  We derive upper and lower bounds on the mass-radius ratio of stable compact
objects in extended gravity theories, in which modifications of the
gravitational dynamics via-{\' a}-vis standard general relativity are described
by an effective contribution to the matter energy-momentum tensor. Our results
include the possibility of a variable coupling between the matter sector and
the gravitational field and are valid for a large class of generalized gravity
models. The generalized continuity and Tolman-Oppenheimer-Volkoff equations are
expressed in terms of the effective mass, density and pressure, given by the
bare values plus additional contributions from the total energy-momentum
tensor, and general theoretical limits for the maximum and minimum mass-radius
ratios are explicitly obtained. As an applications of the formalism developed
herein, we consider compact bosonic objects, described by scalar-tensor
gravitational theories with self-interacting scalar field potentials, and
charged compact objects, respectively. For Higgs type models, we find that
these bounds can be expressed in terms of the value of the potential at the
surface of the compact object. Minimizing the energy with respect to the
radius, we obtain explicit upper and lower bounds on the mass, which admits a
Chandrasekhar type representation. For charged compact objects, we consider the
effects of the Poincar\'{e} stresses on the equilibrium structure and obtain
bounds on the radial and tangential stresses. As a possible astrophysical test
of our results, we obtain the general bound on the gravitational redshift for
compact objects in extended gravity theories. General implications of minimum
mass bounds for the gravitational stability of fundamental particles and for
the existence of holographic duality between bulk and boundary degrees of
freedom are also considered.
