ON ZEROS OF POLYNOMIAL

For a given polynomial P(z) = z(n) + a(n-1)z(n-1) + a(n-2)z(n-2) + ... + a(1)z + a(0) with real or complex coefficients, the Cauchy bound vertical bar z vertical bar < 1+ A, A = max(0 <= j <= n-1) vertical bar a(j)vertical bar does not reflect the fact that for A tending to zero, all the zeros of P (z) approach the origin z = 0. Moreover, Guggenheimer (1964) generalized the Cauchy bound by using a lacunary type polynomial p(z) = z(n) + a(n-p)z(n-p) + a(n-p-1)z(n-p-1) + ... + a(1)z + a(0), 0 < p < n. In this paper we obtain new results related with above facts. Our first result is the best possible. For the case as A tends to zero, it reflects the fact that all the zeros of P(z) approach the origin z = 0; it also sharpens the result obtained by Guggenheimer. The rest of the related results concern zero-free bounds giving some important corollaries. In many cases the new bounds are much better than other well-known bounds.