Let $f \in L^p(0,\infty)$, and consider \begin{equation}F_f(x) = \frac{1}{x} \int_0^x f(y) dy \end{equation} where $F:L^p(0,\infty) \rightarrow L^p(0,\infty)$, show that \begin{equation} \|F\| \leq \frac{p}{p-1} \end{equation}
I can come up with the denominator if I majorate the integral on $F$ by $\|f\|_p$ but i get stuck with $\int_0^\infty x^{-p}$ and this blows up.
I also tried using holder on $\int |f(y)|dy$ but couldn't get anywhere.
Any hints on how to majorate this norm?
