Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$.

I'd like to impose some conditions on $\mu$ so the function

$$f:p\to \frac{\int_0^\infty t^p \mathrm d\mu(t)}{\int_0^\infty t^{p-1}\mathrm d\mu(t) },\quad p>0 $$ is monotone on $(0,\infty)$. For example, if $\mathrm d\mu(t)=e^{-t}\mathrm d t $, then $f(p)=p$. Another example is $\mathrm d\mu(t)= \mathbf{1}_{t\in (0,1)}\mathrm d t $, $f(p)=p/(p+1)$ (monotone, too). If $\mathrm d\mu(t) = e^{-t^2}$, then for $p\in [0,5]$ (I checked numerically) $f(p)$ is monotone. In fact, I can't find a counterexample to the desired result.

Somehow this reminds me the Rieszâ€“Thorin theorem, but I don't see how I can apply it here.

Are there any results on this (or similar) subject? What are possible ways to approach this problem?