Nonexponential band transmission ($-d^2\log \overline{\mathfrak{T}}/d{m}^2=-{\mathrm{Var}}_m(\kappa )$)

A finite-band transmittance $\overline{\mathfrak{T}}=⟨e^{-m{\kappa }_{\nu }}⟩$ averages different monochromatic attenuation rates. Its effective attenuation $a_{\mathrm{eff}}=-d\log \overline{\mathfrak{T}}/dm$ obeys $a_{\mathrm{eff}}^{\prime }=-{\mathrm{Var}}_m(\kappa )\le 0$, where the weighting is proportional to the transmitted spectrum. Strongly absorbing parts saturate first, so surviving radiation samples weaker absorption. Jensen's inequality gives $\overline{\mathfrak{T}}\ge e^{-m⟨\kappa ⟩}$. Equality throughout requires a spectrally constant coefficient.