Barnard's Star
The analysis of radial velocities that eventually led to discovery of the candidate super-Earth orbiting Barnard's Star was also used to set more precise upper mass limits for possible planets, up to and within the habitable zone: a maximum of up to the inner edge and on the outer edge of the optimistic habitable zone, corresponding to orbital periods of up to 10 and 40 days respectively. Therefore, it appears that Barnard's Star indeed does not host Earth-mass planets, or larger, in hot and temperate orbits, unlike other M-dwarf stars that commonly have these type of planets in close-in orbits.