  Properties of nuclear and neutron matter are discussed in a nonlinear
$\sigma$-$\omega$-$\rho$ mean-field approximation with self-interactions and
mixing-interactions of mesons and baryons. The nonlinear interactions are
renormalized by employing the theory of conserving approximations, which
results in a thermodynamically consistent approximation that maintains
Hugenholtz-Van Hove theorem and Landau's requirement of quasiparticles. The
approximation is equivalent to the Hartree approximation with {\it effective
masses} and {\it effective coupling constants} of baryons and mesons. The
effective masses and coupling constants are naturally required by
self-consistency of the theory of conserving approximations. The approximation
is applied to nuclear and neutron matter, which suggests that the lower bound
of nuclear compressibility $K \sim 180$ MeV (with the symmetry energy $a_4 =
35.0$ MeV) be required to be consistent with properties of nuclear matter and
the maximum masses of observed hadronic neutron stars ($M_{max} \ge 2.00$
$M_{\odot}$). The values of the compressibility, symmetry energy together with
effective masses and coupling constants of baryons and mesons will be important
constraints to examine models of nuclear and neutron matter. The accumulating
data and accurate measurements of observables in high density and energy region
will supply significant information in order to testify theoretical consistency
of nuclear models.
