Hölder and Minkowski type inequalities for pseudo-integral

There are proven generalizations of the Hölder's and Minkowski's inequalities for the pseudo-integral. There are considered two cases of the real semiring with pseudo-operations: one, when pseudo-operations are defined by monotone and continuous function g, the second semiring ([a, b], sup,), where is generated and the third semiring where both pseudo-operations are idempotent, i.e., È = sup and = inf. Pseudo-analysis is a generalization of the classical analysis, where instead of the field of real numbers a semiring is defined on a real interval [a, b] & [À1, 1] with pseudo-addition È and with pseudo-multiplication , see [19–24,32]. Based on this structure there were developed the concepts of È-measure (pseudo-additive measure), pseudo-integral, pseudo-convo-lution, pseudo-Laplace transform, etc. The advantages of the pseudo-analysis are that there are covered with one theory, and so with unified methods, problems (usually nonlinear and under uncertainty) from many different fields (system theory, optimization, decision making, control theory, differential equations, difference equations, etc.). Pseudo-analysis uses many mathematical tools from different fields as functional equations, variational calculus, measure theory, functional analysis, optimization theory, semiring theory, etc. Similar ideas were developed independently by Maslov and his collaborators in the framework of idempotent analysis and idempotent mathematics, with important applications [8,9,11]. In particular, idempotent analysis is fundamental for the theory of weak solutions to Hamilton–Jacobi equations with non-smooth Hamiltonians, see [8,9,11] and also [22,23,25] (in the framework of pseudo-analysis). In some cases, this theory enables one to obtain exact solutions in the similar form as for the linear equations. Some further developments relate more general pseudo-operations with applications to nonlinear partial differential equations, see [27]. Recently, these applications have become important in the field of image processing [23,25]. On the other side, more general set functions than pseudo-additive measures, as fuzzy measures and corresponding fuzzy integrals had been investigated in [6,14,21,28,31,15], as aggregation functions with important applications, e.g., given in [30,33]. Recently, there were obtained generalizations of the classical integral inequalities for integrals with respect to The well-known Hölder's and Minkowski's inequality is a part of the classical mathematical analysis, see [29].