On the Erdos-Lax inequality and its various generalizations concerning polynomials

If P(z) is a polynomial of degree n having no zeros in vertical bar z vertical bar < 1, then Erdos conjectured and later Lax (Bull. Amer. Math. Soc. 50 (1944) 509-513) proved that max(vertical bar z vertical bar=1) vertical bar P'(z)vertical bar <= n/2 max(vertical bar z vertical bar=1) vertical bar P(z)vertical bar. Govil (Proc. Nat. Acad. Sci. (India) 50(A) (1980) 50-52) established a generalization of this inequality and proved that if P(z) not equal 0 in vertical bar z vertical bar < k, k <= 1 and Q(z) = z(n )<(P(1/z))over bar>, then max(vertical bar z vertical bar=1) vertical bar P'(z)vertical bar <= n/1 + k(n) max(vertical bar z vertical bar=1) vertical bar P(z)vertical bar, provided vertical bar P'(z)vertical bar and vertical bar'(z)vertical bar attain maximum at the same point on vertical bar z vertical bar = 1. In this work, we establish some new inequalities that relate the uniform norm of a polynomial and its polar derivative, while taking into account the placement of the zeros and the extremal coefficients of the polynomial. The obtained results provide various generalizations and refinements of the above inequalities and related results.