Harmonic measure

Let $\mathfrak {D}$ denote a domain, $\zeta$ be an inner point of $\mathfrak {D}$ and $\mathfrak {B}$ be the set of those boundary points of $\mathfrak {D}$ whose distance to $\zeta$ does not exceed $\varrho$. The circle of radius $\varrho$ and center $\zeta$ is assumed to have an arc that does not belong to $\mathfrak {D}$ and the length of which is not smaller than $\dfrac {2\pi \varrho }{n}$, $n$ integer.

We suppose that the function $f(z)$ is regular and single-valued in the interior of $\mathfrak {D}$ and continuous on the boundary and that, in particular, $\left|f(z)\right|\leq a$ at the points of $\mathfrak {B}$ and $\left|f(z)\right|\leq A$ at the remaining boundary points of $\mathfrak {D}$; $a<A$. Then

[Examine the product $\displaystyle \prod_{\nu =0}^{n-1} f[\zeta +(z-\zeta )\omega^{-\nu }]$ in a suitable domain, $\omega =e^{2\pi i/n}$.]

The domain $\mathfrak {D}$ is supposed to lie in the half-plane $\mathrm{Im}\left(z\right)\geq 0$; the boundary contains a finite number of segments of the real axis; $\Omega$ denotes the sum of the angles under which the segments are seen from an inner point $\zeta$ of $\mathfrak {D}$. Assume that $f(z)$ is regular and single-valued in the interior of $\mathfrak {D}$ and continuous on the boundary of $\mathfrak {D}$, that $\left|f(z)\right|\leq A$ at the real boundary points, $\left|f(z)\right|\leq a$ on the remaining boundary of $\mathfrak {D}$, $0<a<A$. Then

Map the strip $0 < \Im z < D$ onto the circle $\left|w\right| < 1$ in such a way that the point $z=i$ corresponds to the center of the circle $w=0$ ($D > 1$). How large is the arc of the boundary of the circle $\left|w\right|=1$ which in this mapping corresponds to the real axis $\Im z = 0$? (Obvious for $D=2$ and $D=\infty$.) How does the arc in question vary as $D$ increases?

Let two simple closed continuous curves $L_{1}$ and $L_{2}$ have a finite number of common arcs and let the interior of $L_{1}$ be contained in the interior of $L_{2}$. ($L_{1}$ consists of an even number of arcs which run alternately in the interior and on the boundary of the region enclosed by $L_{2}$.)

Map first the interior of $L_{1}$ and then the interior of $L_{2}$ onto the interior of one and the same circle, so that in both cases the same point $O$ in the interior of $L_{1}$ goes over into the center of the circle. The two mappings associate with the distinct segments of $L_{1}$ and $L_{2}$ well-defined circular arcs as their images (see the last remark preceding problem 97).

Prove that the length of the image of any common arc of $L_{1}$ and $L_{2}$ will be smaller in the mapping of the smaller region (bounded by $L_{1}$) than in the mapping of the larger region (bounded by $L_{2}$). Example: problem 130.

Find an interpretation in electrostatics of problem 131.

Let two simple closed, continuous curves $L_{1}$ and $L_{2}$ have only finitely many common points, and let the two regions enclosed by them have a common region $\mathfrak {T}$. Denote by $O$ a point of $\mathfrak {T}$ and by $\mathfrak {T}^{*}$ that connected part of $\mathfrak {T}$ which contains $O$. Such arcs of $L_{1}$ and $L_{2}$ as belong to the boundary of $\mathfrak {T}^{*}$ are termed visible'' from $O$, such arcs as do not belong to the boundary of $\mathfrak {T}^{*}$ are termed hidden'' from $O$. (The terms visible'' and hidden'' have their usual meaning if $L_{1}$ and $L_{2}$ are star-shaped with respect to $O$; see Problem 109 of Part III.)

Map first the interior of $L_{1}$ and then the interior of $L_{2}$ onto the interior of the unit circle, in both cases in such a manner that the center of the circle is the image of $O$. The images of the visible'' parts of $L_{1}$ take up a larger percentage of the boundary of the circle than the images of the hidden'' parts of $L_{2}$. (To visualize this imagine $L_{2}$ as a circle with centre $O$.)

Let $\mathfrak {D}$ be a domain, $\zeta$ an interior point of $\mathfrak {D}$, and $\mathfrak {B}$ the set of the boundary points of $\mathfrak {D}$ whose distance from $\zeta$ is not greater than $\rho$. Let those arcs of the circle of radius $\rho$ and with centre $\zeta$ that do not belong to $\mathfrak {D}$ have total length $\rho \Omega$.

Let the function $f(z)$ be regular and single-valued in the interior of $\mathfrak {D}$ and continuous on the boundary of $\mathfrak {D}$. Further, let $\left|f(z)\right| \leq a$ at points of $\mathfrak {B}$, $\left|f(z)\right| < A$ at the remaining boundary points of $\mathfrak {D}$, $a < A$. Then