Let $\vp:\R_0^+\rar\R_0^+$ be a function such that $\lim_{\to\infty}\vp(r)/r=\infty$. Suppose $\sup\{\E\vp(|X|):X\in A\}\leq C$, then $A$ is uniformly integrable.
Put $h(x)=\vp(|x|)/|x|$. By assumption we find for sufficiently large values of $t$ a number $g(t) > 0$ such that $[|X| > t]\sbe[h(X) > g(t)]$ and $\lim_{t\to\infty}g(t)=+\infty$. Therefore
$$
\E(|X|;|X| > t)
\leq\E(\vp(|X|)/h(X);h(X) > g(t))
\leq\E\vp(|X|)/g(t)
\leq C/g(t)~.
$$