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<p>We generalize a theorem of Housen Li and Markus Haltmeier (which was stated for the arithmetic mean) to the following result: Let <i>M</i> be a strict, continuous and monotonous <i>k</i>-mean on an interval. If each element of a sequence in this interval is bounded from above by the mean of the preceding <i>k</i> elements then the sequence is convergent. Furthermore, we give counterexamples if the mean fails to fulfil the imposed conditions.</p>

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Means and convergent sequences

  • 期刊论文

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  • 发表日期：

    2025年9月17日

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  • Wolfgang Förg-Rob

摘要

We generalize a theorem of Housen Li and Markus Haltmeier (which was stated for the arithmetic mean) to the following result: Let M be a strict, continuous and monotonous k-mean on an interval. If each element of a sequence in this interval is bounded from above by the mean of the preceding k elements then the sequence is convergent. Furthermore, we give counterexamples if the mean fails to fulfil the imposed conditions.