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It is true that the function $\frac{h(s)}{s^\mu}$ is nonincreasing for all $s>0$, where $h(s)=(s+1)^{-\delta}+s^{q}$ for $0<q<\mu$ and $\delta>0$.

Then can one say that the function $h(s)$ is nonincreasing for all $s>0$?

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You can write your function as $$(s+1)^{-\delta}s^{-\mu} + s^{q-\mu}$$ and taking the derivative with respect to $s$ gives you $$-\delta s^{-\mu}(s+1)^{-\delta -1} - \mu s^{-\mu -1}(s+1)^{-\delta} + (q-\mu)s^{q-\mu -1}$$

Since $\delta, \mu >0$, $q-\mu <0$ and $s>0$ this derivative is negative everywhere, and you can conclude that $h(s)/s$ is nonincreasing.

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