  We investigate maximum efficiency at a given power for low-dissipation heat
engines. Close to maximum power, the maximum gain in efficiency scales as a
square root of relative loss in power and this scaling is universal for a broad
class of systems. For the low-dissipation engines, we calculate the maximum
gain in efficiency for an arbitrary fixed power. We show that the engines
working close to maximum power can operate at considerably larger efficiency
compared to the efficiency at maximum power. Furthermore, we introduce
universal bounds on maximum efficiency at a given power for low-dissipation
heat engines. These bounds represent direct generalization of the bounds on
efficiency at maximum power obtained by Esposito et al. Phys. Rev. Lett. 105,
150603 (2010). We derive the bounds analytically in the regime close to maximum
power and for small power values. For the intermediate regime we present strong
numerical evidence for the validity of the bounds.
