Let $f$ and $f_n,n=1,2,\dots$ be real valued function on domain $\Omega \subset \mathbb C$. Then what is mean by $$f(x) \leq \liminf_{n \to \infty}f_n(x)$$ uniformly on compact subsets of $\Omega?$
This appeared in ’Logarithmic potentials with external fields’ by Saff and Totik (Page no. 71).
Without further context I am gonna have to assume "uniform" means "uniform with respect to $n$", in which case it can be rephrased as follows:
For every compact subset $K\subset\Omega$, $$\forall\varepsilon>0,\exists N\in\mathbb Z^+,{\rm s.t. }\ \forall x\in K,\ f(x)\leq\inf_{k\geq N}f_k(x)+\varepsilon$$
"Uniform" means that $N$ is independent of $x$, as long as you only consider $x\in K$. However, $N$ may depend on $K$ and $\varepsilon$.