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 \(\begin{aligned} \max _{|z|=R\ge 1}|p(z)|\le \left( \frac{R^{n}+1}{2}\right) \max _{|z|=1}|p(z)|. \end{aligned}\) This research investigates the polynomial p(z) ensuring it possesses no zero in \(|z|<k\) where \(k\ge 1\) . Simultaneously, we explore the \(s^{th}\) derivative, where \(0\le s<n\) , of this polynomial. In an attempt to obtain an integral setting of the inequalities concerning derivatives of this class of polynomials p(z), we have been able to establish extensions and generalizations of the above Ankeny and Rivlin’s inequality to integral analogs. As a consequence of our result, we obtained an improvement of the result due to Jain [Turk. J. Math., 31 (2007), \(89-94\) ] which we have also compared by considering an example. Further, it is worth to note that we have compared our results by means of this numerical example where the bounds that are in terms of integral means are estimated numerically by numerical integration using Simpson’s \(\frac{1}{3}^{rd}\) rule and illustrate graphically the obtained inequalities as regards sharpness.
