Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>0$ $t^s\in L^1(\mathrm d\mu(t))$, but functions $\mathbf{1}_{t>0} $ and $\mathbf{1}_{t\in (0,1)}$ are not necessarily in $L^1(\mathrm d\mu(t))$.
I'd like to impose some conditions on $\mu$ so the function
$$f:p\to \frac{\int_0^\infty t^p \mathrm d\mu(t)}{\int_0^\infty t^{p-1}\mathrm d\mu(t) },\quad p>1 $$ is monotone on $(1,\infty)$. 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).
Somehow this reminds me the Riesz–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?
For the sake of leaving an answer here, I'll post an idea of proof suggested by Mateusz Wasilewski at Mathoverflow( link): in fact, the result quickly follows from log-convexity of the application $p\to\int_{0}^\infty t^p\mathrm d\mu(t)$ (it can be established by Hölder inequality).