Harmonic Functions

An analytic function that is regular in the closed disk $\mathfrak {R}$ cannot assume real values at all the boundary points of $\mathfrak {R}$ except if it is a real constant.

We denote by $u(x,y)$, $z=x+iy$, a harmonic function that is regular in the disk $(x-x_{0})^{2}+(y-y_{0})^{2}\leq r^{2}$. The value of $u(x,y)$ at the center is

consequently

If a harmonic function is regular in a closed disk its absolute value at the center does not exceed the arithmetic mean of its absolute values on the boundary. Under what conditions is there equality?

A harmonic function is supposed to be regular and single-valued in a domain $\mathfrak {D}$. Then it attains its maximum and its minimum on the boundary and only on the boundary unless it is a constant.

A harmonic function that is regular in the domain $\mathfrak {D}$ and vanishes at all the boundary points of $\mathfrak {D}$ is identically zero.

The equilibrium described in solution 31 is not stable.

A harmonic function is assumed to be single-valued in the domain $\mathfrak {D}$ and regular with the exception of finitely many points at which it becomes $-\infty$ (i.e.the function converges to $-\infty$ as $z$ approaches such a point). Then it assumes its maximum on the boundary of $\mathfrak {D}$.

Let the function $f(z)$ be regular in the disk $\left|z\right|\leq R$. The function $w=f(z)$ maps the disk onto a certain domain of the $w$-plane that is assumed to be star-shaped with respect to $w=0$. Let $f(0)=0$. Then the images of the concentric circles $\left|z\right|=r$, $r<R$, are also star-shaped with respect to the origin.

Suppose that the function $f(z)$ is regular in the disk $\left|z\right|\leq R$ and that it maps the disk onto a convex domain of the $w$-plane. Then the image of an arbitrary circle in the open disk $\left|z\right|<R$ is also convex.

Let $u_{1}(x,y),u_{2}(x,y),\ldots ,u_{n}(x,y)$, $z=x+iy$, be regular harmonic functions in a domain $\mathfrak {D}$. The continuous function

assumes its maximum on the boundary of $\mathfrak {D}$.

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.