Spectral Lipschitz bound from Frobenius distance ($|⟨L_A-L_B,f⟩|\le N^{-1/2}\Vert A-B{\Vert }_F$)

For equally sized Hermitian matrices and a real test function with Lipschitz bound one, the difference of its averages under their empirical spectral measures is at most $N^{-1/2}$ times the Frobenius norm of their difference. Pair the increasing eigenvalues, apply the triangle inequality and Cauchy-Schwarz inequality, then use the Hoffman–Wielandt inequality.