Majorization fails to characterize inequalities among complete homogeneous symmetric functions for all degrees greater than 7.
Timofte, On the positivity of symmetric polynomial functions.: Part I: General results, Journal of Mathematical Analysis and Applications 284(1) (2003) 174-190
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Majorization and Inequalities among Complete Homogeneous Symmetric Functions
Majorization fails to characterize inequalities among complete homogeneous symmetric functions for all degrees greater than 7.