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

We establish sharp Beckner–Onofri on the CR sphere ${\mathbb{S}}^{2N-1}\subset {\mathbb{C}}^N$, $N\ge 2$, under vanishing holomorphic moments through degree $m\ge 1$. The optimal coefficients are $\frac{1}{m+1}$. The function inequality admits a stronger form involving only the energy above degree $m$. We also prove a quantitative concentration estimate in the $1$-Wasserstein distance and show that every maximizing sequence has diverging energy and approaches the family of uniform measures on regular $(m+1)$-polygons. The proof combines subcritical minimization with rigidity of the constrained Euler–Lagrange equation, obtained from an exact radial energy identity and a Hardy space projection estimate. Finally, weighted ball automorphisms yield sharp inequalities without moment constraints, with scalar conformal, barycenter, and holomorphic Gram determinant corrections.