  We focus on weak inhomogeneous models of the Universe at low redshifts,
described by the Lema\^itre-Tolman-Bondi (LTB) metric. The principal aim of
this work is to compare the evolution of inhomogeneous perturbations in the
$\Lambda$CDM cosmological model and $f(R)$ modified gravity theories,
considering a flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric for the
background. More specifically, we adopt the equivalent scalar-tensor formalism
in the Jordan frame, in which the extra degree of freedom of the $f(R)$
function is converted into a non-minimally coupled scalar field. We investigate
the evolution of local inhomogeneities in time and space separately, following
a linear perturbation approach. Then, we obtain spherically symmetric solutions
in both cosmological models. Our results allow us to distinguish between the
presence of a cosmological constant and modified gravity scenarios, since a
peculiar Yukawa-like solution for radial perturbations occurs in the Jordan
frame. Furthermore, the radial profile of perturbations does not depend on a
particular choice of the $f(R)$ function, hence our results are valid for any
$f(R)$ model.
