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Double precision of the Jensen-type operator inequalities for bounded and Lipschitzian functions

The main objective of the present paper is a study of mutual bounds for the Jensen operator inequality and the Lah-Ribari\v c operator inequality for the classes of bounded real-valued functions and Lipschitzian functions. The connection with the classical convexity is also discussed. As an application, our main results are then applied to quasi-arithmetic operator means, with a particular emphasis to power operator means. In particular, we obtain some new reverse relations that correspond to quasi-arithmetic and power operator means.

Jensen operator inequality, Lah-Ribari\v c operator inequality, convexity, operator convexity, bounded function, Lipschitzian function

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