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Disclaimer: I learned the trickery below from N.K.Nikolskii, who was giving us a special topics course in complex analysis when I was a fourth year undergraduate student. I have no idea whether it can …

That is a rather tough puzzle (took me two full days) with a rather short solution. The first step is the differential equation Fred already mentioned: $$ (1-z^2)H'(z)-(1+z)H(z)+2zH(z^2)=0\,. $$ Now d …

OK, here is (I hope) a proof. I prefer to consider the integral $\int_{\mathbb D}\left|\frac 1{z-a}+\frac 1{z-\bar a}\right|\,d\mu(z)$ where $a=e^{i\theta}, \theta\in(0,\pi/2)$. This is just a one-par …

Unfortunately, no, as requested: Take any sequence $\delta_j\in(0,1)$ decaying to $0$, choose small $\mu_j>0$ such that $\prod_j \delta_j^{\mu_j}=e^{-1}$ and put $f_n(z)=e^n\prod_j B_{\delta_j}(z)^{[\ …

Also, the classical Fabry gap theorem tells you that the unit circumference is the natural boundary. Meanwhile, all "elementary" functions can be analytically continued along almost every path on the …

Darij, such stuff is usually Gauss-Lucas in disguise and this case is no exception, though one needs to use once the version for polar derivative $D_1f(z)=(1-z)f'(z)+nf(z)$ of a polynomial $f$ of degr …

Strange that I missed this one. On the other hand I haven't visited MO often lately. Anyway, here is the solution (It is always tempting to renew our old discussion on what problems a minimally intell …

It looks like the inequality is true (at least for the function with the cotangent). The proof (I hope it is correct but, please, check the details: I'm not in my top shape today) is as follows. Claim …

Fix $\xi$ and assume $k>n$ (the other case is similar). Integrate over $x$. You'll get $\int_{\mathbb R+i\xi}H_k(z)P_n(z-i\xi,\xi)e^{-z^2/2-\xi^2}\,dz$ where $P_n$ is some polynomial of 2 variables of …

OK, here are full details. Hope that I don't say any nonsense. The first observation is that $x\mapsto \frac{x+1}{x-1}$ maps the unit disk $|x|\le 1$ to the left half-plane $z\le 0$. You can see it by …

Yes, it is dense. Indeed, if $g$ is an $L^2$ function supported on $[0,1]$ such that $g$ is orthogonal to every $f+\mathscr Hf$ with $f$ compactly supported on $(0,+\infty)$, then $g-\mathscr Hg=0$ on …

In general it is impossible. We can take the square root, map the circle conformally to the half-plane, and arrive at the following problem: given any nonnegative $f\in L^2(\mu)$ with finite logarithm …

$Hello$, Tomasz! (for some reason the MO prohibits saying "Hi" or "Hello" in the normal text mode). Nice to see you back. Apparently you are still asking the same question whether a function $H$ close …

The set can be infinite (but only countable). For an example choose any $t_j$ linearly independent over $\mathbb Q$ and let $V$ be the set of all $v$ such that $vt_j\mod 1 \notin (\frac 12-a_j,\frac 1 …

I did not know at all... I am very sorry but can you tell me any references Ah-oh. It is hard to find a decent reference now because nobody cares about writing down such trivialities any more: they a …