Published online by Cambridge University Press: 20 November 2018
The following theorem is due to J. L. Walsh (see [2, Theorem 17, 2a]):
Theorem. If all the zeros of f1(z)=zn+a1zn-1+ … + an lie in or on the circle C1 with centre c1 and radius r1 and if all the zeros of f2(z)=zn+b1zn-1+ … + bn lie in or on the circle C2 with centre c2 and radius r2, then each zero of the polynomial
lies in at least one of the circles Γk with centre γk and radius ρk, where
and where the ωk (k= 1, 2,…, n) are the nth roots of λ.