We introduce a generalization of the Rearrangement Inequality. For integers m, n, m 3, let a₈, ₁, a₈, ₂, , a₈, ₍ be a permutation of 1, 2, , n \ (i m). We consider the minimum of S= ₉=₁^n ₊=₁^{ma₊, ₉}, denoted as S₍. Since the exact value of S₍ is usually unknown, we focus on its asymptotic behaviour as n tends to infinity, namely computing T₍=S₍n^{m+1}. We provide a lower bound for T₍, using partial inequalities, strengthened proposition and perturbation method. And we conjecture that the lower bound is actually ₍ T₍. Then we provide an upper bound for T₍ by giving examples and that when m=3, for sufficiently large n there holds 0. 0548<T₍<0. 05488. Finally we calculate some S₍ using Python and list the example when m=3, n=201.
Ziheng Gan (Thu,) studied this question.