I've come up with an approach for generating inequalities for real-valued convex functions that allows one to produce inequalities that don't seem to look obvious. What I would like to know is whether they could be proved by elementary methods.
For example, I proved the following result: if $f(x)$ is convex on the interval $[0, 1]$, then
$f(0)+5f(1/6)+5f(1/2)+f(1) \ge 4(f(1/8)+f(1/3)+f(5/8))$.
My question is: can this inequality be proved by well-known/elementary methods/formulas?
Thank you!
It's a direct application of Karamata's inequality.
Just have to check that $\sum x_i \geq \sum y_i$ for the points that were chosen.