Jensen inequalities for functions with higher monotonicities
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We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)ϕ(∫f(x) dμ) ⩽ A ∫ ϕ(f(x) dμ, d∫ dμ = 1. Iff is an arbitrary nonnegativeL x function, this holds ifμ ⩾ 0,ϕ is convex andA = 1. Iff is monotone the measure μ need not be positive for (J) to hold for all convex ϕ withA = 1. If ϕ has higher monotonicity, e.g., ϕ′ is also convex, then we get a version of (J) withA < 1 and measures μ that need not be positive.
AMS (1980) subject classificationPrimary 26D15, 26D20
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