The system of inequalities a_rs > X_r—X_s

Abstract

In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, …, k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr, such that gr = g(xr) has the direction of pr. † Let ur = pr/er, where so that

A permanent inequality for positive functions on the unit square

“…Theorem 1 was discovered independently by the authors in their study of the van der Waerden conjecture; it is very closely related to the linear programming dual of a theorem proved by Garret Birkhoff [1], which states that the doubly stochastic matrices are the convex hull of the permutation matrices. Indeed it was this last fact which persuaded us that Theorem 1 could be applied directly to the van der Waerden conjecture.…”

“…This was established by Marcus and Mine [1] in 1962. Specifically they showed that if Π da is not exceeded by any other term in the permanent expansion, then (1) Σ log α« ^ Σ Σ a io log a id ^ n log n~x .…”

“…This theorem for finite sets S is essentially contained in a paper by S. N. Afriat [1] which appeared in 1963 in connection with a study of empirical preference analysis in economics. Theorem 1 was discovered independently by the authors in their study of the van der Waerden conjecture; it is very closely related to the linear programming dual of a theorem proved by Garret Birkhoff [1], which states that the doubly stochastic matrices are the convex hull of the permutation matrices.…”

“…It is also convenient to state here a partial converse of the Fubini theorem proved by L. Tonelli. A proof of this theorem is in McShane [1]. 1* Proof of theorem 1* Define g(p, q) = f(p, q) for p Φ q and 9(P, V) = 0.…”

“…(See Marcus and Newman [1] and [2], Marcus and Mine [1], among others.) This conjecture states that if A is a doubly stochastic matrix, i.e.…”