Levinson’s inequality for Hilbert space operators

The purpose of this presentation is to consider Levinson's inequality for self-adjoint operators, positive linear mappings and the family K_c(I) or K^._c(I) of functions as follows: Let f in C(I) be a real valued functions on an arbitrary interval I in R and c in I^circ, where I^circ is the interior of I. We say that f in K_c(I) if there exists a constant A such that the function F(x) = f(x)- (A/2) x^2 is concave on I cap (- infty, c] and convex on I cap [c, infty). Moreover, we say that f in K^._c(I) if F is operator concave on I cap (-infty , c] and operator convex on I cap [c, infty).