  If $\rho_E$ denotes the pseudohyperbolic distance on $B_E$, the open unit
ball of a complex finite or infinite dimensional Hilbert space, we prove that,
in general, the expression: $$\frac{\rho_E(zx,zy)}{|z| \rho_E(x,y)}$$ is
unbounded if $x, y$ belongs to $B_E$ and $z \in \mathbb{C}$ such that $zx,zy
\in B_E$. Nevertheless, if $z$ is controlled by:
$\frac{1+\max\{\|x\|,\|y\|\}}{2 \max\{\|x\|,\|y\|\}}$,
  $$(1+\max\{\|x\|,\|y\|\})/(2 \max\{\|x\|,\|y\|\})$$ then the expression above
is bounded by $2$ and this bound is the best possible. As a consequence, if $f$
belongs to the space $\mathcal{B}(B_E)$ of Bloch functions on $B_E$, we prove
that the map defined on $B_E$ given by $x \mapsto (1-\|x\|^2) \mathcal{R}
f(x)$, where $\mathcal{R} f$ denotes the radial derivative of $f$, is Lipschitz
with respect to the pseudohyperbolic distance. This yields some applications.
In particular, we provide results related to interpolating sequences for
$\mathcal{B}(B_E)$ and after a deep study of the automorphisms of $B_E$, we
also supply necessary and suficient conditions for a composition operator on
$\mathcal{B}(B_E)$ to be bounded below.
