Vol. 29, No. 1, 1969

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Some inequalities for starshaped and convex functions

Richard Eugene Barlow, Albert W. Marshall and Frank Proschan

Vol. 29 (1969), No. 1, 19–42

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
Received: 8 September 1967
Published: 1 April 1969
Richard Eugene Barlow
Albert W. Marshall
Frank Proschan