Hadamard's three circles theorem

Prove the theorem: if $f(z)$ is regular for $r < \left|z\right| < R$, and $M(\rho )$ is the maximum modulus of $f(\rho e^{\varphi i})$, then for $r < \rho < R$ the quantity $M(\rho )$ is a convex function of $\ln \rho$ (Hadamard's Three-Circles Theorem).

Let the function $f(z)$ be regular in the disk $\left|z\right|<R$. Suppose

Then

This means that in an orthogonal system of coordinates the graph of $\log M(r)$ as a function of $\log r$ appears as a convex curve. (Hadamard's three circle theorem.) [Examine $z^{\alpha }f(z)$ with suitably chosen $\alpha$.]

(continued). The function $\log M(r)$ is a strictly convex function of $\log r$ unless $f(z)$ is of the form $az^{\alpha }$, $a,\alpha$ constants, $\alpha$ real: this is the only type of function for which the inequality in problem 304 becomes an equality.

Suppose that the function $f(z)$ is regular for $\left|z\right|<R$ but not of the form $cz^{n}$, $c$ constant, $n$ integer; let

denote the arithmetic mean of $\left|f(z)\right|^{2}$ on the circle $\left|z\right|=r$, $r<R$. The function $I_{2}(r)$ is monotone increasing with $r$ and $\log I_{2}(r)$ is a strictly convex function of $\log r$.

Let $f(z)$ be regular for $\left|z\right|<R$;

denotes the geometric mean of $\left|f(z)\right|$ on the circle $\left|z\right|=r$, $r<R$. The function $\mathfrak {G}(r)$ is monotone increasing with $r$ and a convex function of $\log r$ (in the wide sense).

The function $f(z)$ is supposed to be regular for $\left|z\right|<R$ and not a constant. We put

The function $I(r)$ is monotone increasing with $r$ and $\log I(r)$ is a convex function of $\log r$. [Problems 299, 304.]

We suppose that $f(z)$ is regular and not constant for $\left|z\right|<R$ and that $p$ is a positive number. We define

The function $I_{p}(r)$ is monotone increasing with $r$ and $\log I_{p}(r)$ is a convex function of $\log r$. (Cf.problems 306, 308, 307, 267 and 304, for an analogous case see Part IV, problem 19.)

Consider a regular harmonic function in the disk $\left|z\right|<R$. We denote by $A(r)$ its maximum on the circle $\left|z\right|=r$, $r<R$. When $0<r_{1}<r_{2}<r_{3}<R$ we have

i.e.$A(r)$ is a convex function of $\log r$.

Deduce Hadamard's three circle theorem 304 from problem 320 and vice versa problem 320 from Hadamard's three circle theorem.

The function $\eta = \log M(e^{\xi })$ is represented in a rectangular coordinate system $\xi , \eta$ by a curve which is always increasing and is strictly convex, apart from certain exceptional polynomials for which the curve degenerates into a straight line.

Let $\alpha$ be fixed, $0 < \alpha < 1$. The quotient $M(\alpha r)/M(r)$ is strictly decreasing with increasing $r$, apart from certain polynomials (see problem 20).

With the aid of Problem 123 of Part II, obtain a new proof for the Hadamard three-circle theorem (Problem 304 of Part III).

The modular graph (see Volume I, p. 130) of an entire function $g(z)$ is everywhere convex from below in just two cases: Either when

or when

where $a$ and $b$ are arbitrary complex numbers, $n$ a non-negative integer.

Let the function $f(z)$ be regular in the annulus $\rho < \left|z\right| < R$. Denote

Prove that for $\rho < r_{1} < r_{2} < r_{3} < R$ the inequality holds

(Hadamard's three-circle theorem.)