Abstract

Motivated by the classical chain of inequalities $H\le G\le A\le Q$ and recent work by Meštrović [The Teaching of Mathematics 27, 1 (2024), 27--32], we perform a complete structural analysis of the finite family of nested means $$ \mathcal M=\{M_1(M_2,M_3): M_1,M_2,M_3ın\{H,G,A,Q\}\}. $$ We identify the $26$ distinct elements of $\mathcal M$ and determine the full structure of the associated partially ordered set (poset) under the pointwise order. Our investigation shows that the poset has height $19$ and contains exactly $30$ incomparable pairs. Consequently, we construct a maximal chain of length $18$ that contains $8$ nontrivial inequalities, thereby providing affirmative and optimal answers to two open questions posed by Meštrović. The proofs rely on a unified algebraic reduction to a single variable $u=G/A ın (0,1]$, which converts pointwise comparisons into explicit polynomial nonnegativity and factorizations.

Keywords: Nested means; poset; mean inequalities; maximal chain; classical means; polynomial nonnegativity.

DOI: 10.57016/TM-WPXN1976

Pages:  27$-$41     

Volume  XXIX ,  Issue  1 ,  2026