Is $H$ proportional to $B$ or not? I was reading an article saying 

in the cylindrical coordinates a uniform magnetic intensity $H$ along
  the z-axis produces a magnetic field $B$ along the z-axes given by
  $$B\left(r\right)=\frac{H}{1+\frac{GH^{2}r^{2}}{4c^{4}}}$$

I'm a mathematician so not really inside electromagnetism. I went on wikipedia and they simply stase that $H$ is proportional to $B$ so I really do not have a clue of what are talking about. Can anybody give me an insight?
 A: In the presence of matter, the relation between the fields $\bf H$ and $\bf B$, which a trivial proportionality in the vacuum (${\bf B}= {\bf H}$, using cgs units), may be a much more complicate dependence. In particular, the relation may be non-linear and, since we are dealing with relations between vectors, a  component of $\bf B$ along a direction may depend even on components of $\bf H$ in different directions.
If the relation is still linear, this is equivalent to have a magnetic permeability tensor $\mu_{ij}$ instead a single scalar $\mu$.
Your case looks that of non-linear relation between the same cartesian component of the two fields. At low values of $\bf H$ you recover the linear case, but for higher values, deviations from the linear behavior appear. That's something physically possible andmeaningful. if you think in terms of alignment  of microscopic magnetic momenta, it is reasonable that such an effect, at some point should saturate.
added note
I started writing before Alexander's comment. Initially me too I had the impression that your case had to do with a non parallel relation between the two magnetic fields. But that's definitely not your case. You are in a non-linear regime.
