Some Lp Inequalities for Polynomials Not Vanishing inside a Circle

If P(z) is a polynomial of degree n having no zeros in |z|<1, then it is known that, for all α,β∈𝒞 with |α|≤1, |β|≤1, R>r≥1, and p>0, P(Rz)-αP(rz)+β{((R+1)/(r+1))n-|α|}P(rz)p≤([Rn-αrn+β{((R+1)/(r+1))n-|α|}]z+[1-α+β{((R+1)/(r+1))n-α}]p/1+zp)P(z)p. 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 Lp norm of P(z)∀p>0. 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.