Steffensen’s inequality and $L^{1}-L^{∞}$ estimates of weighted integrals

@inproceedings{Rabier2012SteffensensIA,
  title={Steffensen’s inequality and \$L^\{1\}-L^\{∞\}\$ estimates of weighted integrals},
  author={Patrick J. Rabier},
  year={2012}
}
Let Φ : [0,∞) → R be a continuous convex function with Φ(0) = 0. We prove that Φ ( ||f||1 ωN ||f||∞ ) ≤ 1 ωN ||f||∞ ∫ RN |f(x)|Φ′(|x|N )dx for every f ∈ L1(RN ) ∩ L∞(RN ), f = 0, where ωN is the measure of the unit ball of RN . This can be used to obtain lower or upper bounds for weighted integrals ∫ RN |f(x)|η(|x|)dx in terms of the L1 and L∞ norms of f, which are often much sharper than crude estimates that may be obtained, if at all, by a visual inspection of the integrand. The basic… CONTINUE READING

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