Mathematics Problem Archive

Is there a function $f(k)$ of the positive integer $k$, so that the square of every polynomial having at least $f(k)$ terms has a least $k$ terms? Erd...

Let $P(z)$ be a polynomial whose zeros $z_1, z_2, \ldots, z_n$ lie in $|z|\leq1$. Is it true that $P'(z)$ always has a zero in $|z-z_1|\leq1$? (Bl. Se...

If $H_\nu(z)$ is the $\nu$-th Hermite polynomial, so that \[H_\nu(z)e^{-z^2}=(-1)^\nu\big(\frac{d}{dz}\big)^\nu e^{-z^2},\] is it true that the equati...

Let $f(z)=z^n+a_1z^{n-1}+\ldots+a_n$ be a polynomial of degree $n$. Cartan proved that the set $|f(z)|\leq1$, which we call $E^{(n)}_f$ can always be ...

Assume that $E^{(n)}_f$ is connected. Is it true that $$ \max_{z\in E^{(n)}_f}|f'(z)|\leq\frac{1}{2}n^2\,? $$ Pommerenke proved this with $\frac{1}{2}...

Is it true that to every positive $c$, there exists an $A(c)$ independent of $n$, such that $E_f^{(n)}$ can have at most $A(c)$ components of diameter...

Is it true that the length of the curve $|f_n(z)|=1$ is maximal for $f_n(z)=z^n-1$? (P. Erd\"os)...

If $|z_i|\leq1$, estimate from below, the area of $E^{(n)}_f$. Erd\"os, Herzog and Piranian prove that, given positive $\varepsilon$, the area of $E^{...

It is known that there exists a polynomial $P(z)$ \[P(z)=\sum^n_{k=1}\varepsilon_k z^k, \hspace{1cm}\varepsilon_k=\mp1\] for which $$ \max_{|z|=1}|P(z...

Does there exist a polynomial of the type in Problem 4.13, for which $$ \min_{|z|=1}|P(z)|>C_2\sqrt{n} $$ for every $n$? More generally, does there ex...

If again $\varepsilon_k=\mp1$, is it true that, for large $n$, all but $o(2^n)$ polynomials $P(z)=\sum^n_{k=1}\varepsilon_k z^k$ have just $n/2+o(n)$ ...

Is it true that for all but $o(2^n)$ polynomials $P(z)$ \[\min_{|z|=1}|P(z)|<1,\] or, if not, what is the corresponding correct result?...

If $f$ is any polynomial or rational function of degree $N$, find the least upper bound $\phi(N)$ of \[\frac{1}{r}\int^r_0dt\int^\pi_{-\pi}\frac{|f'(r...

Littlewood conjectured that if $n_1, n_2, \ldots, n_k$ are distinct positive integers then $$ \int^{2\pi}_0\Big|\sum^k_{i=1}\cos (n_1 x)\Big|\,dx>c\lo...

If $a_k=\mp1, k=0,\ldots,n$ and \[b_k=a_na_{n-k}+a_{n-1}a_{n-k-1}+\ldots+a_ka_0,\] is it true that \[\sum^n_1|b_k|^2>An^2,\] where $A$ is an absolute ...

Using the notation of Problem 4.7, if $|z_i|\leq1$, Clunie and Netanyahu (personal communication) showed that a path exists joining the origin to $| z...

Some of the Problems 4.7 to 4.12 extend naturally to the space of higher dimensions. Let $x_i$ be a set of $n$ points in $\mathbb{R}^m$ and let $E^{(m...

Let \[P(z)=\sum^n_0a_kz^k\] be a self-inversive polynomial, i.e. if $\zeta$ is a zero of $P(\zeta)$ with multiplicity $m$, then $1/\zeta$ is also a ze...

Determine \[\inf\int^\pi_{-\pi}\big|1-e^{i\theta}\big|^{2\lambda}\big|P(e^{i\theta})\big|^2\,d\theta,\hspace{1cm}\lambda>0,\] where $P(z)$ ranges over...

Let $P_n$ denote the class of polynomials $p(z)$, $p(0) = 1$, of degree at most $n$ and of positive real part in $\mathbb{D}$. Find \[\max_{p\in P_n}\...

Let $p(x)$ be a real polynomial of degree $n$ in the real variable $x$ such that $p(x) = 0$ has $n$ distinct (real) rational roots. Does there necessa...

Suppose that $P$ is a non-linear polynomial with real coefficients. Show that $P^2(z)+P'(z)$ has non-real zeros. We conjecture that the lower bound fo...

Yang claims to prove the following: let $P(z), Q(z)$ be monic polynomials such that $(i)$ $P(z)=0 \iff Q(z)=0$, and $(ii)$ $P'(z)=0 \iff Q'(z)=0$. The...

Let $\mathcal{P}$ denote the set of all polynomials of the form \[p(z)=\prod^n_{\nu=1}(z-\zeta_\nu),\] where $n\geq2$ and $|\zeta_\nu|\leq1$, $\nu=1, ...

