GENERALIZATION OF SOME INEQUALITIES FOR THE POLAR DERIVATIVE OF POLYNOMIALS WITH RESTRICTED ZEROS

If p(z) is a polynomial of degree n, then Govil [N. K. Govil, Some inequalities for derivative of polynomials, J. Approx. Theory, 66 (1991) 29-35.] proved that if p(z) has all its zeros in |z| ≤ k, (k ≥ 1), then max |z|=1 |p 0 (z)| ≥ n 1 + k n max |z|=1 |p(z)| + min |z|=k |p(z)| . In this article, we obtain a generalization of above inequality for the polar derivative of a polynomial. Also we extend some inequalities for a polynomial of the form p(z) = z s a0 + nX−s ν=t aνz ν ! , t ≥ 1, 0 ≤ s ≤ n − 1, which having no zeros in |z| < k, k ≥ 1 except s-fold zeros at the origin.