Annulus containing all the zeros of a polynomial

Recently Dalal and Govil (2013) proved that for any sequence of positive numbers {A(k)}(k-1)(n) such that Sigma(n)(k-1)A(k)-1, a complex polynomial P(z) - Sigma(n)(k-0)a(k)z(k) with a(k) not equal 0, 1 <= k <= n has all its zeros in the annulus C = {z : r(1) <= vertical bar z vertical bar <= r(2)}, where r(1) = min(1 <= k <= n){A(k)vertical bar a(0)/a(k)vertical bar}(1/k) and r(2) = max(1 <= k <= n){1/A(k)vertical bar a(n-k)/a(n)vertical bar}(1/k) They also showed that their result includes as special cases, many known results in this direction. In this paper we prove that the bounds obtained by making choice of different {A(k)}(k-1)(n) 's cannot be in general compared, that is one can always construct examples in which one result gives better bound than the other and vice versa. Also, we provide a result which gives better bounds than the existing results in all cases. Finally, using MATLAB, we compare the result obtained by our theorem with the existing ones to show that our theorem gives sharper bounds than many of the results known in this direction. (C) 2014 Elsevier Inc. All rights reserved.