= Strict monotonicity of Fermi gas entropy per particle
For <density of states> proportional to $\epsilon^a$, $a>-1$, and positive temperature, set $q=\mu/(kT)$, $n(x)=(1+e^{x-q})^{-1}$, $b=a+1$ and $M_j=\int_0^\infty x^{a+j}n(1-n)\,dx$. Differentiation and <integration by parts> give
$$
\frac d{dq}\frac S{kN}=b\left(1-\frac{M_0M_2}{M_1^2}\right)<0.
$$
The inequality is strict <Cauchy-Schwarz inequality>, since $x$ is not constant under this positive weight. Thus fixed entropy per particle forces fixed $\mu/T$. This is the injectivity step behind the adiabatic scaling of an ideal <Fermi gas>.
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