  We study uniform estimates for the family of fundamental Lagrange polynomials
associated with any Leja sequence for the complex unit disk. The main result
claims that all these polynomials are uniformly bounded on the disk, i.e.
independently on the range $N$ of the associated $N$-Leja section. We also give
an improved estimate for special values of $N$. As a first application, we get
an analogous estimate for any compact subset whose boundary is an Alper-smooth
Jordan curve.
  We then deal with some applications in multivariate Lagrange interpolation.
We first give some estimates of the Lebesgue constant for the intertwining
sequence of Leja sequences, a result that also requires an explicit formula for
the associated fundamental Lagrange polynomials with uniform estimates. We
finish by dealing with a method to get explicit bidimensional Leja sequences
and then give a positive example for the bidisk.
