Although M dwarfs are known for high levels of stellar activity, they are ideal targets for the search of low-mass exoplanets with the radial velocity (RV) method. We report the discovery of a planetary-mass companion around LSPM J2116+0234 (M3.0 V) and confirm the existence of a planet orbiting GJ 686 (BD+18 3421; M1.0 V). The discovery of the planet around LSPM J2116+0234 is based on CARMENES RV observations in the visual and near-infrared channels. We confirm the planet orbiting around GJ 686 by analyzing the RV data spanning over two decades of observations from CARMENES VIS, HARPS-N, HARPS, and HIRES. We find planetary signals at 14.44 and 15.53d in the RV data for LSPM J2116+0234 and GJ 686, respectively. Additionally, the RV, photometric time series, and various spectroscopic indicators show hints of variations of 42 d for LSPM J2116+0234 and 37 d for GJ 686, which we attribute to the stellar rotation periods. The orbital parameters of the planets are modeled with Keplerian fits together with correlated noise from the stellar activity. A mini-Neptune with a minimum mass of 11.8M_{Earth} orbits LSPM J2116+0234 producing a RV semi- amplitude of 6.19m/s, while a super-Earth of mass 6.6M{Earth} orbits GJ 686 and produces a RV semi-amplitude of 3.0m/s. Both LSPM J2116+0234 and GJ 686 have planetary companions populating the regime of exoplanets with masses lower than 15M{Earth}_ and orbital periods <20d.