  Let $\gamma_{d}$ be the $d$-dimensional standard Gaussian measure and
$\{Q_{t}\}_{t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on
$L^{1}(\gamma_{d})$. We show that the hypercontractivity of $\{Q_{t}\}_{t\ge
0}$ is equivalent to the property that \begin{align*}
  \left\{
  \int_{\mathbb{R}^{d}}\exp \left(e^{2t}Q_{t}f\right) d\gamma_{d}
  \right\} ^{1/e^{2t}}
  \le \int_{\mathbb{R}^{d}}e^{f}\,d\gamma_{d}, \end{align*} which holds for any
$f\in L^{1}(\gamma_{d})$ with $e^{f}\in L^{1}(\gamma_{d})$ and for any $t\ge
0$. We then derive a family of inequalities that unifies this exponential
variant and the original hypercontractivity, a generalization of the Gaussian
logarithmic Sobolev inequality is obtained as a corollary. A unification of the
reverse hypercontractivity and the exponential variant is also provided.
