The ${{Q}_{p}}$ spaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to the $Q_{p}^{\#}$ classes of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) of $Q_{p}^{\#}$ classes on hyperbolic Riemann surfaces. The same property for ${{Q}_{p}}$ spaces was also established systematically and precisely in earlier work by the authors of this paper.

We introduce and study some new function spaces on Riemann surfaces. For certain parameter values these spaces coincide with the classical Dirichlet space, $\text{BMOA}$, or the recently defined ${{\text{Q}}_{p}}$ space. We establish inclusion relations that generalize earlier known inclusions between the above-mentioned spaces.