Some new inequalities which counterpart Jensen's discrete inequality and improve the recent results from [20] and [22] are given. A related result for generalized means is established. Applications in Information Theory are also provided. [20] S.S. DRAGOMIR, A converse result for Jensen's discrete inequality via Grüss' inequality and applications in information theory [22] S.S. DRAGOMIR, A converse of the Jensen inequality for convex mappings of several variables and applications
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