Root number (xy)≤(x+y)/2= 1/2

Namely: 0

Then: 1/(xy)≥4 (the equation is true if and only if x=y= 1/2).

So:

( 1+ 1/x)( 1+ 1/y)

= 1+ 1/x+ 1/y+ 1/(xy)

= 1+(x+y)/(xy)+ 1/(xy)

= 1+ 2/(xy)

Because 1/(xy)≥4, 2/(xy)≥8.

Namely: 1+ 2/(xy)≥9.

Therefore: (1+1/x) (1+1/y) ≥ 9 (the equation holds if and only if x=y= 1/2).