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I want to determine the subset of $m$ members ($m < n/2$) of the set $e^{i 2\pi k/n}, \ \ k=0,\dots, n-1$, so that the absolute value of its sum is maximal.

Assume that $F(e^{it})=e^{if(t)}$ is a diffeomorphism of the unit circle onto itself and let $A=\left|\int_0^{2\pi}(1-F^2)\,dt\right|$ and $B=\left|\int_0^{2\pi} F^2(1-F^2) \,dt\right|$. It seems that …

Assume that $f$ is an analytic function defined on the unit disk and continuous up to the boundary. How many zeros can $f$ have in the unit disk?

Assume that $$\left<f,g\right>_R=\Re \int_U f(z) \overline{g(z)} \, dx \, dy, f,g \in L^2(U),$$ where $U$ is the unit disk and assume that $A: L^2(U)\to L^2(U)$ is a real-linear operator. Assume also …

Is there a simple construction of a confomral mapping of the half-plane onto a "circular trianagle", i.e. a domain whose sides are the arcs of three circles.

Assume that $g(re^{it}),$ and $h(re^{it})$ are smooth positive functions defined on the annulus $A=A(R,1)=\{z: R<|z|<1\}$. Assume also that $\int_0^{2\pi}h(re^{it})dt\ge 2\pi c$ for every $r\in(R,1)$ …

I wounder what is the value of the following integral $$\displaystyle A(z) = \lim_{\delta \to 0}\int_{|w|<1, |w-z|\ge \delta} \frac{|w|^ndu dv}{w^n(w-z)^2 },$$ where $n$ is an integer and $|z|<1$ and …

Is there is a holomorphic function $g:\mathbb{C}\to \mathbb{C}$ so that $$\frac{|g'(z)|}{1+|g(z)|^2}=\frac{c}{1+|z|^2}$$ for some $c>1$ and all $z\in \mathbb{C}$?