A nearly tight memory-redundancy trade-off for one-pass compression

Let $s$ be a string of length $n$ over an alphabet of constant size $\sigma$ and let $c$ and $\epsilon$ be constants with (1 \geq c \geq 0) and (\epsilon > 0). Using (O (n)) time, (O (n^c)) bits of memory and one pass we can always encode $s$ in (n H_k (s) + O (\sigma^k n^{1 - c + \epsilon})) bits for all integers (k \geq 0) simultaneously. On the other hand, even with unlimited time, using (O (n^c)) bits of memory and one pass we cannot always encode $s$ in (O (n H_k (s) + \sigma^k n^{1 - c - \epsilon})) bits for, e.g., (k = \lceil (c + \epsilon / 2) \log_\sigma n \rceil)