Adaptive inequalities with exact constants for operators that satisfy conditions of \(p\)-convexity or \(p\)-concavity type. (Q1432378)

The author introduces several new notions of \(p\)-convexity and \(p\)-concavity type for operators between quasi-Banach spaces. Basic examples involving linear differential operators on Sobolev spaces and operators of polynomial type are given. For a finite collection of such operators, Theorem~\(1\) states equivalent additive inequalities with exact constants and describes the extremal elements satisfying these inequalities. Theorem~\(2\) is a characteristic special case of Theorem~\(1\) for operators of polynomial type. No proofs are given, and the author does not further illustrate the concluding statement that ``a number of well-known inequalities are special cases of Theorems~\(1\) and~\(2\). These theorems also give some new integral inequalities obtained under a special choice of the spaces, bases, and operators.''