Free-differentiability conditions on the free-energy function implying large deviations

Let $({\mu }_{\alpha })$ be a net of Radon sub-probability measures on the real line, and $(t_{\alpha })$ be a net in $]0,+\mathrm{\infty }[$ converging to 0. Assuming that the generalized log-moment generating function $L(\lambda )$ exists for all $\lambda$ in a nonempty open interval $G$, we give conditions on the left or right derivatives of $L_{\mid G}$, implying vague (and thus narrow when $0\in G$) large deviations. The rate function (which can be nonconvex) is obtained as an abstract Legendre-Fenchel transform. This allows us to strengthen the G"{a}rtner-Ellis theorem by removing the usual differentiability assumption. A related question of R. S. Ellis is solved.