  In this work we investigate the combined finite-size and thermomagnetic
effects on the properties of neutral mesons in a hot medium, in the context of
the Nambu--Jona-Lasinio model. In particular, by using the mean-field
approximation and the Schwinger proper time method in a toroidal topology with
antiperiodic conditions, we evaluate the chiral phase transition, the
constituent quark mass and meson observables like the $\pi ^0$ and $\sigma$
meson masses and pion decay constant under the change of the size, temperature,
chemical potential and strength of external magnetic field. The results
indicate that the observables are strongly affected by the conjoint effects of
relevant variables, and the net result will depend on the balance of these
competing phenomena.
