Lo-Keng Hua inequality and dynamic programming (Q1191795)

This paper states the Lo-Keng Hua inequality as \[ (\delta-x_ 1-\cdots- x_ n)^ 2+\alpha(x^ 2_ 1+\cdots+x^ 2_ n)\geq k_ n\delta^ 2 \tag{1} \] with equality if and only if \(x_ 1=\cdots=x_ n=h_ n\delta\), where \(k^ 2=\alpha(n+\alpha)^{-1}\), \(h_ n=(n+\alpha)^{- 1}\), \(\delta,\alpha>0\). Moreover, (1) is generalized to the inequality \[ (\delta-x_ 1-\cdots-x_ n)^ p+\alpha^{p-1}(x^ p_ 1+\cdots+x^ p_ n)\geq k_ n^{p-1}\delta^ p,\;p\geq 0, \] and its continuous counterpart. Two proofs for either inequality are presented.