Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative axis. Is $u(z)$ subharmonic on $D\setminus\lbrace z\in D: f(z)=0\rbrace?$ When applying the maximum principle to $u(z)$ must one check values in $D$ along the negative axis?
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Assuming you are using the principal branch of the $s$'th root, $u(z)$ is indeed a subharmonic function. You do need to "check the branch cut", but it works out ok. Consider a point $p$ where $f(p)$ is on the negative real axis (so $p$ is on a branch cut of $f(z)^{1/s}$), and let $v(z)$ be the version of $\Re f(z)^{1/s}$ obtained by moving the branch cut slightly away from $p$ in some direction. Then $v(z)$, being the real part of an analytic function, is harmonic near $p$, and $u(z) \ge v(z)$. So for small $r > 0$, $$u(p) = v(p) = \frac{1}{2\pi} \int_0^{2\pi} v(p + r e^{i\theta})\ d\theta \le \frac{1}{2\pi} \int_0^{2\pi} u(p + r e^{i\theta})\ d\theta $$ 

