  Using the general structure of the vacuum polarization tensor
$\Pi_{\mu\nu}(k_{0},\mathbf{k})$ in the infrared (IR) limit, $k_{0}\to 0$, the
ring contribution to QED effective potential at finite temperature and non-zero
magnetic field is determined beyond the static limit, $(k_{0}\to
0,\mathbf{k}\to \mathbf{0})$. The resulting ring potential is then studied in
weak and strong magnetic field limit. In the limit of weak magnetic field, at
high temperature and for $\alpha\to 0$, the improved ring potential consists of
a term proportional to $T^{4}\alpha^{5/2}$, in addition to the expected
$T^{4}\alpha^{3/2}$ term arising from the static limit. Here, $\alpha$ is the
fine structure constant. In the limit of strong magnetic field, where QED
dynamics is dominated by the lowest Landau level (LLL), the ring potential
includes a novel term consisting of dilogarithmic function
$(eB){Li}_{2}(-\frac{2\alpha}{\pi}\frac{eB}{m^{2}})$. Using the ring improved
(one-loop) effective potential including the one-loop effective potential and
ring potential in the IR limit, the dynamical chiral symmetry breaking of QED
is studied at finite temperature and in the presence of strong magnetic field.
The gap equation, the dynamical mass and the critical temperature of QED in the
regime of LLL dominance are determined in the improved IR as well as in the
static limit. For a given value of magnetic field, the improved ring potential
is shown to be more efficient in decreasing the critical temperature arising
from one-loop effective potential.
