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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).

Levinson's inequality; self-adjoint operator; positive linear mapping; convex function

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