A generalization of the Mulholland inequality for continuous Archimedean t-norms
Motivated by the study of the dominance relation between Archimedean \(t\)-norms the authors make an interesting contribution generalizing Mulholland inequality. More precisely, the following result is proved: Theorem. Consider a function \(h:[0, \infty ] \to [0 , \infty]\) and some fixed value \(t \in ]0, \infty[\) such that (h1) \(h\) is continuous on \([0,t)\); (h2) \(h\) is strictly increasing on \([0,t]\) and \( h(x) \geq h(t) \) whenever \( x \geq t\); (h3) \(h(0)=0\); (h4) \(h\) is convex on \(]0,t[\); (h5) \(h\) is geo-convex on \(]0,t[\). Define the functions \(g: [0, \infty]\to [0, \infty]\) and \(H:[0, \infty]^2 \to [0, \infty] \) by \[ g(x)= \begin{cases} h^{-1}(x) , &{\text{if }} x \in [0, h(t)], \cr t, & {\text{otherwise }}\end{cases} \] \[ H(x,y) = g(h(x) + h(y)). \] Then the following inequality holds for all \(a,b,c,d \in [0,\infty]\): \[ H(a+b, c+d) \leq H(a,c) + H(b,d). \]
scientific article; zbMATH DE number 5299464