  We present natural analogues of strong Haagerup inequalities on non-Kac free
orthogonal quantum groups $O_F^+$ in which $L^p$-analytic problems are harder
due to their non-tracial nature. Furthermore, we prove optimality of the
inequalities, and apply the obtained results to compute the optimal time for
ultracontractivity of the heat semigroup and to distinguish the complex
interpolation space $L^p(O_F^+)$ and the real interpolation space
$L^{p,p}(O_F^+)$.
