Integral means of analytic functions

Let the function $f$ be analytic in the punctured disc $0<\left|z\right|<r_{0}$, with Laurent series

Assume there is a positive number $M$ such that

Prove that $c_{n} = 0$ for $n<-2$.

Let

be analytic in $D = \left\{ \left|z\right| < 1\right\}$ and assume that the integral

is finite.

Let $f$ be holomorphic in a neighborhood of $\closure U$, $U = \left\{ z : \left|z\right| < 1\right\}$. Let $f(z) = \sum_{k=0}^{\infty } a_k z^k$ and $s_n(z) = \sum_{k=0}^{n} a_k z^k$. Show that, among all polynomials $P$ of degree $n$ or less, the integral $\dfrac {1}{2\pi }\displaystyle \int_0^{2\pi } \left|f(e^{i\theta }) - P(e^{i\theta })\right|^2\, d\theta$ attains its minimum for $P = s_n$.

Prove the theorem: if $f(z)$ is regular for $r < \left|z\right| < R$, then for $r < \rho < R$ the identity

holds, where $a_{n}$ is a coefficient of the Laurent series.

Prove the theorem: if $f(z) = \sum_{n=0}^{\infty } a_{n}z^{n}$ converges for $\left|z\right| < R$, and moreover $\left|f(z)\right| \leqslant M$, then $\sum \left|a_{n}\right|^{2}R^{2n} \leqslant M^{2}$.

Prove that if the identity $w = \sum_{n=0}^{\infty } a_{n}z^{n}$ gives a mapping of the disk $\left|z\right| < R$ onto an area $S$, then $S = \pi \sum_{n=0}^{\infty } n\left|a_{n}\right|^{2}R^{2n}$.

Prove the theorem: if $f(z)$ is regular in the disk $\left|z\right| < R$ and is not constant, then the function

increases together with $r$ for $0 < r < R$.

Prove the theorem: among all functions $f(z)$, regular inside the disk $\left|z\right| < R$ and satisfying the conditions $\int_{0}^{2\pi } \left|f(Re^{\varphi i})\right|^{2}\, d\varphi = M$, the linear function gives the smallest area onto which the disk $\left|z\right| < R$ is mapped.

Prove the theorem: for a function $f(z)$, regular for $r < \left|z\right| < R$, the quantity

is a convex function of $\ln \rho$, i.e., the inequality holds (for $m > 0$, $n > 0$):

Prove the theorem: if $f(z)$, besides the previous conditions (problem 5652), is also nonzero for $r < \left|z\right| < R$, then the quantity

is likewise, for any $\sigma > 0$, a convex function of $\ln \rho$.

Show that the condition $f(z) \neq 0$ in the previous problem (problem 5653) can be dispensed with.

Prove the theorem: if $f(z)$ is regular for $\left|z\right| < R$, then the quantity $I_{\sigma }(\rho )$ (problem 5653) increases together with $\rho$.

With $z=e^{i\vartheta }$ we find for $k,l=0,1,2,3,\ldots$

The functions $1,z,z^{2},z^{3},\ldots$ form an orthogonal system on the unit circle.

Let $f(z)$ denote a regular function on the disk $\left|z\right|\leq r$. The arithmetic mean [Part II, problem 48] of $f(z)$ on the circle $\left|z\right|=r$ is defined by

where $\omega_{n}=e^{\frac{2\pi i}{n}}$. Show that

This means: If an analytic function is regular at every point of a closed disk its value at the center of the disk is equal to the arithmetic mean of its values on the bounding circle.

Let the function $f(z)$ be regular and different from zero for $\left|z\right|\leq r$. Prove that the geometric mean of $\left|f(z)\right|$ on the circle $\left|z\right|=r$

The function $f(z)$ is regular in the disk $\left|z\right|\leq r$ and does not vanish for $z=0$; the zeros of $f(z)$ in the disk are $z_{1},z_{2},\ldots ,z_{n}$, where each multiple zero is represented with its multiplicity. Then the geometric mean of $\left|f(z)\right|$ on the circle $\left|z\right|=r$ is

Under the hypothesis of problem 120 the geometric mean of $\left|f(z)\right|$ on the disk $\left|z\right|\leq r$,

is always smaller than the geometric mean of $\left|f(z)\right|$ on the circle $\left|z\right|=r$,

