ABSTRACT
The main objective of this chapter is the derivation of Bogolubov's inequality for the polaron reduced free energy by means of the algebraic T-product method. This inequality is a source of various upper bounds for the polaron ground-state energy. Feynman's well-known inequality is reproduced as a particular case of Bogolubov's inequality. A weak-interaction systematic finite-temperature perturbation scheme, based on the same T-product formalism, as well as the adiabatic perturbation approach, valid for the strong-coupling case, are also outlined.
