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On bounds for the range of ordered variates II

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Letx 1, ⋯,x n be real numbers with 1 n x j =0, |x 1 |≦|x 2 |≦⋯≦|x n |, and ∑ 1 n f(|x i |)=A>0, wheref is a continuous, strictly increasing function on [0, ∞) withf(0)=0. Using a generalized Chebycheff inequality (or directly) it is easy to see that an upper bound for |x m | isf −1 (A/(n−m+1)). If (n−m+1) is even, this bound is best possible, but not otherwise. Best upper bounds are obtained in case (n−m+1) is odd provided either (i)f is strictly convex on [0, ∞), or (ii)f is strictly concave on [0, ∞). Explicit best bounds are given as examples of (i) and (ii), namely the casesf(x)=x p forp>1 and 0<p<1 respectively.

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Beesack, P.R. On bounds for the range of ordered variates II. Aeq. Math. 14, 293–301 (1976). https://doi.org/10.1007/BF01835979

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AMS (1970) subject classification

  • Primary 62G30, 26A51
  • Secondary 26A87