Monotonicity and convergence of the sequence I_n/I_{n−1} in Pleijel bounds on Hn

Investigate whether the sequence I_n/I_{n−1}, where I_n is the explicit upper bound used to estimate the Pleijel constant y(Hn), is decreasing and convergent as n increases. Provide a rigorous proof of monotonicity and convergence suggested by numerical evidence.

Background

The authors bound y(Hn) using Sobolev constants and Weyl asymptotics, reducing the problem to explicit constants Cn and derived quantities In. Numerical computations indicate a decreasing and convergent behavior of In/In−1, which would strengthen and generalize their asymptotic control.

A proof of this monotonicity/convergence would refine the understanding of Pleijel bounds across dimensions and potentially simplify or sharpen results for low-dimensional Heisenberg groups.