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Necessary and sufficient conditions are obtained on a function G of bounded variation such that ( (t)dG(t)) S [ \) ) \(x(t))dG(t) for all increasing x for which x(f 0 ) = 0 for some specified U, and all convex for which (0) -0; the conditions are otherwise independent of and x. Similar results are obtained when the inequality is reversed. Necessary and sufficient conditions for both directions of inequality are also obtained when is starshaped and (0) = 0.
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Barlow et al. (1969) studied this question.
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