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In this paper, we present some generalizations of an inequality of Hardy-Littlewood-Pólya. We give the n-exponential convexity and log-convexity of the functions associated with the linear functionals defined as the non-negative differences of the generalized inequalities and prove the monotonicity property of the generalized Cauchy means obtained via these functionals. Finally, we give several examples of the families of functions for which the results can be applied

3-convex functions and generalizations of an inequality of Hardy-Littlewood-Pólya