If the product of three positive real numbers is 1 and their sum is greater than sum of their reciprocals, then
exactly one of them exceeds 1
Let the numbers be a, b, 1ab. We are given
a+b+1ab>1a+1b+abNow,(a−1)(b−1)(1ab−1)=1+(a+b+1ab)−(ab+1a+1b)−1>0
⇒ either all three of a-1, b-1, 1ab−1 are positive or exactly one of them is positive.
But all three of a-1, b-1, 1ab−1 cannot be positive.