We present in this paper a necessary and sufficient condition to establish the inequality between generalized weighted means which share the same sequence of numbers but differ in the weights. We first present a sufficient condition, and then obtain the more general, necessary and sufficient, condition. Our results were motivated by an inequality, involving harmonic means, found in the study of multiple importance sampling Monte Carlo technique. We present new proofs of Chebyshev’s sum inequality, Cauchy-Schwartz, and the rearrangement inequality, and derive several interesting inequalities, some of them related to the Shannon entropy, the Tsallis, and the Rényi entropy with different entropic indices, and to logsumexp mean. Those inequalities are obtained as particular cases of our general inequality, and show the potential and practical interest of our approach. We show too the relationship of our inequality with sequence majorization

The authors are funded in part by grants TIN2013-47276-C6-1-R from Spanish Government and by grant number 2014-SGR-1232 from Catalan Government. The authors acknowledge the comments by David Juher to an earlier draft, and comments by anonymous reviewers that helped to improve the final version of the paper and suggested simplified proofs for Lemma 2 and Theorem 14