Question

10分求高手证明下列不等式！谢谢啦！

若x,y,z∈R，且x+y+z=xyz,
求证：(y+z)/x+(z+x)/y+(x+y)/z≥2(1/x+1/y+1/z)^2

Answer 1

设
1／x＝p
1／y＝q
1／z＝r
则pq+qr+pr=1

(y+x)/z+(y+z)/x+(z+x)/y≥2(1/x+1/y+1/z)^2
为(pq+qr+pr)[r/p+r/q+q/r+q/p+p/r+p/q]>=2(p+q+r)^2
即2(r^2+p^2+q^2+pq+qr+rp)+rrq/p+rrp/q+qqr/p+qqp/r+ppr/q+ppq/r>=2(p+q+r)^2
即rrq/p+rrp/q+qqr/p+qqp/r+ppr/q+ppq/r>=2(pq+qr+pr)

又因为rrp／q＋rrq/p>=2rr
所以只需证rr+pp+qq>＝pq+qr+pr
(r-p)^2+(p-q)^2+(q-r)^2>=0