Moduli of holomorphic functions and logarithmically convex radial weights

Let $H(D)$ denote the space of holomorphic functions on the unit disk $D$. We characterize those radial weights $w$ on $D$, for which there exist functions $f,g\in H(D)$ such that the sum $\mathrm{\mid }f\mathrm{\mid }+\mathrm{\mid }g\mathrm{\mid }$ is equivalent to $w$. Also, we obtain similar results in several complex variables for circular, strictly convex domains with smooth boundary.