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Inequalities among symmetric functions are fundamental questions inmathematics and have various applications in science and engineering. In thispaper, we tackle a conjecture about inequalities among the complete homogeneoussymmetric function $H_{n,\lambda}$, that is, the inequality $H_{n,\lambda}\leqH_{n,\mu}$ implies majorization order $\lambda\preceq\mu$. This conjecture wasproposed by Cuttler, Greene and Skandera in 2011. The conjecture is a closeanalogy with other known results on Muirhead-type inequalities. In 2021, Heatonand Shankar disproved the conjecture by showing a counterexample for degree$d=8$ and number of variables $n=3$. They then asked whether the conjecture istrue when~ the number of variables, $n$, is large enough? In this paper, weanswer 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 $H_{n,\lambda}\leqH_{n,\mu}$.

A family of Counterexamples on Inequality among Symmetric Functions