$\newcommand\C{\mathbb C}\newcommand\R{\mathbb R}$Yes, this follows by Loewner's theorem on monotone matrix functions (see e.g. [Theorem 1.6][1]), which in particular implies the following: 

Let $M_n$ denote the set of all analytic functions $f\colon\C\setminus(-\infty,0]\to\C$ such that $f((0,\infty))\subseteq\R$ and 
$$A\le B\implies f(A)\le f(B)$$
for all $n\times n$ positive-definite matrices $A$ and $B$, where $A\le B$ means that $B-A$ is positive semidefinite. 
Then $f\in M_n$ for all natural $n$ iff 
$$\Im z>0\implies \Im f(z)>0.$$ 

The above conditions on $f$ hold if $f(z)=z^\alpha$ for $\alpha\in[0,1]$ and all $z\in\C\setminus(-\infty,0]$. 

So, your desired result immediately follows.


  [1]: https://link.springer.com/book/10.1007/978-3-030-22422-6