Relative Growth of a Complex polynomial with Restricted Zeros

Abstract

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros in |z| ≥ k or in |z| ≤ k, k > 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles |z| = r and |z| = R and obtain inequalities about the dependence of |p(rz)| on |p(Rz)|, where |z| = 1, for 0 < r ≤ R ≤ k or 0 < k ≤ R ≤ r while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to graphically illustrate and compare the obtained inequalities with some recent results. Mathematics Subject Classification (2010). 30A10, 30C10, 30D15.