What does "hold uniformly" mean in the context of asymptotic analysis?
We say a bound on a family of objects holds uniformity if there is a single bound that applies to all objects in the family, as opposed to a bound that is dependent upon the parameter that determines which element of the family is under discussion.

For example, $f_n(x)= \sin(nx)$ is uniformly bounded by $1$, while $g_n(x)=n\cos(nx)$ is not. In fact, $g_n$ isn’t uniformly bounded by anything, as any bound on it needs to depend on $n$.