I'm wondering if an upper bound in the order of $O(\log T)$ can be gotten for this sum:
$\sum_{t=1}^{T}\sqrt{\sum_{j=1}^{d}\frac{a_j(t+1)-a_j(t)}{a_j^2(t)}}$,
where $d>0$ is a fixed integer, $a_j(t+1)\ge a_j(t)>1$, $\sum_j a_j(t+1)-a_j(t)\le 1$ and $\sum_j a_j(t)\le t+d$.