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Let $f(w)=\frac13+\frac12 w+\frac16 w^3$. If $\vert f(w)\vert\leq1$ or simply $\vert f(w)\vert=1$, show that $\vert \frac{w}2 f(\frac{w}2)\vert\leq1$. Here, $w$ is a complex number. What happens if w …

Let $\omega_n=e^{\frac{\pi i}{2n+1}}$. I've an experimental encounter with certain relation involving roots of unity. Question. Is this true? If yes, any proof? For $p\geq0$ an integer, we have th …

Let $D\subset\mathbb{C}$ be the open unit disk. Suppose $f,g,F,G:D\rightarrow\mathbb{C}$ are analytic functions linked by $$\vert f(z)\vert^2+\vert g(z)\vert^2=\vert F(z)\vert^2+\vert G(z)\vert^2; \q …

Let $z_1,z_2,\dots,z_n\in\Bbb{C}$ be distinct and $w_1,w_2,\dots,w_n\in\Bbb{C}$ be arbitrary. Suppose $f, g$ are two polynomials of degree less than $n$ such that $$f(z_j)=w_j,\qquad g(z_j)=\bar{w}_j …

Question. If $f(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$, is it true that $$\max\{\,\vert f(x)\vert: \, -1\leq x\leq 1\}\geq 2^{1-n} \,\, ?$$ Context. This came up while working …

The following polynomial (after harmless factors dropped) is found in the paper entitled Mock theta functions and quantum modular forms by Folsom-Ono-Rhoades (see Theorem 1.1) $$Q_k(z)=\sum_{n=0}^{k-1 …

Let $f(x)$ be a polynomial in the ring $\mathbb{R}[x]$, the roots are all real and $f(0)=1$. Write the Taylor series of $1/f(x)$ around the origin as $$\frac1{f(x)}=\sum_{k=0}^{\infty}a_kx^k,$$ and de …

Recall $\text{sinc}(x)=\frac{\sin x}x$. It's a familiar exercise that $\int_0^{\infty}\text{sinc}(x)\,dx=\frac{\pi}2$. But, at present, I wish to ask about the following claim on a "sinc-ing" produc …

Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It a …

Denote $\pmb{a}=(a_1,\dots,a_d)\in\mathbb{R}^d$ and consider the set $$\mathcal{E}_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a_dx^{d-1}+\cdots+a_2x+a_1=0$ lies in $\vert\xi\vert<1$}\}. …

Let $p(x)$ be a polynomial of degree $n>2$, with roots $x_1,x_2,\dots,x_n$ (including multiplicities). Let $m$ be a positive even integer. Define the following mapping $$V_m(p)=\sum_{1\leq i<j\leq n}( …

Define the sequence $$a_s=(-1)^{\binom{s-1}2}\left(\frac{\pi}2\right)^s\frac1{2\cdot s!}\begin{cases} s\,E_{s-1}, \qquad \text{if $s$ is odd} \\ 2^{2s}B_s, \qquad \,\,\text{if $s$ is even};\end{cases} …

While working with a generating function for the Catalan numbers, I came across the integral representation $$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)} …

Let $(a_n)$ be a (double-sided) sequence of complex numbers satisfying $$\sum_{\mathbb{Z}}\vert n\vert\,\vert a_n\vert^2<\infty, \tag1$$ $$\sum_{\mathbb{Z}}a_n\bar{a}_{n+k}=\delta_0(k), \qquad \forall …

Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$. Question. Is it true that $D^nf(z)=0$ has only real root …