设n>=2,x1,x2,……xn为正数,并且x1+x2+……+xn=1,求证：求和xi/(1-xi)^(1/2)＞＝（求和xi^(1/2)）/(n-1)^(1/2)

Question

设n>=2,x1,x2,……xn为正数,并且x1+x2+……+ xn=1,求证：求和xi/(1-xi)^(1/2)＞＝（求和xi^(1/2)）/(n-1)^(1/2)

Answer 1

∑(i=1,..,n)[xi/√(1-xi)]>=(∑(i=1,..,n)√xi)/√(n-1)由柯西不等式∑(i=1,..,n)√xi=√n/√(n-1)由柯西不等式{∑(i=1,..,n)[1/√(1-xi)]}{∑(i=1,..,n)[√(1-xi)]}>=n^2∑(i=1,..,n)√(1-xi)=n^2/{∑(i=1,..,n)[...