I suppose the following result follows
from Ambrose-Singer theorem, but I cannot
find a reference, and the arguments I found
in the literature are usually weaker. The idea
is that holonomy over a null-homotopic loop is bounded
by the supremum of the curvature times the area of the
2-dimensional surface segment bounded by the loop.


THEOREM-CONJECTURE  

Let $D$ be a unit disk, and $(B, \nabla)$ a trivial
vector bundle on $D$ with connection (not necessarily
orthogonal). Assume that the curvature
$R$ of $\nabla$ is uniformly bounded,
that is, $R(x, y)$ belongs to a compact subset $K$
in $End(B\_m)$ for all $x, y \in T\_m D$ of length 1.
Then the holonomy of $\nabla$ around the boundary
of $D$ is bounded by a uniform constant which
depends on $K$ only.


I think I can prove this, but there are
some segments of the proof which are tricky
and take too much effort.


Can someone please point me to a reference,
or to some relevant papers. Many thanks in advance.



Holonomy bounded in terms of area and the curvature
There are in fact more precise versions, expressing the parallel translation around a loop as the identity map plus a curvature integral over a homotopy. References:


Section 3.1 of Werner Ballmann's lecture notes on vector bundles:
<http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/conncurv1999.pdf>


Deane Yang's notes "Holonomy equals curvature" on his web page
[https://cims.nyu.edu/~yangd/papers/holonomy.pdf](https://cims.nyu.edu/%7Eyangd/papers/holonomy.pdf)


Buser-Karcher, Gromov's almost flat manifolds, page 92.


