Average prefix code length of every 4-sized frequency vector is bounded at 2
Hint: The code generated by Huffman's algorithm has the minimal average word length, and in particular is at least as good as any other code. Show that there is a "universal" code (of length $n=4$) whose average code length is exactly $2$ on any distribution.

The more general theorem states that the average word length for a code over an alphabet of size at most $2^a+b$ is at most $a + \frac{b}{2^a+b}$; to get the best bound for a specific $n$, write $n = 2^a+b$ with $a$ as large as possible. When $b=0$ there is even a universal code as in your particular case.