On a generalized Minkowski inequality and its relation to dominates for t-norms

Abstract

Leth:ℝ⁺ → ℝ⁺ be a continuous strictly increasing function withh(0) = 0. Such functionsh give rise to a generalization of the Minkowski inequality; namely,

$$h^{ - 1} (h(a + b) + h(c + d)) \leqq h^{ - 1} (h(a + c) + h(b + d))$$

((1))

wherea, b, c, andd are arbitrary non-negative real numbers.

Theorem 1 shows that, ifh and logh′ (e ^x) are both convex functions, thenh satisfies (1). Theorem 2, the major result, demonstrates that, if bothh ₁ andh ₂ satisfy the hypotheses of Theorem 1, then the composition ofh ₁ withh ₂ also satisfies the hypotheses of Theorem 1 and hence the inequality (1).

The remainder of the paper shows how (1) and Theorems 1 and 2 impinge on the dominates relation for strict t-norms. In particular, Theorem 3 shows how (1) can be interpreted as equivalent to the dominates relation for two strict t-norms. Theorem 4 shows how to use Theorems 1 and 3 to construct a strict t-norm which dominates a given strict t-norm. And, Theorem 5 shows how Theorem 2 can be used to give a qualified answer of yes to the open question of whether or not the dominates relation is a transitive relation.