  Combining ferroelectricity with other properties such as visible light
absorption or long-range magnetic order requires the discovery of new families
of ferroelectric materials. Here, through the analysis of a high-throughput
database of phonon band structures, we identify a new structural family of
anti-Ruddlesden-Popper phases A$_4$X$_2$O (A=Ca, Sr, Ba, Eu, X=Sb, P, As, Bi)
showing ferroelectric and anti-ferroelectric behaviors. The discovered
ferroelectrics belong to the new class of hyperferroelectrics which polarize
even under open-circuit boundary conditions. The polar distortion involves the
movement of O anions against apical A cations and is driven by geometric
effects resulting from internal chemical strains. Within this new structural
family, we show that Eu$_4$Sb$_2$O combines coupled ferromagnetic and
ferroelectric order at the same atomic site, a very rare occurrence in
materials physics.
