Hadamard three-lines and three-circles theorems

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Problem 2.12.7 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.12 "The Maximum Modulus Principle"

In the proof of Theorem 12.8 it was tacitly assumed that $M(a) > 0$ and $M(b) > 0$ . Show that the theorem is true if $M(a) = 0$ , and that then $f(z) = 0$ for all $z \in \Omega$ .

Problem 2.12.8 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.12 "The Maximum Modulus Principle"

If $0 < R_{1} < R_{2} < \infty$ , let $A\left(R_{1}, R_{2}\right)$ denote the annulus

\left\{ z: R_{1} < \left|z\right| < R_{2}\right\}

There is a vertical strip which the exponential function maps onto $A\left(R_{1}, R_{2}\right)$ . Use this to prove Hadamard's three-circle theorem: If $f \in H\left(A\left(R_{1}, R_{2}\right)\right)$ , if

M(r) = \max _{\theta }\left|f\left(r e^{i \theta }\right)\right| \quad \left(R_{1} < r < R_{2}\right)

and if $R_{1} < a < r < b < R_{2}$ , then

\log M(r) \leq \frac{\log (b / r)}{\log (b / a)} \log M(a)+\frac{\log (r / a)}{\log (b / a)} \log M(b)

For which $f$ does equality hold in this inequality?

Remark.

In other words, $\log M(r)$ is a convex function of $\log r$ .

References

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.