An upper bound on the second fiber coefficient of the fiber cones

Let (R, m) be a Cohen-Macaulay local ring of dimension d > 0, I an m-primary ideal of R and K an ideal containing I. When depth G(I) ≥ d− 1 and r(I|K) < ∞, we present an upper bound on the second fiber coefficient f2(I, K) of the fiber cones FK(I), and also provide a characterization, in terms of f2(I, K), of the condition depth FK(I) ≥ d− 2.