# What are all functions $f(x)$ that ensure $\int_{a}^{\infty} \frac{f(x)}{\sqrt{x^2-a^2}} \, \mathrm{d} x \le 0$ for all $a$ where $0 \le a \le \infty$

I'm looking to find a set of functions $$f(x)$$ such that members of the set satisfy the condition

$$\int_{a}^{\infty} \frac{f(x)}{\sqrt{x^2-a^2}} \, \mathrm{d} x \le 0 \qquad \textrm{for all }0 \le a \le \infty,$$

and that all functions outside this set contradict the condition. $$f(x)$$ is defined for $$0 \le x \le \infty$$.

Is there a more concise condition on $$f(x)$$ that be expressed without requoting the integral inequality? Is it possible to infer an equivalent condition on $$f(x)$$ that does not require testing the function over a range of values, such as how the above condition involves verifying over all $$a$$?