Certain interesting implications of T. J. Rivlin\u27s result on maximum modulus of a polynomial

Let f(z) be an arbitrary entire function and M(f,r) = max|z|=r |f(z)|. For a polynomial p(z) of degree n, having no zeros in |z| < 1, Rivlin had obtained M(p,r) ≥ ((r+1)/2)n M(p,1), r ≤ 1. Using this and various associated results, we have obtained precise relationship between M(p,ρ), and M(p,1) for polynomials p(z) having all its zeros on |z| = k, k > 0. Some sort of a converse of Rivlin\u27s result has also been obtained