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).
Mir et al. (Thu,) studied this question.