We have in fact

[Cf.M. A. Parseval: Mem. par divers savants Vol. 1, pp.639--648 (1805); A. Gutzmer: Math. Ann. Vol. 32, pp.596--600 (1888).] Let $f(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n}+\cdots$ be a regular function for $\left|z\right|\leq r$. The arithmetic mean of $\left|f(z)\right|^{2}$ on the circle $\left|z\right|=r$ is

We assume that the function

is regular for $\left|z\right|\leq r$. The mapping $w=f(z)$ transforms the disk $\left|z\right|\leq r$ into a domain of the $w$-plane. The $w$-values to which several $z$-values, $\left|z\right|\leq r$, correspond have to be counted with the proper multiplicity. The area of the image is

(The area is the additive combination of the areas of the images of the disk $\left|z\right|=r$ under the functions $w=a_{n}z^{n}$, $n=0,1,2,\ldots$.)

Let

be regular in the annulus $r\leq \left|z\right|\leq R$. The area of the image (count covering with the proper multiplicity) is equal to

Suppose that the function

is regular and establishes a one to one relationship between the points of the circle $\left|z\right|=r$ and its image $L$. The area of the domain bounded by $L$ is

The area is considered positive or negative depending on whether the moving image $w$ of the point $z$ describing the circle in positive sense leaves the area enclosed by $L$ on the left or on the right.

Let $f(z)$ be regular on the disk $\left|z\right|\leq r$ and let $J(\varrho )$ denote the area of the image of the disk $\left|z\right|\leq \varrho$, $0\leq \varrho \leq r$, under the mapping $w=f(z)$. Then

We assume that the given points $P_{1},P_{2},\ldots ,P_{n}$ are all inside a circle of radius $R$ and $P$ is moving along this circle. Then

(the geometric mean of the $n$ distances $\overline{PP_{\nu }}$) attains a maximum $>R$ and a minimum $<R$ unless all the $P_{\nu }$'s coincide with the center of the circle.

(continued). The same statement as in problem 139 is true for the maximum of the arithmetic mean

of the $n$ distances $\overline{PP_{\nu }}$, but it is not true for the minimum.

(continued). The same statement as in problem 139 holds for the minimum of the harmonic mean

of the $n$ distances but not for the maximum.

The function $f(z)$ is assumed to be analytic in the open disk $\left|z\right|<1$, bounded on the closed disk $\left|z\right|\leq 1$ and continuous, possibly with the exception of a finite number of points. Then

(More general than problem 118.)

If an analytic function is regular in a closed disk and not a constant its absolute value at the center of the disk is smaller than the arithmetic mean of its absolute value on the boundary of the disk.

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.]

Let $f(z)$ be regular and not a constant in the disk $\left|z\right|<R$. The function $w=f(z)$ maps the circle $\left|z\right|=r$, $r<R$, in the $z$-plane onto a curve in the $w$-plane with length $l(r)$. The ratio $\dfrac {l(r)}{2\pi r}$ is monotone increasing with $r$.

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.)

Let $f(z)$ be an entire function and let $\mathfrak {G}(r)$ and $\mathfrak {g}(r)$ be defined as the geometric means of $\left|f(z)\right|$ on the circle $\left|z\right| = r$, and in the circular disc $\left|z\right| \leq r$, respectively (this is the notation of Problem 121 of Part III). If the moduli of the zeros of $f(z)$ are regularly spaced (this is the notation of Chapter 4, Section 1 of Part II), then

where $\lambda$, $0 < \lambda < \infty$, denotes the exponent of convergence of the zeros. For polynomials the above limit also exists and equals $e^{-1/2}$.

(Continued from problem 64.) Without assuming the regular spacing of the zeros we have

Let $f(z)$ be an entire function of finite order $\lambda$ and $\mathfrak {M}(r)$ and $\mathrm{m}(r)$ the arithmetic means of $\left|f(z)\right|^{2}$ on the circle $\left|z\right| = r$ and in the circular disc $\left|z\right| \leq r$ respectively. Then

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$, and let $m$ be a positive integer. Prove that if the function $f(z)$ is not identically constant, then the function

monotonically increases for $0 \leqslant r < R$.

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

is, for any $\alpha > 0$, a monotonically increasing function of $r$ on the interval $0 \leqslant r < R$ (if $f(z) \not\equiv$ const).

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