We present a new proof of Janson’s strong hypercontractivity inequality for the Ornstein-Uhlenbeck semigroup in holomorphic algebras associated with CAR (canonical anticommutation relations) algebras. In the one generator case we calculate optimal bounds for t such that $U_t$ is a contraction as a map $L₂\left(\right)\to L_p\left(\right)$ for arbitrary p ≥ 2. We also prove a logarithmic Sobolev inequality.