  The evolution of the energy difference between the neutron states $1i_{13/2}$
and $1h_{9/2}$ in the $N=82$ isotones and that between the proton states
$1h_{11/2}$ and $1g_{7/2}$ in the $Z = 50$ isotopes are investigated within the
framework of the relativistic Hartree-Fock theory, using the density-dependent
effective interactions PKA1 and PKO$i$ ($i = 1 $, $ 2 $, $ 3 $). By identifying
the contributions of the tensor force, which is naturally induced via the Fock
terms, we find that the tensor force plays crucial roles in the evolution of
the shell structure. The strength of the tensor force is also explored. It is
found that moderately increasing the coupling strength of pion-nucleon
coupling, i.e., $f_{\pi}$, will significantly improve the description of the
shell-structure evolution. In particular, reducing the density dependence of
$f_{\pi}$ is shown to be more preferable, in comparison to enlarging $f_{\pi}$
with a factor. This is in consistence with the idea of "tensor renormalization
persistency" and provides valuable guidance for the development of nuclear
energy density functional in the relativistic framework.
