Monotonicity in the central dimension of the Pleijel upper-bound constant for H-type groups

Show that for every H-type group G ≅ R^{2n}_x × R^m_t, the explicit Pleijel upper-bound constant γ_{n,m} = (Sobolev_{n,m})^{-Q/2} · W_{n,m}^{-1}, with Q = 2n + 2m, is a decreasing function of m for each fixed n ≥ 1. Here W_{n,m} is the Weyl constant with c_{n,m} = ∑_{k≥0} binomial(k+n−1,k)/(2k+n)^{n+m}, and Sobolev_{n,m} is Yang’s sharp L^2-Sobolev constant for H-type groups. Establish γ_{n,m} ≤ γ_{n,m−1} for all integers n ≥ 1 and m ≥ 2 without resorting to truncations of c_{n,m}.

Background

The paper derives an explicit upper bound γ{n,m} for the Pleijel constant on H-type groups using Yang’s sharp L^2-Sobolev inequality and a computed Weyl constant expressed via a convergent series c{n,m}. The authors prove that γ{n,m} decreases with n but only obtain monotonicity in m for a truncated bound \bar{γ}{n,m} that uses a first-term lower bound on c_{n,m}.

Numerical evidence suggests that the exact γ_{n,m} also decreases with m uniformly in n, but the authors could not adapt their argument due to a term-by-term quotient that lacks a lower bound. Establishing the full monotonicity would strengthen their results and potentially reduce manual verification of exceptional cases.