0
votes
1answer
19 views

An inequality concerning an harmonic function

Let $h$ be a positive harmonic function on $\Delta (0,\rho )=\lbrace z\in\mathbb{C} : |z|\leq \rho \rbrace$. I wish to show that $|\nabla h(z)|\leq \frac{2\rho}{\rho ^2-|z|^2}h(z)$. Since $h$ is ...
3
votes
2answers
72 views

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
0
votes
0answers
27 views

Prove that $\int_{{B(0,\epsilon)}\setminus \{z_1=0\}}\det\left(\text{Hessian}_u(z)\right)\mathrm{d}V=\infty$

I have a problem: For $u(z_1,z_2)=\left (-\log\left | z_1 \right | \right )^\alpha\cdot \left ( \left | z_2 \right |^2-1 \right )$, where $\alpha \in \left (0,1 \right )$. Prove that if $\alpha ...
2
votes
2answers
260 views

An inequality about the gradient of a harmonic function

Let $G$ a open and connected set. Consider a function $z=2R^{-\alpha}v-v^2$ with $R$ that will be chosen suitably small, where $v$ is a harmonic function in $G$, and satisfies $$|x|^\alpha\leqslant ...