12 problems

Let $D$ be a planar regular region with finitely many boundary components that are analytic Jordan curves. Let $c_{\beta}(z)$ denote the logarithmic capacity and let $B(z)$ denote…

Ramaswamy's conjecture. The fail points occur either at the contact points of the largest inscribed circle or at points of minimal curvature.

Saint-Venant's conjecture. Fail points occur at the contact points of the largest inscribed circle.

Multiplicity-two conjecture.

Let $\Omega\subset\mathbb C$ be an $m$-connected domain without degenerate boundary components. For each $z\in\Omega$ and $j=0,\ldots,m-1$, let $f_{z,j}$ be the unique conformal ma…

Let $\Lambda_{\epsilon}$ be a bounded connected planar annulus with two boundary components, and let $\lambda_{1}(\Lambda_{\epsilon})$ be its first nontrivial Steklov eigenvalue, g…

Let $D$ be a planar regular region whose boundary consists of $n$ analytic Jordan curves. Let $\hat{R}(z,\bar{w})$ be its conjugate Hardy $H^{2}$ kernel and $B(z,\bar{w})$ its Berg…

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with piecewise smooth boundary $\partial\Omega$, and let $\lambda_i$ be the $i$-th Dirichlet eigenvalue. Let…

Zeta-invariant inequalities. Inequalities (3.1) hold for every real function $a\in C^\infty({\mathbb S})$.

Let $G$ be an open subset, with components whose harmonic measures are considered. A positive measure $\mu$ on $G$ is a Carleson measure when it satisfies the relevant Carleson emb…

Let $G$ be a domain conformally equivalent to a circular domain $W$, and let $\phi:G\to W$ be a conformal map. A positive measure on $G$ is a Carleson measure if it satisfies the C…

For $0<s<t$, let $G_s$ and $G_t$ be the planar domains … For points $z_1,z_2\in G_s$, write $k_{G_s}(z_1,z_2)$ and $k_{G_t}(z_1,z_2)$ for their quasihyperbolic distances in the res…