Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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Inequality involving Pochhammer symbols

Let $m,S$ be integers satisfying $2\leq m\leq S$. I would like to show that $$h_1\left(x\right) h_3\left(x\right) \leq h_2^2\left(x\right)$$ for all $x\geq 0$ where $$h_k\left(x\right) \equiv ...