Generalized and Extended Versions of Ankeny–Rivlin and Improved, Generalized, and Extended Versions of Rivlin Type Inequalities for the sth Derivative of a Polynomial

Let p(z) be a polynomial of degree n having no zeros in |z|<1. In this paper, we generalize and extend a well-known result proven by Ankeny and Rivlin for the sth derivative of the polynomial. Furthermore, another well-known result proven by Rivlin is also improved, generalized and extended for the sth derivative of p(z). Our results also give a number of interesting consequences as special cases.