Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk

We prove that, for every $\alpha >-1$, the pull-back measure $\phi ({\cal A}\alpha)$ofthemeasure$d{\cal A}\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha , d{\cal A} (z)$,where${\cal A}$ is the normalized area measure on the unit disk $\text{D}$, by every analytic self-map $\phi \colon \D \to \D$isnotonlyan$(\alpha + 2)$-Carleson measure, but that the measure of the Carleson windows of size $\text{eps}h$ is controlled by ${\text{eps}}^{\alpha +2}$ times the measure of the corresponding window of size $h$. This means that the property of being an $(\alpha +2)$-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces.