On sharpening and generalization of Rivlin’s inequality

Abstract

An inequality due to T. J. Rivlin from 1960 states that if P(z) is a polynomial of degree n having no zeros in |z| < 1, then $underset{left | z right |=r}{mathrm{max}}left | P(z) right |geq left (frac{1+r}{2}right )^{n}underset{left | z right |=1}{mathrm{max}}left | P(z) right |$ for 0 ≤ r ≤ 1. In this paper, we prove some generalizations of the above Rivlin’s inequality which sharpens Rivlin’s inequality as a special case. Some important consequences of these results are also discussed and some related inequalities are obtained.