Do Holder continuous functions preserve rate of uniform convergence sequence? Suppose $g$ is $\alpha$-Hölder continuous and $f_n$ converges uniformly to $f$ at rate $a_n$, so that

$a_n \sup_{x}|f_n(x) - f(x)| \to 0 $ as $n \to \infty$

Then does $g \circ f_n$ converges uniformly to $g\circ f$ at rate $a_n$?
The claim is false in general. To see it, 
take $f_n(x) = x + \frac 1n$, $f(x) = x$, and set $a_n = n^{1/2}$ (for instance), let also $g(x)= x^{1/2}$, where $x\geq 0$. 

We have that $g$ is Holder-$\frac 12$, but the claim in your question fails for these data.

Indeed, $f_n$ converges to $f$ uniformly on $\mathbb{R}$, in particular 
$$
n^{1/2} \sup_{x \in [0,1]} |f_n(x) - f(x)| \to 0.
$$
On the other hand,
$$
n^{1/2} \sup_{x\in [0,1]} |g(f_n(x)) - g(f(x))| = n^{1/2} \sup_{x\in [0,1]} \left| \frac{ f_n(x) - f(x) }{\sqrt{f_n(x)} + \sqrt{f(x)}} \right| = \\
n^{1/2} \frac 1n \sup_{x\in [0,1]} \left| \frac{ 1 }{\sqrt{x+1/n} + \sqrt{x}} \right| \geq  (\text{ take } x=0) \\ n^{-1/2} \frac{1}{\sqrt{1/n}} = 1,
$$
so the convergence fails.
No. 

We need $a_n \sup_x|f_n(x) - f(x)|^\alpha \to 0$

From the Hölder continuity of $g$, there are non-negative constant $C$, $\alpha$ such that

$ a_n|g(x) - g(y)| \le a_n  C |x - y |^{\alpha} $

For $\alpha &gt; 1$, the function $g$ is constant and the convergence is automatic. For $\alpha &lt; 1$, we have, from the uniform convergence of $f_n$ at rate $a_n$, that for any $\epsilon &gt; 0$, $\alpha \le 1$, $x$, there exists a $N &gt; N_*$ such that for $n  &gt; N$

$a_nC|f_n(x) - f(x)|^\alpha &lt;  \epsilon $

Which implies that for any $\epsilon &gt; 0$ and $x$, we have a $n$ so that

$ a_n|g(f_n(x)) - g(f(x))| \le a_n  C |f_n(x) - f(x) |^{\alpha} &lt; \epsilon$

which proves the result.