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Solution of an open problem for Schur convexity or concavity of the Gini mean values

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Abstract

The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ ℝ × ℝ is still open. In this paper, we prove that S(a, b; x, y) is Schur convex with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): a ⩾ 0, b ⩾ 0, a + b ⩾ 1}, and Schur concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): b ⩽ 0, b ⩽ a, a + b ⩽ 1} ∩ {(a, b): a ⩽ 0, a ⩽ b, a + b ⩽ 1}.

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Authors and Affiliations

  1. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China

    YuMing Chu

  2. School of Teacher Education, Huzhou Teachers College, Huzhou, 313000, China

    WeiFeng Xia

Authors
  1. YuMing Chu
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  2. WeiFeng Xia
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Correspondence to YuMing Chu.

Additional information

This work was supported by National Natural Science Foundation of China (Grant Nos. 60850005, 10771195) and the Natural Science Foundation of Zhejiang Province (Grant Nos. D7080080, Y607128, Y7080185)

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Chu, Y., Xia, W. Solution of an open problem for Schur convexity or concavity of the Gini mean values. Sci. China Ser. A-Math. 52, 2099–2106 (2009). https://doi.org/10.1007/s11425-009-0116-5

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  • DOI: https://doi.org/10.1007/s11425-009-0116-5

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