Mathematics Problem Archive

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 Edrei has proved (unpublished)...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...

Let $E_\rho$ be the linear space of entire functions $f$ such that \mbox{$|f(z)|\leq B\exp(A|z|^\rho)$} for some positive $A$ and $B$. Let $K_\rho$ be...

If $A, B$ are countable dense subsets of $\mathbb{R}$, $\mathbb{C}$ respectively, does there necessarily exist a transcendental entire function that m...

If $f(z)$ is a transcendental entire function, we define \[M=\{z:|f(z)|=M(|z|,f)\}.\] Tyler has shown that $M$ can have isolated points, and that, giv...

Characterise those entire functions having at least one continuous maximum modulus path going from $0$ to $\infty$. (W. Al-Katifi)...

Suppose that an entire function $f$ has exactly one curve $\Gamma$ of maximum modulus (that is, $\Gamma$ is connected, joins $0$ to $\infty$, and $f$ ...

What is the best function $g(\sigma)$, $\sigma\geq 0$ such that, for a non-constant entire function $f(z)$ with maximum and minimum modulus $M(r,f)$ a...

For entire or, more generally, meromorphic functions $f$ and $g$, let `$f\leq g$' mean that, for any sequence $\{z_n\}^\infty_1$ for which $|f(z_n)|\t...

Let $E$ be a closed set in $\mathbb{C}$, with the following properties: $(1)$ there exists a transcendental entire function $f(z)$ that is bounded on ...

Let $f_i(z)$, $i=1, 2, 3$ be non-constant entire functions of one complex variable, and \[V=\{z:z=(z_1,z_2,z_3)\in\mathbb{C}^3,f_1(z_1)+f_2(z_2)+f_3(z...

Prove or disprove the conjecture that an entire function $f$ of $n$ complex variables is an $L$-atom (where this is defined in a way analogous to the ...

If $f$ is an entire function such that $\log M(r,f)=O(\log r)^2$ as $r\to\infty$, then Hayman has shown that $\log |f(re^{i\theta})|\sim\log M(r,f)$, ...

Suppose that $f$ is entire with a non-zero Picard exceptional value $\alpha$. Then $f$ has $\alpha$ as an asymptotic value. It can be shown that $f\to...

(A width conjecture) Given a power series $\sum^\infty_{k=0}a_kz^k$, suppose that there is a non-negative $\rho$ such that all of the partial sums $S_...

Let $\sum^\infty_{k=0}a_kz^k$ be a non-vanishing entire function, and let \mbox{$S_n(z)=\sum^n_{k=0}a_kz^k$}. Given $\varepsilon>0$, must there exist ...

Let $\Gamma$ be a rectifiable curve. Suppose $f$ is a continuous function on the plane satisfying \[\int_{\sigma(\Gamma)}f(z)dz=0\hspace{1cm}\text{ fo...

Let $f$ denote a rational or entire function of a complex variable, and $f^n, n=1, 2, \ldots$, the $n$-th iterate of $f$, so that $f^1=f, f^{n+1}=f\ci...

Let $f$ be a rational function and $C$ be as in Problem 2.62. We say that $g$ is a limit function for $f$ if $g$ is defined in some component $G$ of $...

Since the knowledge of the zeros of an entire function $f$ leaves an unknown factor, $e^h$ say, in the Hadamard product for $f$, one can ask if $f$ is...

Given a countable number of entire functions, one can find an entire function growing faster than any of these. Without making any assumption about th...

Let $f$ be an entire function, and let $D$ be a component of the set in $\mathbb{C}$ where the family of iterates $\{f_n\}$ is normal. Can this family...

Let $f$ be an entire function satisfying the condition \[\log M(r,f)\leq(1+o(1))r^\rho,\hspace{1cm}\text{ as }r\to\infty.\] Suppose that there exists ...