Mathematics Problem Archive

Find criteria for and/or give explicit methods for the construction of meromorphic functions $f$ in $\mathbb{C}$ with the following properties: [(a)] ...

[(a)] ; Let $f$ be non-constant and meromorphic in the open unit disc $\mathbb{D}$, with $\alpha<+\infty$, and define $$ \alpha = \limsup_{r\to1}\frac...

Let $f$ be a function meromorphic in $\mathbb{D}$, for which $\alpha<+\infty$ in ([source label: alphadef]). [(a)] ; Shea and Sons have shown that if ...

Let $f$ be a function meromorphic in $\mathbb{D}$ of finite order $\rho$. Shea and Sons have shown that \[\sum_{a\neq\infty}\delta(a,f)\leq\delta(0,f'...

Let $f$ be a function meromorphic in $\mathbb{D}$, for which $\alpha=+\infty$ in ([source label: alphadef]). Then it is known that \[\sum_{a\in\mathbb...

Let $f$ be meromorphic in $\mathbb{C}$, and suppose that the function \[F(z)=f^{(k)}(z)+\sum^{k-2}_{j=0}a_j(z)f^{(j)}(z)\] is non-constant, where $k\g...

Let $f$ be a meromorphic function of lower order $\lambda$. Let \[m_0(r,f)=\inf\{|f(z)|:|z|=r\}\] and \[M(r,f)=\sup\{|f(z)|:|z|=r\}\] and suppose that...

Produce a general method for constructing an entire function of finite order, and in fact, minimal growth, which tends to different asymptotic values ...

If $\phi(z)$ is an entire function growing slowly compared with the function $f(z)$, we can consider $\phi(z)$ to be an asymptotic function of $f(z)$,...

Suppose that $f(z)$ is a meromorphic function in the plane, and that for some $\theta$, $0\leq\theta<2\pi$, $f(z)$ assumes every value infinitely ofte...

What can we say about the set $E$ of values $a$ which an entire function $f(z)$ assumes infinitely often in every angle? Simple examples show that $E$...

Let $f(z)$ be an entire function. Then Boas (unpublished) proved that there exists a path $\Gamma_\infty$ such that, for every $n$, $$ \left|\frac{f(z...

If $f(z)$ of finite order, can anything be asserted about the length of $\Gamma_\infty$, which is the path on which $f(z)$ tends to $\infty$, or the p...

Does ([source label: 2.1]) remain true if the number $n(r)$ of poles of $f(z)$ in $|z|<r$ satisfies $n(r)=O(r^k)$, where $k<\frac{1}{2}<\lambda$, and ...

We ask the analogues of Problems 2.6, 2.7 and 2.8 if, in addition, $f(z)$ has another finite Picard value, e.g. $f(z)\neq0$. In this case, if $\infty$...

If $f(z)=\sum a_nz^{\lambda_n}$ is an entire function, and $\sum(1/\lambda_n)$ converges, is it true that: [(a)] ; $f(z)$ has no finite asymptotic val...

If the entire function $f(z)$ has finite order $\rho$, and the maximal density of non-zero coefficients is $\Delta$, is it true that if $\rho\Delta<\f...

If $f(z)=\sum a_n z^{\lambda_n}$ is an entire function, and $\lambda_n/n\to\infty$, is it true that $f(z)$ has [(a)] ; no Picard value, ; no Borel exc...

[(a)] ; Let $f(z)=\sum a_n z^n$ be entire and $m(r)=\max_n |a_n|r^n$. If $C>\frac{1}{2}$ then does there exist an entire $f$ with \[m(r)/M(r,f)\to C ?...

(Blumenthal's conjecture) Let $w=f_1(z), f_2(z)$ be entire functions. Is it true that if \[M(r,f_1)=M(r,f_2),\hspace{1cm}0<r<\infty,\] then $f_1(z), f...

Let $\nu(r)$ be the number of points on $|z|=r$, such that \mbox{$|f(z)|=M(r,f)$}. Can we have [(a)] ; $\limsup_{r\to\infty}\nu(r)=\infty$\,? ; $\limi...

If $f(z)$ is a non-constant entire function and \[b(r)=\left(r\frac{d}{dr}\right)^2\log M(r,f),\] then $$ \limsup_{r\to\infty} b(r)\geq A $$ where $A$...

Consider the function $b(r)$ of Problem 2.17. Since $\log M(r,f)$ is an analytic function of $r$, except for isolated points, $b(r)$ exists except at ...

If $f(z)$ is an entire function of exponential type, i.e. satisfying \mbox{$|f(z)|\leq Me^{K|z|}$} for some constants $M$, $K$, and if, further, $|f(x...

