Improvement and generalization of polynomial inequality of T. J. Rivlin

Let p(z) be a polynomial of degree n having no zero in vertical bar z vertical bar < 1, then for r <= 1, the famous result proved by Rivlin (Am. Math. Mon., 67:251-253, 1960) is (max)(vertical bar z vertical bar=r) vertical bar p(z)vertical bar >= (1 + r/2)(n) (max)(vertical bar z vertical bar=1) vertical bar p(z)vertical bar. The inequality is sharp. In this paper, by involving some coefficients of the polynomial, we prove an inequality which not only improves as well as generalizes this inequality but also has some interesting consequences.