A family of Counterexamples on Inequality among Symmetric Functions
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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,λ, that is, the inequality H_n,λ≤ H_n,μ implies majorization order λ≼μ. 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≥8 and n≥2. A crucial step of the proof relies on variables reduction. Inspired by this, we propose a new conjecture for H_n,λ≤ H_n,μ.
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