Upper Bounds for the Abscissa of Stability of a Stable Polynomial

Let n be a positive integer, let a sub 1, a sub 2, ..., a sub n be real numbers, and let p be the polynomial p:z (arrow) z to the nth power + (a sub 1) z to the (n-1) power +...+ a sub n. If the zeros of p are denoted by zeta sub 1, ..., zeta sub n, we call sigma: = max (1= or < i = or < n) Re (zeta sub i) the abscissa of stability of p. The polynomial p is called stable if and only if sigma < 0. Several lower bounds for the abscissa of stability have been given by G. F. Schrack. The purpose of this paper is to exhibit some negative upper bounds for the abscissa of stability of a polynomial that is already known to be stable. These bounds are elementary functions of the coefficients. (Author)