In this note, the existence of translation-universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets. Corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of
harmonic functions on the plane. Moreover, it is shown the existence of entire functions which are bounded on many rays and, simultaneously, are universal with respect to a prescribed infinite-order differential operator.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superior (DGES). EspañaMinisterio de Ciencia y Tecnología (MCYT). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER
