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Questions tagged [poisson-integrals]

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Rado–Kneser–Choquet Theorem proof

I was reading the proof of the Rado–Kneser–Choquet Theorem. The statement is there in the image (taken from the book "Harmonic Mapping in the Plane, Duren page-$30$": In the proof, he shows that ...
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Poisson integral formula for boundary value problem

I have gotten stuck on a boundary value problem which I believe is to be solved using the Poisson Integral Formula. The problem is: $$\nabla^{2}\psi=0, \psi(x,0)=0, |x|>1 ; |\psi(x,y)|<|x|, |x|\...
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An estimate for the conjugate Poisson Integral on the unit disk.

Let $f \in L^1(\Bbb{T})$ and $Q_r(\theta)=\frac{2r\sin{\theta}}{1-2r \cos{\theta}+r^2},r \in[0,1]$ the conjugate Poisson kernels and $P_r(\theta)=\frac{1-r^2}{1-2r \cos{\theta}+r^2},r \in[0,1]$ and $\...