Optimal Self-Dual Inequalities to Order Polarized BECs

We prove $1 - {\left( {1 - {x^M}} \right)^{{2^M}}} > {\left( {1 - {{(1 - x)}^M}} \right)^{{2^M}}}$ for all x ∈ [0, 1] and all M > 1. This confirms a conjecture about polar code, made by Wu and Siegel in 2019, that ${W^{{0^m}{1^M}}}$ is more reliable than ${W^{{1^m}}}{0^M}$, where W is any binary erasure channel and M = 2
^m
. The proof relies on a relaxation that m needs not be an integer, the solvability of a hexavariate ordinary differential equation, and a generalization of Green’s theorem that concerns function composition. The resulting inequality is optimal, M cannot be 2
^m
− 1, witnessing how far polar code deviates from Reed–Muller code.