Inequalities among symmetric functions are fundamental in various branches of mathematics, thus motivating a systematic study of their structure. Majorization has been shown to characterize inequalities among commonly used symmetric functions, except for complete homogeneous symmetric functions (shortened as CHs). In 2011, Cuttler, Greene, and Skandera posed a natural question: Can majorization also characterize inequalities among CHs? Their work demonstrated that majorization characterizes inequalities among CHs up to degree 7 and suggested exploring its validity for higher degrees. In this paper, we show that, for every degree greater than 7, majorization does not characterize inequalities among CHs.

翻译：对称函数之间的不等式是数学各个分支的基础，因此激励了对其结构的系统性研究。主化已被证明能够刻画常用对称函数间的不等式，但完全齐次对称函数（简称CHs）除外。2011年，Cuttler、Greene和Skandera提出了一个自然问题：主化是否也能刻画CHs间的不等式？他们的工作表明，主化在次数不超过7时能刻画CHs间的不等式，并建议探讨其在更高次数下的有效性。在本文中，我们证明：对于每个大于7的次数，主化不能刻画CHs间的不等式。