In the question, $O()$ is the big O notation https://en.wikipedia.org/wiki/Big_O_notation .
For me, since $\frac{d}{m}>1$ always holds, so $\frac{d^2}{m}<\frac{d^3}{m^2}$ holds. Then $O(\frac{d^3}{m^2}+\frac{d^2}{m})= O(\frac{d^3}{m^2})$.
Then, I have another confusion. Assume an estimation has an error=$O(\frac{d^3}{m^2})$ with $1<m<d$. So fix $d$, we can say the error decreases with $\frac{1}{m^2}$. However, error=$O(\frac{d^3}{m^2})\leq O(\frac{d^4}{m^3})$ because $1<m<d$, then fix $d$, we will say the error decreases with a better convergence rate $\frac{1}{m^3}$. Thus, we actually do nothing to improve the estimation method, but the convergence rate on $m$ improves... ...
Why am I wrong and what is right? Thanks.
BTW, how about for the constraint $1<m\ll d$?