Some Lp Inequalities for Polynomials Not Vanishing inside a Circle

If is a polynomial of degree having no zeros in , then it is known that, for all with , , , and , . In this paper, we will prove a result which not only generalizes the above inequality but also generalize and refines the various results pertaining to the norm of . We will also prove a result which extends and refines a result of Boas Jr. and Rahman (1962). Also we will see that our results lead to some striking conclusions giving refinements and generalizations of other well-known results.