Miscellaneous Problems

If the function $f(z)$ is regular, single valued and bounded in the strip $0 \leq \Re z \leq \pi$, and vanishes there at the points $z_{1}, z_{2}, z_{3}, \ldots , z_{n}, \ldots$ ($z_{n}=x_{n}+iy_{n}$) then either the series of positive terms

is convergent, or $f(z)$ is identically equal to $0$.

Assume that the coefficient $f_{1}(z)$ of the linear homogeneous differential equation

is an entire function. The necessary and sufficient condition that the general solution of this differential equation be an entire function is that the remaining coefficients $f_{2}(z), f_{3}(z), \ldots , f_{n}(z)$ are also entire functions.

Let the function $w=\varphi (z)$ be regular (possibly many-valued) in the annular region $0 < \left|z\right| < \rho$, $\rho > 0$, and satisfy there identically an equation of the form

where $F_{0}(z), F_{1}(z), \ldots , F_{l-1}(z), F_{l}(z)$ are regular in a neighbourhood of the point $z=0$. If there exists a power series $c_{0}+c_{1}z+c_{2}z^{2}+\cdots$ such that the function $(\varphi (z)-c_{0}-c_{1}z-c_{2}z^{2}-\cdots -c_{n-1}z^{n-1})z^{-n}$ remains bounded in the neighbourhood of $z=0$ for infinitely many values of $n$, then $\varphi (z)$ is regular in the neighbourhood of $z=0$, and we have $\varphi (z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots +c_{n}z^{n}+\cdots$.

Let the function $f(z)$ be regular and single-valued for $R \leq \left|z\right| < \infty$, $R > 0$. Then there exists an integer $p \geq 0$, an entire function $G(z)$ and a power series

convergent for $\left|z\right| \geq R$, such that for $R \leq \left|z\right| < \infty$ we have

If in the assumption as well as in the conclusion we replaced the inequality $R \leq \left|z\right| < \infty$ by the inequality $R < \left|z\right| < \infty$, the theorem obtained would be invalid.

Let $f(z)=f(x+iy)$ be a meromorphic periodic function of period $2\pi$ that has in the strip $0 \leq \Re z < 2\pi$ only a finite number of zeros and poles. Denote by $M(y)$ the maximum and by $\mu (y)$ the minimum of $\left|f(x+iy)\right|$ for $0 \leq x \leq 2\pi$. If it is the case that

then $f(z)$ must be a rational function of $e^{iz}$.

Let $f(z)$ be an analytic function, $z=x+iy$, and let the square of the absolute value of $f(z)$

be an algebraic function of the real variables $x$ and $y$. Then the function $f(z)$ is itself an algebraic function of $z$.

There does not exist a function which is regular analytic along the real axis and which assumes for real values of the variable every value in the interior of a fixed circle. In brief: There does not exist an analytic Peano-curve.

Find, if possible, $n+1$ analytic functions $f_{1}(z), f_{2}(z), \ldots , f_{n}(z), f(z)$, which differ from each other not merely by constant factors, are regular in a domain $\mathfrak {D}$ and for which the relation

holds in $\mathfrak {D}$.

Find, if possible, two non-constant analytic functions $f(z)$ and $g(z)$, which are regular in a domain $\mathfrak {D}$ and for which the relation

holds in $\mathfrak {D}$.

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.

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.

We say that two entire functions $f(z)$ and $g(z)$ have the same $a$-points if the two functions

are both entire. Find two distinct entire functions, which have the same $a$-, $b$- and $c$-points. (Naturally we require $b\neq c$, $c\neq a$, $a\neq b$.)

Does there exist an entire function $G(z)$ which satisfies the equations

if the sequence of numbers $a_{0}, a_{1}, a_{2}, \ldots$ is prescribed arbitrarily?

Given the numbers

for $n=1,2,3,\ldots$ subject to the conditions:

$m_{n}$ is a positive integer, however the $w_{n,k}$ are unrestricted.

