The origin problem is as follows:
let are positive real numbers,and prove:
The solution is easy enough, which is to plug into to and then move it to , notice:
which would finally lead to the proof of the problem.
However, when I tried to solve the problem, I applied the Lagrange identical equation to the of the inequality, then I left out some quadratic term which finally leads to the inequality:
Which seems quite right, but I'm not really sure if it is. However, numerical tests imply that it is right. But I'm still not sure.
That stuff is kind of like Chebyshev inequality but it can't be directly used here.
From another point of view, my question is whether the following strengthening of the origin inequality is right or not:
let are positive real numbers,and then: