We show that the classical Hölder inequality between means of order $\alpha$, $0<\alpha\leqq 1$, can be improved on the assumption that the terms are not too often of comparable size. As an application, we derive a general, optimal bound for the entropy of a probability distribution.
"Entropy: An Inequality." Tokyo J. Math. 11 (2) 323 - 328, December 1988. https://doi.org/10.3836/tjm/1270133978