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Properties of mappings generated with inequalities for isotonic linear functionals

We consider mappings generated by inequalities for isotonic linear functionals such as the inequalities of Chebyshev, Beckenbach-Dresher, Jensen-Mercer, Jensen, Hölder, Minkowski and their reversed versions. Properties like quasilinearity, boundedness and monotonicity are proved. Also, properties of the composite functional x 7→ h(v(x))Φ g(x) v(x) are mentioned, where g and v are functions associated with the mappings generated by the inequalities and Φ is a h-concave monotone function.