Then there exists an entire function such that

for $n=0,1,2,\ldots$, $0\leq k<m_{n}$.

Given the numbers

for $n=1,2,3,\ldots$ subject to the conditions:

$m_{n}$ is a positive integer, however the $w_{n,k}$ are unrestricted.

Then there exists an entire function such that

for $n=0,1,2,\ldots$, $0\leq k<m_{n}$.

Suppose that $F(z)$ and $G(z)$ are entire functions that have no common zeros. Then there exist two entire functions $f(z)$ and $g(z)$ such that

Suppose that $F(z)$ and $G(z)$ are entire functions that have no common zeros. Then there exist two entire functions $f(z)$ and $g(z)$ such that

Let $f(z) = a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n}+\cdots$ be regular for $\left|z\right|<R$ and set $\mathfrak {M}(r) = \left|a_{0}\right|+\left|a_{1}\right|r+\left|a_{2}\right|r^{2}+\cdots +\left|a_{n}\right|r^{n}+\cdots$. Then we have

(continued). If $\mathfrak {M}_{n}(r)$ is defined for the function $[f(z)]^{n}$ in the same manner as $\mathfrak {M}(r)$ for $f(z)$, $n=1,2,3,\ldots$, then we have

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

Prove the following generalization of problem 160: If $\omega_{n}$ is defined as in problem 157, then under the assumptions of problem 160 we have the following estimate of the coefficients

Assume the Bernstein theorem for trigonometric polynomials (Problem 82 of Part VI) in the following imprecise form: There exists an absolute constant $K$, $K>0$ with the property that, if $\varphi (\vartheta )$ denotes a trigonometric polynomial of $n$th order whose absolute value does not exceed $1$, then we have

Find a method of deducing from this estimate the sharper estimate

There exist entire functions that increase arbitrarily rapidly as $z\to \infty$ along a ray. More precisely: If $\varphi (x)$ is an arbitrary, strictly positive, monotonically increasing function defined for $x\geq 0$, there exists an entire function $g(z)$, which assumes real values for real $z$ and satisfies for $x\geq 0$ the inequality $g(x)>\varphi (x)$. (See Problem 290 of Part III.)

There exist entire functions that tend to $0$ along the positive real axis, but tend to $\infty$ along any other ray emanating from the origin as $z\to \infty$. Can a rational entire function behave in this manner?

There exist transcendental entire functions that tend to $\infty$ along every ray emanating from the origin. --- Can this happen uniformly?

There exist entire functions that are real-valued along the positive real axis and along it tend to $+\infty$ while tending to $0$ along any other ray emanating from the origin. --- Can an entire function of finite order behave in this fashion?

There exist not identically vanishing entire functions that tend to $0$ along every ray emanating from the origin.

There exist entire functions that tend to $+1$ along all rays running from the origin into the interior of the upper half-plane, and that tend to $-1$ along all rays running from the origin into the interior of the lower half-plane. The convergence is in fact uniform in every sector $\varepsilon <\arg z<\pi -\varepsilon$, or $-\pi +\varepsilon <\arg z<-\varepsilon$, where $\varepsilon >0$.

Let the whole plane be divided into $n$ sectors by $n$ rays emanating from the origin. There exists an entire function that tends in these sectors (more precisely, as in problem 185) to $a_{1}, a_{2}, \ldots , a_{n}$ respectively, where $a_{1}, a_{2}, \ldots , a_{n}$ are arbitrarily assigned complex numbers.

Let the rays emanating from the origin be divided in any manner into two classes. Does there exist, corresponding to every such division, an entire function that tends to $0$ along the rays of the first class, and that tends to $\infty$ along the rays of the second class?

The numbers $0$ and $\infty$ are the only asymptotic values of $e^{z}$.

The numbers $\sqrt{\pi /2}, -\sqrt{\pi /2}$ and $\infty$ are the only asymptotic values of the function $\int_{0}^{z} e^{-x^{2}/2} \, dx$.

Let $n$ be a positive integer. The integral function of order $n$

has exactly $2n$ distinct finite asymptotic values.

