Inequalities among symmetric functions are fundamental questions in mathematics and have various applications in science and engineering. In this paper, we tackle a conjecture about inequalities among the complete homogeneous symmetric function $H_{n,\lambda}$, that is, the inequality $H_{n,\lambda}\leq H_{n,\mu}$ implies majorization order $\lambda\preceq\mu$. This conjecture was proposed by Cuttler, Greene and Skandera in 2011. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for degree $d=8$ and number of variables $n=3$. They then asked whether the conjecture is true when~ the number of variables, $n$, is large enough? In this paper, we answer the question by proving that the conjecture does not hold when $d\geq8$ and $n\geq2$. A crucial step of the proof relies on variables reduction. Inspired by this, we propose a new conjecture for $H_{n,\lambda}\leq H_{n,\mu}$.

翻译：对称函数的不等式是数学中的基本问题，在科学与工程领域有广泛应用。本文研究关于完全齐次对称函数 $H_{n,\lambda}$ 的一个猜想，即不等式 $H_{n,\lambda}\leq H_{n,\mu}$ 蕴含优超序 $\lambda\preceq\mu$。该猜想由Cuttler、Greene和Skandera于2011年提出，与Muirhead型不等式的已知结果高度相似。2021年，Heaton和Shankar通过给出次数 $d=8$、变量数 $n=3$ 的反例否定了该猜想，并进一步提问：当变量数 $n$ 足够大时猜想是否成立？本文通过证明 $d\geq8$ 且 $n\geq2$ 时猜想不成立，回答了该问题。证明的关键步骤依赖于变量约化。受此启发，我们针对 $H_{n,\lambda}\leq H_{n,\mu}$ 提出一个新猜想。