Moment generating function bound from detailed fluctuation theorem

A famous consequence of the detailed fluctuation theorem (FT), $p(\mathrm{\Sigma })\mathrm{/}p(-\mathrm{\Sigma })=\exp (\mathrm{\Sigma })$, is the integral FT $⟨\exp (-\mathrm{\Sigma })⟩=1$ for a random variable $\mathrm{\Sigma }$ and a distribution $p(\mathrm{\Sigma })$. When $\mathrm{\Sigma }$ represents the entropy production in thermodynamics, the main outcome of the integral FT is the second law, $⟨\mathrm{\Sigma }⟩\ge 0$. However, a full description of the fluctuations of $\mathrm{\Sigma }$ might require knowledge of the moment generating function (MGF), $G(\alpha ):=⟨\exp (\alpha \mathrm{\Sigma })⟩$. In the context of the detailed FT, we show the MGF is lower bounded in the form $G(\alpha )\ge B(\alpha ,⟨\mathrm{\Sigma }⟩)$ for a given mean $⟨\mathrm{\Sigma }⟩$. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a weakly coupled bosonic mode and a qubit swap engine.