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In the following article, it is shown that Gage's inequality (a form of isoperimetric inequality) can be generalised as the following integral inequality which is presented as a conjecture in the article:

Suppose $p(\theta)$ is a $C^2$ periodic function with period $2\pi$ with $p(\theta)>0$ and $p(\theta)+p''(\theta)>0$. Then

$$\int_0^{2\pi}p(\theta)(p(\theta)+p''(\theta))d\theta \int_0^{2\pi}\frac{d\theta}{p(\theta)+p''(\theta)}\geq 2\pi \int_0^{2\pi}p(\theta)d\theta $$

Since the inequality is presented there as a conjecture, I guess it cannot be attacked by regular integral inequalities. I tried to solve it but didn't have any progress with it.

[edit] The question is to prove the inequality.

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What is the question? –  Aryabhata Jun 2 '11 at 8:39
Is is simply asking to show that the LHS is $\geq$ to the RHS? –  night owl Jun 2 '11 at 8:50

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