This work displays the min-max and max-min principles involved in solving different 2 + 1 Dirac fermion problems. The first problem concerns a charged fermionic oscillator in an external homogeneous magnetic field, for which a quantum phase transition was predicted (Bermudez et al., 2008). The 'zeromass' fermion near Dirac point in graphene monolayer constitutes the second example. Here, the minmax and max-min theorems arise from lattice symmetry, being otherwise independent of the trial function. The min-max and max-min principles hold for the trial solution of the fermion with 'nearly zero' mass in diatomic materials with graphene-type lattice. (C) 2017 Elsevier B.V. All rights reserved.