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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.doi:10.1186/s13660-020-02355-3 doaj:6565c9169dc14f86b78c089ab0677ad6 fatcat:hcvjv7mvpzat3nqwohe4m6jzci