Theorem 6.5 from Garnett's 2007 book Bounded Analytic Functions is as follows. I quote it verbatim, because I am concerned about the possibility of there being a typo:
Theorem 6.5. Let $v\left(z\right)$ be a subharmonic function in the unit disk $D$. Assume $v(z)\not\equiv-\infty$. For $0<r<1$, let $$v_{r}(z)=\begin{cases} v(z), & \lvert z\rvert\leq r,\\ \frac{1}{2\pi}\int P_{z/r}(\theta)v(re^{i\theta})\mathrm d\theta, & \lvert z\rvert<r. \end{cases}$$ Then $v_{r}(z)$ is a subharmonic function in $D$, $v_{r}(z)$ is harmonic on $\lvert z\rvert<r$, $v_{r}(z)\geq v(z)$, $z\in D$, and $v_{r}(z)$ is an increasing function of $r$.
Am I correct in assuming that the second condition $\lvert z\rvert<r$ in the definition of $v_{r}(z)$ is incorrect, and that it should read $r<\lvert z\rvert<1$? If so, please include "Yes, you are correct" in your response. If am not correct, please include "No, you are incorrect" in your response and tell me exactly how I am supposed to interpret this result. This detail is very important for my research, so I want to make sure I get it exactly right.
