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Let (Ω, S, μ) be a probabilistic measure space, " ∈ R, δ ≥ 0, p > 0 be given numbers and let P ⊂ R be an open interval. We consider a class of functions f : P → R, satisfying the inequality f(EX) ≤ E(f ◦ X) + "E(|X − EX|p) + δ for each S-measurable simple function X : Ω → P.
We show that if additionally the set of values of μ is equal to [0, 1] then f : P → R satisfies the above condition if and only if f(tx+(1−t)y) ≤ tf(x)+(1−t)f(y)+" [(1 − t)pt + tp(1 − t)] |x−y|p+δ for x, y ∈ P, t ∈ [0, 1].
We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.
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참고문헌 신청
J. Acz´el / 1947 / A generalization of the notion of convex functions / Norske Vid. Selsk. Forhdl. Trondheim 19(24):87~90
G. Aumann / 1933 / Convexe Functionen und die Induktion bei Ungleichungen zwischen Mit-telwerten / S.-B. Math. Natur. Abt. Bayer. Akad. Wiss. M¨unchen :403~415
D. H. Hyers / 1952 / Approximately convex functions / Proc. Amer. Math. Soc 3:821~828
M. Kuczma / 1985 / An Introduction to the Theory of Functional Equations and Inequalities / PWN-Uniwersytet ´Sl¸aski, Warszawa - Krak´ow - Katowice
J. Matkowski / 2003 / Convex and affine functions with respect to a mean and a characterization of the weighted quasi-arithmetic means / Real Anal. Exchange 29:229~246
G. Aumann / 1933 / Convexe Functionen und die Induktion bei Ungleichungen zwischen Mit-telwerten / S.-B. Math. Natur. Abt. Bayer. Akad. Wiss. M¨unchen :403~415
D. H. Hyers / 1952 / Approximately convex functions / Proc. Amer. Math. Soc 3:821~828
M. Kuczma / 1985 / An Introduction to the Theory of Functional Equations and Inequalities / PWN-Uniwersytet ´Sl¸aski, Warszawa - Krak´ow - Katowice
J. Matkowski / 2003 / Convex and affine functions with respect to a mean and a characterization of the weighted quasi-arithmetic means / Real Anal. Exchange 29:229~246
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