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

Abstract

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|