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<p>Based on some classical means with a focus on Bajraktarević means and Cauchy means, we develop their set-valued extensions. By analyzing the monotonicity and convexity of derived functions, we get the sufficient conditions for the validity of these two classes of set-valued means. When the derivatives of the denominator functions are equal, by applying the equivalence conditions under which the inequalities of these two types of means hold, we derive the necessary and sufficient conditions for the set-valued mean to be valid. Some examples of set-valued means have been constructed based on Gini means and Stolarsky means. We also investigate sufficient conditions under which they satisfy the Schur-concavity property.</p>

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Some Set-Valued Means and Their Schur-Concavity

  • 期刊论文

  • 发表日期：

    2026年7月16日

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  • Qian Zhang,
  • Fen Wang,
  • Yong-Guo Shi

摘要

Based on some classical means with a focus on Bajraktarević means and Cauchy means, we develop their set-valued extensions. By analyzing the monotonicity and convexity of derived functions, we get the sufficient conditions for the validity of these two classes of set-valued means. When the derivatives of the denominator functions are equal, by applying the equivalence conditions under which the inequalities of these two types of means hold, we derive the necessary and sufficient conditions for the set-valued mean to be valid. Some examples of set-valued means have been constructed based on Gini means and Stolarsky means. We also investigate sufficient conditions under which they satisfy the Schur-concavity property.