证：
a>1,b>1,设m=a-1>0,n=b-1>0,则
a=1+m,b=1+n
(m-n)^2≥0,2(m-n)^2≥0,m^2+n^2≥2mn
m*(n-1)^2≥0,n*(m-1)^2≥0
mn^2+m≥2mn.(1)
nm^2+n≥2mn.(2)
(1)+(2),得
(mn^2+m)+(nm^2+n)≥4mn
mn^2+nm^2+m+n≥4mn
(m+n)*mn+(m+n)≥4mn
(m+n)*(2mn-mn)+(m+n)≥4mn
(m+n)*(m^2+n^2-mn)+(m+n)≥4mn
m^3+n^3+m+n≥4mn
m^3+n^3+m+n+2(m-n)^2≥4mn
m^3+n^3+m+n+2m^2+2n^2-4mn≥4mn
m^3+n^3+m+n+2m^2+2n^2≥8mn
（m^3+2m^2+m）+（n^3+2n^2+n）≥8mn
m(1+m)^2+n(1+n)^2≥8mn
(1+m)^2/n+(1+n)^2/m≥8
a^2/(b-1)+b^2/(a-1)≥8