aequationes mathematicae

, Volume 40, Issue 1, pp 26–43

# Jensen inequalities for functions with higher monotonicities

• A. M. Fink
• M. JodeitJr
Research Papers

## Summary

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 classification

Primary 26D15, 26D20

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## References

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Fink, A. M. andJodeit Jr. Max,On Chebyshev's other inequality. InInequalities in Statistics and Probability (Lecture Notes IMS, No. 5). Inst. Math. Statist., Hayward, Calif., 1984, pp. 115–120.Google Scholar
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Fink, A. M.,Steffensen type inequalities. Rocky Mountain J. Math.12 (1982), 785–793.Google Scholar

© Birkhäuser Verlag 1990

## Authors and Affiliations

• A. M. Fink
• 1
• M. JodeitJr
• 2
1. 1.Department of MathematicsIowa State UniversityAmesUSA
2. 2.Department of MathematicsUniversity of MinnesotaMinneapolisUSA