RELATIVE GROWTH OF A COMPLEX POLYNOMIAL WITH RESTRICTED ZEROS

Let p (z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be in | z | >= k or in | z | <= 5 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 illustrate and compare graphically the obtained inequalities with some recent results.