Mathematics Problem Archive

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 ...

Hayman has shown that $$ \liminf_{r\to\infty}\frac{T(r,f)}{T(r,f')}\leq1 $$ for transcendental entire functions $f$ of lower order zero. Toppila has s...

Let $H$ be an entire function, let $f_1, f_2$ be linearly independent solutions of the differential equation $w'' + Hw = 0$, and let $E = f_1 f_2$. Cl...

It is shown by Hellerstein and Rossi , and Gundersen that if $f_1$ and $f_2$ are two linearly independent solutions to the differential equation $w'' ...

Let $\{f_1,\ldots,f_n\}$ be a fundamental system for the differential equation $$ L_n(w)\equiv w^{(n)}+a_{n-1}(z)w^{(n-1)}+\ldots+a_0(z)=0, $$ where $...

Let $F(z, a, b)$ be an entire function of three complex variables, and suppose that $F$ is not of the form $$ F(z,a,b) = G(z,H(a,b)) $$ for any entire...

Suppose that $f(z) = 1 + a_1z + a_2 z^2 +\ldots \in U_{2p}$. If $p = 0$ (so that $f\in U_0)$ and if $f$ is not a polynomial, it is well-known that $f$...

Suppose that $f$ is entire of proximate order $\rho(r)$, and that $f$ has a representation as a Dirichlet series \[f( z ) = \sum^\infty_{n=1}a_ne^{\la...

Let $\Omega$ be a component of the normal set of an entire function (under iteration). Is $\dim(\partial\Omega) > 1$? Or is $\partial\Omega$ a circle/...