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Rudin's Real and Complex Analysis, 3rd edition, page 236 : 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 $$ \frac{R-r}{R+r} \leq \frac{R^2 - r^2}{R^2 -2rR \cos(\theta -t) + r^2} \leq \frac{R+r}{R-r}$$

M.G
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  • What do the biggest and smallest values for the Poisson kernel's denominator look like? – Steve D May 16 '16 at 00:50
  • There seems to be a square missing in the denominator? I'd be surprised if it shouldn't be $r^2$. – joriki May 23 '16 at 14:30
  • Yeah, I spotted this mistake just a few days ago. It should be $r^2$ indeed. – M.G May 23 '16 at 14:34
  • It would have made sense to mention that in the question. – joriki May 23 '16 at 14:53
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    For the unit disk, it was shown in 11.9 that a harmonic function that is continuous on the closed unit disk is the Poisson integral of its boundary values. For general disks, translation and scaling give the same result (11.10). But for the centre of the disk, the Poisson integral is just the mean value of the boundary values. – Daniel Fischer May 23 '16 at 14:56
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    @ Joriki : it was. I must have deleted it inadventently while rewording the question this morning. My apologies. – M.G May 23 '16 at 15:02

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