Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 5 down vote accepted

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 $1.000000078542232$.

share|improve this answer
I feel like I am constantly being disappointed by false inequalities :( –  Jand Sep 15 '11 at 20:24

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.