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Some inequalities for
starshaped and convex functions
Richard Eugene Barlow, Albert W. Marshall and Frank Proschan |
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Vol. 29 (1969), No. 1, 19–42
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Abstract |
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Necessary and sufficient conditions are obtained on a function G of bounded variation such that ϕ(∫ x(t)dG(t)) ≦∫ ϕ(x(t))dG(t) for all increasing x for which x(t0) = 0 for some specified t0, and all convex ϕ for which ϕ(0) = 0; the conditions are otherwise independent of ϕ and x. Similar results are obtained when the inequality is reversed. Necessary and sufficient conditions for both directions of inequality are also obtained when ϕ is starshaped and ϕ(0) = 0. The relationship to previous results is sketched. Applications to statistical tolerance limits are indicated. |
Mathematical Subject Classification
Primary: 34.90
Secondary: 52.40
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Milestones
Received: 8 September 1967
Published: 1 April 1969
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Authors |
| Richard Eugene Barlow | |
| Albert W. Marshall | |
| Frank Proschan | |
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