Rudin's Real and Complex Analysis, 3rd edition, page 236 :
How did Rudin get to $$ \frac{R-r}{R+r}u_n(a) \leq u_n(a+re^{i\theta}) \leq \frac{R+r}{R-r}u_n(a).$$
It seems to me that he used the mean value property, but this what only introduced later in the book.
Caution : The first deplayed equation should be read $$ \frac{R-r}{R+r} \leq \frac{R^2 - r^2}{R^2 -2rR \cos(\theta -t) + r^2} \leq \frac{R+r}{R-r}$$