Let the sequence of positive numbers

be chosen such that the series

converges uniformly in every finite domain of the $z$-plane and thus represents an entire function $g(z)$. (Set for example $a_{m}=\exp (-m^{8^{m}})$.)

Show that the set of asymptotic values of the entire function $g(z)$ thus defined has the power of the continuum. More precisely: All numbers of the form

are asymptotic values.

If an entire function converges to $n$ distinct finite asymptotic values along $n$ distinct rays emanating from the origin, then its order is not less than $\dfrac {n}{2}$.

For every entire function (that is not a constant) the number $\infty$ is an asymptotic value.

Let $m$ be the number of zeros of $g(z)-a$. Determine the polynomial $P(z)$ of degree $m$ such that $[P(z)]/[g(z)-a]$ is regular in the whole $z$-plane, and then the polynomial $Q(z)$ of degree at most $m$ such that also

is regular in the whole $z$-plane. There hence exists (using problem 193) a continuous curve going to infinity such that at points sufficiently far along it we have

Let $f(z)$ be a function that can be analytically continued everywhere in the annular region $0<\left|z\right|<1$. If in this process $f(z)$ and all its derivatives $f'(z), f''(z), \ldots$ remain bounded, then $f(z)$ is single-valued in this region and regular at the point $z=0$.

If $g(z)$ is an entire function of genus $0$ and $\varepsilon$ is an arbitrary positive number, then we have

on all circles with sufficiently large radii, and

on certain circles with arbitrarily large radii.

Let $M(r)$ be the maximum and $m(r)$ the minimum of the absolute value of an entire function on the circumference of the circle $\left|z\right|=r$. If

and $\lambda <\tfrac {1}{2}$, then we have also

Let $\lambda_{1}, \lambda_{2}, \lambda_{3}, \ldots , \lambda_{n}, \ldots$ be distinct positive numbers such that the series

is divergent. If a function $h(t)$ which is properly integrable over the interval $0\leq t\leq 1$ satisfies the condition

then $h(t)=0$ at every point of continuity $t$. (A generalization of Problem 139 of Part II.)

Let $g(z)$ be a transcendental entire function of order $\lambda <\tfrac {1}{2}$. Let its coefficients be different from $0$ and its zeros $w_{1}, w_{2}, w_{3}, \ldots , w_{n}, \ldots$ be distinct, $w_{k}\neq w_{l}$ for $k\neq l$. If a function $h(t)$ which is defined and properly integrable over the interval $0\leq t\leq 1$ satisfies the condition

then $h(t)=0$ at every point of continuity $t$. (See Problem 139 of Part II.)

Let $g(z)$ be a transcendental entire function of genus $0$ with all its zeros $w_{1}, w_{2}, w_{3}, \ldots , w_{n}, \ldots$, real and distinct. If a function $h(t)$ defined and properly integrable in the interval $0\leq t\leq 1$ satisfies the condition

then $h(t)=0$ at every point of continuity $t$. (We may, for example, set $g(z)=J_{0}(\sqrt{z})$ or $\cos \pi \sqrt{z}$.)

Set

Assume that there exist two positive constants $\rho$ and $M$ such that

1. the sequence $a_{0}, a_{1}\rho^{-1}, a_{2}\rho^{-2}, \ldots , a_{n}\rho^{-n} \ldots$ is bounded, and

2. $\left|F(z)\right|\leq M$ for all real values of $z$. Then we have for all real values $z$ also

and equality holds only if $F(z)=A\cos \rho z+B\sin \rho z$, where $A,B$ are constants. (A generalization of Problem 82 of Part VI.)

(continued). If $d$ is the distance of the point $z$ from the real axis ($d=\left|\Im z\right|$), then

Let the entire function $G(z)$ satisfy the same conditions as the function $F(z)$ in problem 201, and in addition let $G(z)$ be an odd function, $G(-z)=-G(z)$. Then we have for real $z$

Equality holds only if $G(z)=cM\sin \rho z$, $\left|c\right|=1$ and if $z=0$. (Analogue of Problem 81 of Part VI.)

