The upper and lower bounds on the mass of the Higgs boson are numerically investigated in an SU(2) L ⊗ SU(2) R symmetric Yukawa model, which describes two heavy degenerate fermion doublets in the limit of vanishing gauge couplings. The values for the bounds are in good agreement with those obtained from an integration of the one-loop β-functions, although the Yukawa couplings are strong. Simulations in a U(1) L ⊗ U(1) R symmetric Yukawa model with decoupled mirror fermions and a single fermion doublet have been performed. This theory, whose interacting sector is truly chiral, has a complex fermion determinant.