For an increasing absolutely continuous function $\varphi:(0,1)\to(0,+\infty)$ we define the spaces $H(p,q,\varphi)$, $p>0$, and $h(p,q,\varphi)$, $p\geq 1$, (of analytic and harmonic functions $f$ on the unit disc, respectively) by the requirement that the function $r\to\varphi(1-r)M_p(r,f)$, $0