If $f(z)$ is an entire function, the iterates $f_n(z), n=1,2,\ldots$ are defined inductively by \[f_{n+1}(z)=f(f_n(z)),\hspace{1cm}f_1(z)=f(z).\] A po...

If, in the terminology of Problem 2.20, $z_0$ is a fixed point of exact order $n$ for $f(z)$, the fixed point is called repelling if $|{f_n}'(z_0)|>1$...

With the terminology of Problem 2.20, denote by $\mathcal{F}(f)$ the set of points where the sequence $\{f_n(z)\}$ is not normal. Fatou asks if there ...

Baker has proved that if $f(z)$ is a transcendental entire function, then $\mathcal{F}(f)$ is not restricted to a straight line in the plane. This imp...

Can an entire function have all its zeros and ones on two distinct straight lines, having infinitely many on each line? Edrei has proved (unpublished)...

If $f, g$ are linearly independent entire functions of order $\rho$, which is not a positive multiple of $\frac{1}{2}$, can $fg'-gf'$ have order less ...

What is the least integer $k=k(N)$, such that every entire function $f(z)$ can be written as \[f(z)=\sum^k_{\nu=1}[f_\nu(z)]^N,\] where $f(z)$ and $f_...

Let $\phi_1, \ldots, \phi_n$ denote entire functions of the form $$ \phi(z)=\sum e^{f_\nu(z)}/\sum e^{g_\nu(z)} $$ where $f_\nu(z), g_\nu(z)$ are enti...

A meromorphic function $f(z)$ in the plane, is said to be of bounded value distribution (b.v.d.) if, for every positive $r$, there exists a fixed cons...

Is it possible to give an analogous characterisation of the solutions of ([source label: 2.5]) in the case where the $f_\nu(z)$ are polynomials? (P. T...

Let $S_k, k=1, 2, \ldots$ be sets which have no finite limit points. Does there exist a sequence $n_k$ and an entire function $f(z)$, so that whenever...

Let $A, B$ be two countable dense sets in the plane. Does there exist an entire function $f(z)$, so that $f(z)\in B$, if and only if $z\in A$? If the ...

Let $f(z)=\sum^\infty_{n=0} a_nz^n$ be a transcendental entire function where $a_n\geq0$ for $n\geq0$, and set \[p_n(z)=\frac{a_nz^n}{f(z)}.\] Then \[...

Is it possible to obtain the exact value of $C_\infty$, or the asymptotic behaviour of $\frac{C_\lambda}{\log\lambda}$ as $\lambda\to\infty$? The ques...

Is it possible to say something more precise about $C(\lambda)$ when $\lambda$ is just greater than $1$? In particular, is it true that $C(\lambda)=-1...

If $\Gamma$ is a continuum that recedes to $\infty$, it is known (see Hayman ) that as $z\to\infty$ on $\Gamma$, \[\limsup_{r\to\infty}\frac{\log |f(z...

Suppose that $0<\rho<\alpha\leq1$, where $\rho$ is the order of an entire function $f$. Let $E_\alpha$ be the set of $r$ for which $\log m_0(r,f)>\cos...

Let $r_n$ be a sequence of P\'olya peaks (as defined by Edrei ) of order $\rho$. Then Edrei showed that there exists $K=K(\alpha,\rho)$ such that $\lo...

It was shown by Kjellberg that if $0<\alpha<1$ and \[\log m_0(r,f)<\cos (\phi\alpha)\log M(r,f)+O(1),\hspace{1cm}\text{ as }r\to\infty,\] then \[\lim_...

We can also compare $m_0(r,f)$ with the characteristic $T(r)$. We have \[\limsup_{r\to\infty}\frac{\log m_0(r,f)}{T(r)}\geq D(\lambda)\] and ask for t...

Let $f(z)$ be a non-constant entire function, and assume that for some constant $c$ the plane measure of the set $E(c)$ where $|f(z)|>c$ is finite. Wh...

Let $f(z)$ be an entire function (of sufficiently high order) with $l$, $l\geq2$ different asymptotic values $a_k$, $k=1,\ldots, l$. Suppose that $\ga...

Let $f(z)$ be a transcendental entire function which permutes the integers, i.e. gives an injective mapping of the integers onto themselves. Is it tru...

For $f(z)$ entire of order $\rho$, and non-constant, let $\nu(r)$ be the number of points on $|z|=r$ where $|f(z)|=1$. Is it true that \[\limsup_{r\to...

Let $J_0(z)$ be the Bessel function of order zero. Is it true that the equation $J_0(z)=1$ has at most one solution on each ray from the origin? An af...

Let $\{f_\alpha(z)\}$ be a family of entire functions, and assume that for every $z_0$, there are only denumerably many distinct values of $f_\alpha(z...