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Weighted L Φ integral inequalities for operators of Hardy type

Steven Bloom, Ron Kerman (1994)

Studia Mathematica

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for Φ 2 - 1 ( ʃ Φ 2 ( w ( x ) | T f ( x ) | ) t ( x ) d x ) ≤ Φ 1 - 1 ( ʃ Φ 1 ( C u ( x ) | f ( x ) | ) v ( x ) d x ) to hold when Φ 1 and Φ 2 are N-functions with Φ 2 ∘ Φ 1 - 1 convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

Weighted means and weighting functions

Radko Mesiar, Jana Špirková (2006)

Kybernetika

We present some properties of mixture and generalized mixture operators, with special stress on their monotonicity. We introduce new sufficient conditions for weighting functions to ensure the monotonicity of the corresponding operators. However, mixture operators, generalized mixture operators neither quasi-arithmetic means weighted by a weighting function need not be non- decreasing operators, in general.

Weighted multidimensional inequalities for monotone functions

Sorina Barza, Lars-Erik Persson (1999)

Mathematica Bohemica

We discuss the characterization of the inequality (RN+ fq u)1/q C (RN+ fp v )1/p,   0<q, p <, for monotone functions f ≥ 0 and nonnegative weights u and v and N ≥ 1 . We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions.

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