A radially weighted Besov space H is a space of holomorphic functions on the unit ball \(\mathbb {B}_d \subseteq \mathbb {C}^d\) whose N-th radial derivative is square integrable with respect to a given admissible radial measure. We write Mult(H) for its multiplier algebra. The cyclic vectors in H are those functions f whose multiplier multiples are dense in H. We call a multiplier \(f \in Mult(H)\) weak* sequentially cyclic if its multiplier multiples are weak* sequentially dense in Mult(H). It is immediate that every weak* sequentially cyclic multiplier is cyclic, and it turns out that the two notions coincide whenever H has the complete Pick property. However, in more general radially weighted Besov spaces there may be multipliers that are cyclic, but not weak* sequentially cyclic. For bounded holomorphic functions f with no zeros in \(\mathbb {B}_d\) , we obtain a condition on \(\log f\) that implies the cyclicity of f in H and yields invertibility properties for 1/f within an associated Smirnov-type class. This condition is formulated in terms of weak* sequentially cyclic multipliers and can often be verified using a comparison principle: if \(f, g \in Mult(H)\) satisfy \(|f| \le |g|\) and if f is weak* sequentially cyclic, then g is also weak* sequentially cyclic. These results provide new insights into cyclicity phenomena in radially weighted Besov spaces in settings where H fails to be a complete Pick space.
