Some lower bounds for the derivative of certain polynomials

Abstract

For the class of polynomials having no zero outside the disc $$\{z:|z| \le k \},$$ there is considerable amount of literature concerning the estimation of lower bound of maximum of $$|P^{\prime }(z)|$$ in terms of maximum of |P(z)|. However no such results are available when the polynomials have some zeros outside the disc $$\{z:|z| \le k \}$$ . In this paper we propose to obtain some estimates for the lower bound of maximum modulus of the derivative of a polynomial in terms of maximum modulus of the polynomial when some zeros of the polynomial lie outside the disc $$\{z:|z| \le k \}$$ .