Abstract In this paper, we present a new proof of the improved Moser–Trudinger–Onofri inequality under constraints, originally established by Aubin 2, 3 and further developed by Chang and Hang 6, 14 on S^n. Our approach provides deeper insight into the relationship between the optimal constant in these inequalities and the number of potential blow-up concentration points. We also extend the inequalities to non-spherical domains, including the torus, annulus, and rectangle, where similar intriguing phenomena emerge. Additionally, we formulate higher-order versions of these inequalities involving general GJMS operators. Besides, we establish a refinement of the concentration compactness principle for the Adams inequality, as well as an improved Riesz-type Adams inequality under constraints. Furthermore, we prove an improved almost sharp Sobolev inequality on general compact manifolds and establish an improved almost sharp higher-order Sobolev–Beckner inequality on S^n.
Sun et al. (Fri,) studied this question.