I am still trying to find a true statement :-(
Assume that for $x(\alpha,\beta) > 0$, $y(\alpha,\beta) > 0$ and $\alpha > 0$ and and $\beta > 0$ and for all $q>1$ holds $\alpha \leq \frac{\beta(x(\alpha,\beta)^q) + y(\alpha,\beta)^q}{x+y}$. It follows that $\alpha \leq \frac{f(x(f)) +f(y(f))}{x+y}$ holds for all strictly increasing and strictly convex functions f(x) with $f(0) \geq 0$.
Is the above statement true? If yes, can you give a source for it?
My intuition is that $x^q$ with $q→1$ is the flattest possible strictly convex function. Therefore, the statement should be true.
Thank you