Lower bounds for thermodynamic quantities of $d$-dimensional classical one-component plasmas with $d$-dimensional Coulomb interactions ($d=1,2, \mathrm{and} 3$)

An exact lower bound for the correlation energy of a three-dimensional classical one-component plasma (OCP), based on Mermin's inequality for the structure factor and a trivial inequality for the pair correlation function, is generalized to the cases of $d$-dimensional OCP's with $d$-dimensional Coulomb interaction where $d=1,2, \mathrm{and} 3$. For $d=1\mathrm{}$ and $d=3\mathrm{}$, this lower bound gives values close to the known exact values and the results of numerical experiments, respectively. In the case of $d=2\mathrm{}$, where the interaction potential is logarithmic, this lower bound improves upon the known one in the domain $\frac{e^2}{T}<25.0\mathrm{}$, $e$ and $T$ being the charge and the temperature in energy units.