Bounded superposition operators between weighted Banach spaces of analytic functions

We characterize all entire functions that transform a weighted Banach spaces of analytic functions ${\mathcal{H}}_{{\mu }_1}^{\mathrm{\infty }}$ into another space of the same kind ${\mathcal{H}}_{{\mu }_2}^{\mathrm{\infty }}$ by superposition for very general weights ${\mu }_1$ and ${\mu }_2$.