There have been several articles in literature that study the spectral behaviour of special classes ofmatrices such as the matrices based on power function t -> t(r) given by the matrices P-r = [(p(i) + p(j))(r)], the matrices B-r = [vertical bar p(i) - p(j)vertical bar(r)] for positive values of r and positive real numbers p(1), p(2), ..., p(n). Bhatia and Jain in 2015 and Dyn, Goodman and Micchelli in 1986 have studied the spectral behaviour of P-r and Br, respectively, for all real values of r. The power function t. tr is operator monotone when 0 <= r <= 1 and operator convex when 1 <= r <= 2. It is natural to study the inertia of all these matrices when the power function is replaced by any operator monotone or operator convex function. In the present work, inertia of the matrices [f(p(i) + p(j))] and [f (vertical bar p(i) - p(j)vertical bar)] is discussed for non-negative operator monotone and operator convex function f, which further leads to many known and new results.