Erd\"os and Newman conjectured that if $$ f(z)=\sum^n_{k=0}a_kz^k,\hspace{1cm} |a_k|=1, \hspace{1cm}0\leq k\leq n, $$ then there is an absolute consta...

Is it true that ([source label: 5.1]) implies \[I_1(r,f)=O(1-r)^{-1-\varepsilon}\] and \[|a_n|=O(n^{1+\varepsilon})\,?\]...

Is it true that ([source label: 5.3]) implies that $$ I_1(r,f)=O(1-r)^{-1} $$ and $$ |a_n|=O(n)? $$...

An even stronger hypothesis than ([source label: 5.3]) is that $f(z)$ is weakly univalent (see Hayman ) i.e. for every $r$ with $0<r<\infty$, either $...

If the sequence $w_n$ satisfies $$ \arg w_n=O\big(|w_n|^{\frac{1}{2}}\big) $$ and $$ |w_{n+1}- w_n|=O(|w_n|^{\frac{1}{2}}) $$ then it is known (see Ha...

If $f(z)=u+iv$ assumes only values in the right half-plane, then subordination shows that $$ a_n=O(1). $$ It is of interest to ask what other hypothes...

It is known that there exist functions which fail to take any of the values $2\pi ik$, $-\infty<k<+\infty$ and which do not satisfy ([source label: 5....

If $c_k$ is a sequence of positive numbers such that \[\sum c_k=S<+\infty,\] and $n_k$ is an arbitrary sequence of positive integers, then \[f(z)=\sum...

Suppose that $f(z)=z+a_2z^2+\ldots$ is analytic in $\mathbb{D}$. Then $f(z)$ maps some sub-domain of $\mathbb{D}$ univalently into a disc of radius at...

With the hypotheses of Problem 5.8, it follows that $f(z)$ assumes all values in some disc of radius $L$, $L\geq B$. What is the value of $L$? The bes...

If, in addition, $f(z)$ is univalent in $\mathbb{D}$, the conclusions of Problem 5.8 and Problem 5.9 follow with a constant $S$, $S\geq L$, known as t...

$f(z)$ meromorphic in $\mathbb{D}$, $f(z)\neq0, f^{(l)}(z)\neq1$, where $l\geq1$....

$f(z)$ meromorphic in $\mathbb{D}$, $f'(z)f(z)^n\neq1$, for $n\geq3$....

$f'-f^n\neq a$, where $a$ is some complex number, and $n\geq 5$ if $f$ is meromorphic, $n\geq3$ if $f$ is entire. The corresponding results for functi...

Is it possible to remove the restriction that $D^*$ is simply connected in $(a)$ and $(b)$ above? It might be possible to start with the case when $D$...

Do corresponding results to Problem 5.15(a) apply to the means \[I_\lambda(r,f)=\Big\{\frac{1}{2\pi}\int^{2\pi}_0\big|f(re^{i\theta})\big|^\lambda \,d...

Let $D=D_0$ be a domain, $g(z,a_0)$ be the Green's function of $D$ with respect to a point $a_0$ on the positive real axis, and let $D_\lambda$ be the...

Let $f(z)=\lambda+a_1z+\ldots$ be analytic in $\mathbb{D}$, where $0<\lambda<1$. Find the best constant $B(\lambda)$ such that if \[F(r)=\lambda+|a_1|...

A function meromorphic in $\mathbb{D}$ which has no asymptotic value, assumes every value infinitely often in the disc. Every point of the circumferen...

Plessner proves after Privaloff that if $f$ is analytic in $\mathbb{D}$, almost all points $P$ of the boundary are of two kinds. Either [(a)] ; $f$ te...

Corresponding to each function $f$ analytic in $\mathbb{D}$, and each value $w$, with $(|w|<1)$, write \[f_w(z)=f\Big(\frac{z-w}{1-wz}\Big)=\sum^\inft...

Let $H^p$ be the space of functions $f(z)=\sum^\infty_{n=0}a_nz^n$ analytic in $\mathbb{D}$, and such that \[\int^{2\pi}_0\big|f(re^{i\theta})\big|^p\...

Describe similarly the coefficient multipliers from $S$ to $S$, where $S$ is the class of functions $\sum^\infty_{n=1} a_nz^n$ univalent in $\mathbb{D...

Is the intersection of two finitely generated ideals in $H^\infty$ finitely generated? (L. A. Rubel)...

Let $W^+$ be the Banach algebra of power series $f(z)=\sum^\infty_{n=0}a_nz^n$ absolutely convergent in $|z|\leq1$, with $\|f\|=\sum^\infty_{n=0}|a_n|...

(The corona conjecture) Let $D$ be an arbitrary domain in the plane that supports non-constant bounded analytic functions. Suppose that $f_1(z),\ldots...