Optimal constant in the universal Dirichlet-tree bound λk/λj ≤ C·(k2/j2)

Determine the smallest constant C such that for every compact Dirichlet tree Γ and all integers k>j≥1 the inequality λk(Γ)/λj(Γ) ≤ C·(k2/j2) holds; establish whether the optimal C is strictly smaller than 4 and identify its exact value.

Background

The authors prove for Dirichlet trees the bound λk/λj ≤ 4^{⌈log2(k/j)⌉} ≤ 4·(k2/j2), which respects the Weyl asymptotics up to a universal constant. They also show the constant C must be at least 9/4 using equilateral stars.

They conjecture the constant 4 may not be optimal, suggesting 9/4 as a plausible candidate based on extremal behavior of equilateral stars and known extremizers in related spectral gap problems.