# Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true:

$$\int_0^11-(1-x^{c^n})(1-x)^{c-1}-[F(F(x))]^{c^{n-1}}+(c^n-c^{n-1})(1-F(x))\beta_{[F(F(x))]^{\frac{1}{c}}}\left(c^n,\frac{c-1}{c}\right) \,dx \geq 0$$

where $$F(x) = (1-(1-x)^c)^c$$ and $$\beta_{[F(F(x))]^{\frac{1}{c}}}\left(c^n,\frac{c-1}{c}\right) = \text{ the incomplete Euler Beta Function } = \int_0^{[F(F(x))]^{\frac{1}{c}}}x^{c^n-1}(1-x)^{\frac{-1}{c}} \, dx$$ and and $c$ is an integer > 1 and $n$ is an integer $\geq$ 0.

I have strong reason to believe the inequality is true. Some exploration with plotting suggests is might be true without the integral (for all $x$ in $(0,1)$) but I'm skeptical about that.

Does anyone have any ideas on approaches to proving this? My math skills just aren't strong enough and I haven't been able to do it. This question on MO might be helpful.

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I have strong reason to believe the inequality is true. Good to know. Care to share what these strong reason(s) are? – Did Nov 27 '11 at 8:34