= Nonexponential band transmission
{title2=$-d^2\log\overline{\mathfrak T}/dm^2=-\operatorname{Var}_{m}(\kappa)$}
A finite-band <transmittance> $\overline{\mathfrak T}=\langle e^{-m\kappa_\nu}\rangle$ averages different monochromatic attenuation rates. Its effective attenuation $a_{\rm eff}=-d\log\overline{\mathfrak T}/dm$ obeys $a_{\rm eff}'=-\operatorname{Var}_m(\kappa)\le0$, where the weighting is proportional to the transmitted spectrum. Strongly absorbing parts saturate first, so surviving radiation samples weaker absorption. <Jensen's inequality> gives $\overline{\mathfrak T}\ge e^{-m\langle\kappa\rangle}$. Equality throughout requires a spectrally constant coefficient.
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