Some refinements of inequalities for polynomials

If P(z) is a polynomial of degree n; having all its zeros in vertical bar z vertical bar <= 1; then Turan [18] 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 We prove a generalization of the above inequality to the class of polynomials having all their zeros in vertical bar z vertical bar <= K, K >= 1. We also prove an inequality for the derivative of a polynomial P(z) having no zeros in the disc vertical bar z vertical bar < K, K <= 1 whenever vertical bar P'(z)vertical bar, and vertical bar d(z(n)<(P(1/<(z)over bar>)over bar>))/dz vertical bar attain maximum at a same point on vertical bar z vertical bar = 1. Both the results generalize and sharpen several of the known results in this direction. We also present two examples to show that in some cases the bounds obtained by our results can be considerably sharper than the known bounds. Further, these results have been extended to polar derivatives of polynomials also.