Do there exist two functions $f$ and $g$ continuous and strictly increasing $[0,1] \to \mathbf{R}$ such that
$$ f^{-1}\left(\frac{1}{3} f(x) + \frac{2}{3} f(y)\right)&lt;g^{-1}\left(\frac{1}{3} g(x) + \frac{2}{3} g(y)\right)$$
and
$$ f^{-1}\left(\frac{2}{3} f(x) + \frac{1}{3} f(y)\right)&gt;g^{-1}\left(\frac{2}{3} g(x) + \frac{1}{3} g(y)\right)$$
for all $0\le x &lt; y \le 1$?



This question, related to generalized means (characterized by Kolmogorov-Nagumo and de Finetti), arises from the fact that such system has a [infinitely many] solution in the case that the sign of the inequalities are the same, e.g. $(f(t),g(t))=(t^\alpha,t^\beta)$ for some $\alpha \neq \beta$. What happens if the signs are reversed?


Question: Freeform timing out on return page
Answer: This sounds like it's an issue on your server since it was working and now is not.

I'd first test to see if you are having timeouts when sending from the Communicate page in EE. If you can't send from the Communicate page, enable Email Debugging to see what's going on.

Also check your PHP error logs to see what you see. An error related to the timeout may exist there.

If necessary, contact your host to see if they have changed anything on the server related to PHPMail/SMTP (I'm not clear from your comment which one you are using exactly)... Send them the info you find in the debugging and logs.