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In this paper, we present some refinements and reverses for some inequalities involving the weighted arithmetic mean and the weighted geometric mean of two convex functionals. Inequalities involving the Heinz functional mean are also obtained. As applications, we give some refinements and reverses for the relative entropy and the Tsallis relative entropy involving operator or functional arguments.

Refining and reversing the weighted arithmetic–geometric mean inequality involving convex functionals and application for the functional entropy