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In [Univ. Beograd Publ. Elektrotehn. Fak. Ser. Math. 15 (2004), 85-86], we proved a new inequality for the Lebesgue measure and gave some applications. Here, we present as it new application new short and simple proof of de Haan's uniform convergence theorem. A measurable function g : (0, +∞) → (0, +∞) is translational O-regularly varying if (1) lim s→∞ g(s + t) g(s) < +∞ for each t ∈ R. For properties and applications of this class of mappings see Tasković . Let λ be a Lebesgue measure onfatcat:6rzljavoxrd6rduxyc3jw7g6iy