= Solution
{c}
A <quasi-Fermi level> is the <chemical potential> parameter of an approximately thermalized carrier population within one <energy band>. Rapid intraband scattering can establish a <Fermi-Dirac distribution> in each <energy band> before slower interband recombination equilibrates their populations. Thus separate $E_{\rm Fc}$ and $E_{\rm Fv}$ describe a pumped <population inversion> even though the entire <semiconductor> is far from <thermal equilibrium>.
At a common positive <temperature>, the <Fermi-Dirac distribution> is strictly decreasing in $E-E_F$. Consequently
$$
\begin{aligned}
f_c(E_2)>f_v(E_1)
&\iff E_2-E_{\rm Fc}<E_1-E_{\rm Fv}\\
&\iff \hbar\omega<E_{\rm Fc}-E_{\rm Fv}.
\end{aligned}
$$
The <joint density of states> is nonzero only above the <band gap>, so
$$
\boxed{E_g<\hbar\omega<E_{\rm Fc}-E_{\rm Fv}.}
$$
In particular the <quasi-Fermi-level splitting> must exceed $E_g$. The upper boundary is transparency, where <stimulated emission> balances <absorption>; the lower boundary has zero <joint density of states> in this ideal three-dimensional parabolic model. The strict occupation comparison uses $T>0$; at exactly zero <temperature> occupations become step functions, and their supports should be checked explicitly, as in the next part.
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