Sharp CR Beckner-Onofri inequalities under high-order holomorphic moment constraints

We establish sharp Beckner--Onofri on the CR sphere $\mathbb{S}^{2N-1}\subset\mathbb C^N$, $N\ge2$, under vanishing holomorphic moments through degree $m\ge1$. The optimal coefficients are $\frac{1}{m+1}$. The function inequality admits a stronger form involving only the energy above degree $m$. We