What does it mean for a pdf to have this property? 
What does it mean for a probability density function $f(x)$ to have the following property?
  $$1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$ 

I have tried a lot to simplify this condition and see what it means (in terms of moments of $f(x)$, etc), but no luck yet. Do you have any idea?

That can be written as
$$ \int_{0}^{+\infty} x^2 \cdot\frac{d^2}{dx^2}\log(f(x))\cdot f(x)\,dx < 1 $$
that is a constraint that depends on minimizing a Kullback-Leibler divergence.
It essentially gives that your distribution has to be close to a normal distribution (in the KL sense).
