Let me make the question in the title more precise.
Let $f:X\to $ Spec $k$ be a smooth projective connected variety over a field $k$ of characteristic zero. Let $\mathcal L$ be a line bundle on $X$. In my set-up this line bundle has many positivity properties, e.g., it is ample.
Is the function $m\mapsto $ rank of $f_\ast\mathcal L^{\otimes m}$ a non-decreasing function?
That is, does the inequality rank$f_\ast(\mathcal L^{\otimes m}) \leq $ rank$f_\ast(\mathcal L^{m+1})$ hold for all $m\geq 1$?
