  The contributions to the anomalous magnetic moment of the lepton $L$ ($L=e\ ,
\mu $ or $\tau$) generated by a specific class of QED diagrams are evaluated
analytically up to the eighth order of the electromagnetic coupling constant.
The considered class of the Feynman diagrams involves the vacuum polarization
insertions into the electromagnetic vertex of the lepton $L$ up to three closed
lepton loops. The corresponding analytical expressions are obtained as
functions of the mass ratios $r=m_l/m_L$ in the whole region $0 < r < \infty$.
Our consideration is based on a combined use of the dispersion relations for
the polarization operators and the Mellin-Barnes integral transform for the
Feynman parametric integrals. This method is widely used in the literature in
multi-loop calculations in relativistic quantum field theories. For each order
of the radiative correction, we derive analytical expressions as functions of
$r$, separately at $r<1$ and $r>1$. We argue that in spite of the obtained
explicit expressions in these intervals which are quite different, at first
glance, they represent two branches of the same analytical function.
Consequently, for each order of corrections there is a unique analytical
function defined in the whole range of~$r\in (0,\infty)$.
  The results of numerical calculations of the $4th$, $6th$ and $8th$ order
corrections to the anomalous magnetic moments of leptons ($L=e ,\mu , \tau$)
with all possible vacuum polarization insertions are represented as functions
of the ratio $r=m_l/m_L$. Whenever pertinent, we compare our analytical
expressions and the corresponding asymptotic expansions with the known results
available in the literature.
