Two-point estimates for the logarithmic p-flux of the first Dirichlet p-eigenfunction

Let \(u>0\) be the first Dirichlet \(p\)-eigenfunction on a bounded convex domain \(Ω\subset\mathbb R^N\), and set \[ X_Ω:=|\nabla\log u|^{p-2}\nabla\log u. \] We study whether the sharp one-dimensional two-point modulus for \(X_Ω\), which for \(p=2\) reduces to the logarithmic-gradient estimate of