Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain and $W_0^{1,2}(\Omega)$ be the usual Sobolev space. Let $\beta$, $0\leq\beta1$, $$\lambda_{p,\beta}(\Omega)=\inf_{u\in W_0^{1,2}(\Omega),\,u\not\equiv 0}{\|\nabla u\|_2^2}/{\|u\|_{p,\beta}^2},$$ where $\|\cdot\|_2$ denotes the standard $L^2$-norm in $\Omega$ and $\|u\|_{p,\beta}=({\int_{\Omega}|x|^{-\beta}|u|^pdx})^{1/p}$. Suppose that $\gamma$ satisfies $\f{\gamma}{4\pi}+\f{\beta}{2}=1$. Using a rearrangement argument, the author proves that $$\sup_{u\in W_0^{1,2}(\Omega), \|\nabla u\|_2\leq 1}\int_{\Omega} |x|^{-\beta}e^{\gamma u^2 \le(1+\alpha\|u\|_{p,\beta}^2\ri) }dx$$ is finite for any $\alpha$, $0\leq\alpha