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In this paper, we establish an improved Hardy–Littlewood–Sobolev inequality on S n under higher-order moments constraint. Moreover, by constructing precise test functions, using improved Hardy–Littlewood–Sobolev inequality on S n , we show such inequality is almost optimal in critical case. As an application, we give a simpler proof of the existence of the maximizer for conformal Hardy–Littlewood–Sobolev inequality. [ABSTRACT FROM AUTHOR]