已知x+y+z=1……

您好！

(x²／(x＋y)＋y²/(y+z)+z²/(x+z))*(x+y+y+z+x+z)≥(x+y+z)^2
即x²／(x＋y)＋y²/(y+z)+z²/(x+z)≥0.5
当且仅当x=y=z=1/3时取等号

所以x²／(x＋y)＋y²/(y+z)+z²/(x+z)≥0.5

由柯西不等式得：
(x²／(x＋y)＋y²/(y+z)+z²/(x+z))*(x+y+y+z+x+z)≥(x+y+z)^2
即x²／(x＋y)＋y²/(y+z)+z²/(x+z)≥0.5
当且仅当x=y=z=1/3时取等号

x²／(x＋y)＋y²/(y+z)+z²/(x+z)≥x²／(2x)＋y²/(2y)+z²/(2z)
=1/2(x+y+z)=1/2

cc