This is a really tough inequality (at least for me).
Can anyone help me show: $\frac{1}{c}(1-(1-x)^c)^{c^{n}} + \frac{c-1}{c}(1-(1-x)^c)^c + (1-x)^{c-1}(1-x^{c^{n}}) \leq 1$ within the range $0
I have plotted it over $x = 0 \text{ to } 1$ and it looks like this is completely true for all values I enter of $c$ and $n$, so long as $c$ is $\geq$ 4.
It is related to this question in that I believe proving this inequality here is sufficient for proving a small variation on the linked question over a subset of the desired range, and I can manually calculate the rest. I am posting it as a separate question, however, since it's not really the same thing.