Let satisfy for some p with . By applying the Gershgorin theorem to a matrix which is similar to the companion matrix of P(z), the authors show that for each all zeros of P(z) lie in . From this the authors derive the following main results. Theorem 1. If for some the above P(z) satisfies , then all zeros of P(z) lie in , where is the largest positive root of . Theorem 4. If for some and some P(z), satisfies
then all zeros of P(z) lie in
Next, by using Bernstein’s inequality and a special case of Theorem 1, the authors derive Theorem 2. If P(z) are given and if satisfies , then P(z) holds for . Finally, several special cases are discussed and generalizations of Theorem 4 are stated.