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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 $a$?

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