# A tough inequality, for a bounded range of three variables

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 $0<x<1$ and $c \geq 4$ and $n \geq 2$ (and $c$ and $n$ are both integers).

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.

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Plots can be misleading. The Taylor series of the left hand side of your inequality around $x=1$ is $$1 + \left( c - 1 \right) \left( c^{n - 1} - 1 \right) \left( 1 - x \right)^c + O \left( y^{c + 1} \right)$$ so it will be larger than one for x a bit lower than one. For example, for $c = 4$, $n = 2$ and $x = 0.99$ I get $1.000000078542232$.