  We show that the distillable coherence---which is equal to the relative
entropy of coherence---is, up to a constant factor, always bounded by the
$\ell_1$-norm measure of coherence (defined as the sum of absolute values of
off diagonals). Thus the latter plays a similar role as logarithmic negativity
plays in entanglement theory and this is the best operational interpretation
from a resource-theoretic viewpoint. Consequently the two measures are
intimately connected to another operational measure, the robustness of
coherence. We find also relationships between these measures, which are tight
for general states, and the tightest possible for pure and qubit states. For a
given robustness, we construct a state having minimum distillable coherence.
