A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals
and convex functions are proved and compared with results in . As applications some results for the means are pointed out.
Moreover, further inequalities of Hölder type are presented.
An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL.
A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism.
These properties were introduced by Schmidt in .
In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic
to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM.
An analogous theorem is derived for Riesz spaces with extension property (E2).