Let $\mu$ be a finite measure on $X \subseteq \mathbb{R}^n$.
Consider the Uniformly Integrable family $\{ f_n(\cdot) \}_{n \in \mathbb{N}}$ of functions $f_n : X \rightarrow \mathbb{R}_{\geq 0}$.
Consider a continuous, strictly-increasing function $\phi:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$, with $\phi(0)=0$ such that $\phi(f_n(\cdot))$ is integrable for all $n$.
Is the family $\{ \phi( f_n(\cdot) ) \}_{n \in \mathbb{N}}$ Uniformly Integrable as well?
More generally, with functions $g_n : X \rightarrow \mathbb{R}^n$ such that the family $\{ |g_n(\cdot)| \}_{n \in \mathbb{N}}$ is Uniformly Integrable, and a continuous $\psi: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$, is $\{ \psi(g_n(\cdot)) \}_{n \in \mathbb{N}}$ Uniformly Integrable?