopus, TRDizin)
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 (Formula presented.) This research examines the polynomial p(z), ensuring that it has no zero in the disk jzj < 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 rule and illustrate graphically the obtained inequalities as regards sharpness.