(continued). Deduce problem 201 from problem 203.

Let $f(z)$ be an entire function of order $\lambda$, $\lambda \geq \tfrac {1}{2}$, that is bounded on the positive real axis. If $\varepsilon$ denotes an arbitrary positive number, then as $x$ tends to infinity through positive values we have

We assume that the sequence of complex numbers $z_{1}, z_{2}, \ldots , z_{k}, \ldots$ has the following three properties:

1. The series

   $$\sum _{k=1}^{\infty } z_{k}, \qquad \sum _{k=1}^{\infty } z_{k}^{2}, \qquad \ldots , \qquad \sum _{k=1}^{\infty } z_{k}^{n}$$

   are convergent.

2. $\left|z_{1}\right| \geq \left|z_{2}\right| \geq \cdots \geq \left|z_{k}\right| \geq \cdots$.

3. There exist two real numbers $\alpha$ and $\beta$ such that

   $$0 < \beta -\alpha < \frac{2\pi n}{n+1}, \qquad \alpha \leq \arg z_{k} \leq \beta , \qquad \text{for } k=1,2,3,\ldots .$$

Then $\left|z_{1}\right|^{n}+\left|z_{2}\right|^{n}+\cdots +\left|z_{k}\right|^{n}+\cdots$ is convergent.

The conclusion holds under a wider assumption: The series listed in (1) need not be convergent; it is sufficient to assume that their partial sums are bounded. Yet the conclusion becomes invalid if we drop assumption (2) or if we require less than the first part of (3), namely that

The functions $f_{1}(z), f_{2}(z), \ldots , f_{n}(z)$; $g_{1}(z), g_{2}(z), \ldots , g_{n}(z)$ are regular analytic in a region of the $z$-plane where

If $f_{1}(z), f_{2}(z), \ldots , f_{n}(z)$ are linearly independent, then so also are $g_{1}(z), g_{2}(z), \ldots , g_{n}(z)$, and

for $\nu =1,2,\ldots ,n$; the constants $c_{\nu \lambda }$ form a unitary matrix.

We consider the approximation of the entire function $g(z)$ by polynomials. We define the integer

(termed the degree of approximation of the function $g(z)$ on the circle $\left|z\right|=r$ to accuracy $\alpha$) as the degree of the polynomial $P(z)$ of the lowest possible degree for which

($r>0$, $\alpha >1$). Then

where we keep $r$ fixed when computing the first limit and $\alpha$ fixed when computing the second limit; $\rho$ is the order of $g(z)$.

Let $g(z)$ be an entire function whose growth does not exceed the minimum type of order $1$; that is, $g(z)e^{-\varepsilon \left|z\right|}$ remains bounded in the whole $z$-plane provided that $\varepsilon$ is a fixed positive number. Assume that the sequence

is bounded. Then $g(z)$ reduces to a constant.

We consider square matrices with $n$ rows and $n$ columns, and let capital letters denote such matrices. Let

be an entire function and $Z$ a matrix. Then the series

converges to a well-defined matrix.

Given the matrix $A$, when does there exist a matrix $Z$ such that

Let $f(z)$ be a single-valued analytic function which is regular everywhere including the point $z=\infty$, except at the point $z=1$. In a neighborhood of the point $z=0$ it has the expansion

and in a neighborhood of the point $z=\infty$ the expansion

If there exists a constant $k$ such that both $a_{n}n^{-k}$ and $b_{n}n^{-k}$ are bounded ($n=1,2,3,\ldots$), then $f(z)$ is a rational function.

Let the numbers $a, b, m_{1}, m_{2}, m_{3}, \ldots$ be real, and satisfy

Furthermore, let $f(x)$ be continuous in the interval $[a,b]$. Then it follows from

for $n=1,2,3,\ldots$, that $f(x)$ vanishes identically. (A fraction of the whole system of Fourier functions is closed over the corresponding fraction of the whole interval.)