I encountered the following computation in a paper:
The highlighted $t$ is mysterious to me, how did it end up there? For some guidance: the first line is fundamental theorem of calculus and change of variables. The last equality is change of variables and fubini. Thanks!
Jensen’s inequality should do the trick. Replacing $dr$ with the probability measure $dr/t$ on the interval $[0,t]$ and using convexity of $x \mapsto x^2$ furnishes the inequality $$ \left( \int_{[0, t]} f(r) \frac{dr}{t} \right)^2 \leq \int_{[0, t]} f(r)^2 \frac{dr}{t}. $$ Pulling out the factors of $t$ and rearranging gives the desired inequality, $$ \left( \int_{[0, t]} f(r) dr \right)^2 \leq t \int_{[0, t]} f(r)^2 dr$$ where $f(r) = \nabla u (r \xi) \cdot \xi$.