Abstract
We give new results on Levinson’s operator inequality and its converse for normalized positive linear mappings and some large class of ‘3-convex functions at a point c’.
Highlights
Introduction and preliminary resultsLet B(H) be the algebra of all bounded linear operators on a complex Hilbert space H
We considered order among quasi-arithmetic means under similar conditions
For convenience we introduce some abbreviations
Summary
Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. Witkowski [ , ] extended this result in several ways He showed that Levinson’s inequality can be stated in a more general setting with random variables. We give Levinson’s operator inequality for unital fields of positive linear mappings and classes of functions given by Definition. Jensen’s inequality for an operator concave function implies It follows that f (X) – f (X) ≤ C. If f ∈ K c ((a, b)) and D ≥ D holds, the reverse inequalities are valid in ( ) Proof This result is proven directly in [ ], Theorem , using Jensen’s operator inequality on the sum of the operators.
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