On the zeros of a polynomial and its derivative

Abstract

Let P( z ) be a polynomial of degree n and P′( z ) be its derivative. Given a zero of P′( z ) , we shall determine regions which contains at least one zero of P( z ) . In particular, it will be shown that if all the zeros of P( z ) lie in |z| < 1 and W 1 , W 2 , …, W n −1 are the zeros of P ′( z ), then each of the disks |( z /2)– w j | < ½ and | z – W j | < 1, j = 1, 2, …, n −1 contains at least one zero of P ( z ). We shall also determine regions which contain at least one zero of the polynomials mP ( z ) + zP′( z ) and P′( z ) under some appropriate assumptions. Finally some other results of similar nature will be obtained.

Locations

• Bulletin of the Australian Mathematical Society - View - PDF