# A nonincreasing function

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 $$0 and $$\delta>0$$.

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

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.