  In the papers [6, 7] we have established linear and quadratic bounds, in $k$,
on the growth of the Lebesgue constants associated with the $k$-sections of
Leja sequences on the unit disc $\mathcal{U}$ and $\Re$-Leja sequences obtained
from the latter by projection into $[-1, 1]$. In this paper, we improve these
bounds and derive sub-linear and sub-quadratic bounds. The main novelty is the
introduction of a "quadratic" Lebesgue function for Leja sequences on
$\mathcal{U}$ which exploits perfectly the binary structure of such sequences
and can be sharply bounded. This yields new bounds on the Lebesgue constants of
such sequences, that are almost of order $\sqrt{k}$ when $k$ has a sparse
binary expansion. It also yields an improvement on the Lebesgue constants
associated with $\Re$-Leja sequences.
