Carleson measures and some classes of meromorphic functions

For $|a|<1$ let $\varphi _{a}$ be the Möbius transformation defined by $\varphi _{a}(z)=\frac{a-z}{1-\bar az}$, and let $g(z,a)=\log |\frac{1-\bar az}{z-a}|$ be the Green's function of the unit disk $\mathcal{D}$. We construct an analytic function $f$ belonging to $M_{p}^{\#}=\{f:\text{$f$ meromorphic in $\mathcal{D}$ and\,} \sup _{a\in \mathcal{D}} \iint _{\mathcal{D}}(f^{\#}(z))^{2}(1-|\varphi _{a}(z)|^{2})^{p}\,dA(z)<\infty \}$ for all $p$, $0<p<\infty$, but not belonging to $Q_{p}^{\#}=\{f:f$ meromorphic in $\mathcal{D}$ and $\sup _{a\in \mathcal{D}}\iint _{\mathcal{D}}(f^{\#}(z))^{2}(g(z,a))^{p}\,dA(z)<\infty \}$ for any $p$, $0<p<\infty$. This gives a clear difference as compared to the analytic case where the corresponding function spaces ($M_{p}$ and $Q_{p}$) are same.