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What are the probability density functions $p_X(x)$, with support $\subset [0,\infty)$, that for all $a<1$, $\frac{p_X(ax)}{p_X(x)}$ is a non-decreasing function of $x$ over the support of $P_X (x)$.

Based on my calculations, the generalized gamma distribution is a large class containing such probability functions. Can I find the complete class that contains all of probability functions with such a property? Any other thing?

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