$f$ has a bounded variation. $\vert f\vert ^{1.5}$ also has a bounded variation. $\vert f \vert $ is a bounded variation function, as well as integer powers of $\vert f \vert$. $\vert f \vert ^{1.5}$ is not an integer product, I don't think I can rely on these facts. $\sqrt{\vert f\vert}$ is not always a bounded variation function.
How can the variation of $f$ help me calculate $\sum_{k=1}^n \vert \vert f(x_{k+1}) \vert^{1.5} -\vert f(x_k) \vert ^{1.5}\vert$?