Vol. 29, No. 1, 1969

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ISSN: 0030-8730
Some inequalities for starshaped and convex functions

Richard Eugene Barlow, Albert W. Marshall and Frank Proschan

Vol. 29 (1969), No. 1, 19–42
Abstract

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