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 vertical bar z vertical bar <= k, (k >= 1), then max(vertical bar z vertical bar=1) vertical bar P'(z)vertical bar >= 1 n/1 + k(n) {max(vertical bar z vertical bar=1) vertical bar P(z)vertical bar + min(vertical bar z vertical bar=1) vertical bar P(z)vertical bar} 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) (a(0) + Sigma(n-s)(v=t) a(nu)z(nu)), t >= 1, 0 <= s <= n - 1, which having no zeros in vertical bar z vertical bar < k, k >= 1 except s-fold zeros at the origin.