  Wang and Ye conjectured in [22]:
  Let $\Omega$ be a regular, bounded and convex domain in $\mathbb{R}^{2}$.
There exists a finite constant $C({\Omega})>0$ such that \[
\int_{\Omega}e^{\frac{4\pi u^{2}}{H_{d}(u)}}dxdy\le C(\Omega),\;\;\forall u\in
C^{\infty}_{0}(\Omega), \] where $H_{d}=\int_{\Omega}|\nabla
u|^{2}dxdy-\frac{1}{4}\int_{\Omega}\frac{u^{2}}{d(z,\partial\Omega)^{2}}dxdy$
and $d(z,\partial\Omega)=\min\limits_{z_{1}\in\partial\Omega}|z-z_{1}|$.}
  The main purpose of this paper is to confirm that this conjecture indeed
holds for any bounded and convex domain in $\mathbb{R}^{2}$ via the Riemann
mapping theorem (the smoothness of the boundary of the domain is thus
irrelevant).
  We also give a rearrangement-free argument for the following Trudinger-Moser
inequality on the hyperbolic space
$\mathbb{B}=\{z=x+iy:|z|=\sqrt{x^{2}+y^{2}}<1\}$: \[
\sup_{\|u\|_{\mathcal{H}}\leq 1} \int_{\mathbb{B}}(e^{4\pi u^{2}}-1-4\pi
u^{2})dV=\sup_{\|u\|_{\mathcal{H}}\leq 1}\int_{\mathbb{B}}\frac{(e^{4\pi
u^{2}}-1-4\pi u^{2})}{(1-|z|^{2})^{2}}dxdy< \infty, \] by using the method
employed earlier by Lam and the first author [9, 10], where $\mathcal{H}$
denotes the closure of $C^{\infty}_{0}(\mathbb{B})$ with respect to the norm
$$\|u\|_{\mathcal{H}}=\int_{\mathbb{B}}|\nabla
u|^{2}dxdy-\int_{\mathbb{B}}\frac{u^{2}}{(1-|z|^{2})^{2}}dxdy.$$ Using this
strengthened Trudinger-Moser inequality, we also give a simpler proof of the
Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].
