Vol. 34, No. 3, 1970

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ISSN: 0030-8730
Rearrangement inequalities involving convex functions

David London

Vol. 34 (1970), No. 3, 749–753
Abstract

Let a = (a1,,an) and b = (b1,,bn) be n-tuples of nonnegative numbers. Then

∏n  ′   ′   ∏n      ′   ∏n  ∗   ′
(ai + bi) ≦ (ai + bi) ≦  (ai + bi)
i=1          i=1          i=1
(1)

and

∑n       ∑n       ∑n
a∗ib′i ≦   aib′i ≦   a′ib′i.
i=1       i=1      i=1
(2)

a= (ai,,an) and a = (a1,,an) are respectively the rearrangement of a in a nondecreasing or nonincreasing order. (1) was recently found by Minc and (2) is well known. In this note we show that these inequalities are special cases of rearrangement inequalities valid for functions having some convex properties.

Mathematical Subject Classification
Primary: 26.70
Milestones
Received: 26 February 1970
Published: 1 September 1970
Authors
David London