  We investigate scaling phenomena at first-order quantum transitions, when the
boundary conditions favor one of the two phases. We show that the corresponding
finite-size scaling behavior, arising from the interplay between the driving
parameter and the finite size of the system, is more complex than that emerging
when boundary conditions do not favor any phase. We discuss this issue in the
framework of the paradigmatic one-dimensional quantum Ising model, along its
first-order quantum transition line driven by an external longitudinal field.
