Inclusion-exclusion inequalities

Abstract

Ifμ is a positive measure, andA 2, ...,A n are measurable sets, the sequencesS 0, ...,S n andP [0], ...,P [n] are related by the inclusion-exclusion equalities. Inequalities among theS i are based on the obviousP [k]≧0. Letting\(M_k = S_k /\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)\)=the average average measure of the intersection ofk of the setsA i , it is shown that (−1)k Δ k M i ≧0 fori+kn. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS 0=1,\(\left( {\begin{array}{*{20}c} {S_1 } \\ N \\ \end{array} } \right)\) whenS 1N−1, and\(\left( {\begin{array}{*{20}c} \nu \\ {k - 1} \\ \end{array} } \right)S_N \geqq \left( {\begin{array}{*{20}c} \nu \\ {N - 1} \\ \end{array} } \right)S_k + \left( {\begin{array}{*{20}c} \nu \\ N \\ \end{array} } \right)\left( {\begin{array}{*{20}c} \nu \\ {k - 1} \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} v \\ {N - 1} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} \nu \\ k \\ \end{array} } \right)\) for 1≦k<Nn andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN,\(\sum\limits_{i = 0}^N {a_i M_i } \geqq 0\) for all sequencesM 0, ...,M n of sufficiently large length if and only if\(\sum\limits_{i = 0}^N {a_i t^i } > 0\) for 0<t<1.

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References

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Hausner, M. Inclusion-exclusion inequalities. Combinatorica 5, 215–225 (1985). https://doi.org/10.1007/BF02579364

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AMS subject classification (1980)

  • 05 A 20
  • 05 A 05