Exponential integrability for log-concave measures

Talagrand observed that finiteness of $\mathbb{E}\, e^{\frac{1}{2}|\nabla f(X)|^{2}}$ implies finiteness of $\mathbb{E}\, e^{\, f(X)}$ where $X$ is the standard Gaussian vector in $\mathbb{R}^{n}$ and $f$ is a smooth function with zero average. However, in this paper we show that finiteness of $\mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-1}$ implies finiteness of $\mathbb{E}\, e^{\, f(X)}$, and we also obtain quantitative bounds \begin{align*} \log\, \mathbb{E}\, e^{\, f} \leq 10\, \mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-1}. \end{align*} Moreover, the extra factor $(1+|\nabla f|)^{-1}$ is the best possible in the sense that there is smooth $f$ with $\mathbb{E}\, e^{\,f} =\infty$ but $\mathbb{E}\, e^{\frac{1}{2}|\nabla f|^{2}} (1+|\nabla f|)^{-c}<\infty$ for all $c>1$. As an application we show corresponding dual inequalities for the discrete time dyadic martingales and its quadratic variations.