We characterize compact composition operators acting on weighted Bergman-Orlicz spaces ${}_{\alpha }^{\psi }=f\in H\left(\right):{\int }_{}\psi \left(\right|f\left(z\right)\left|\right)d{A}_{\alpha }\left(z\right)<\infty$, where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition $li{m}_{t\to \infty }\psi \left(t\right)/t=\infty$ and the Δ₂-condition. In fact, we prove that $C_{\phi }$ is compact on ${}_{\alpha }^{\psi }$ if and only if it is compact on the weighted Bergman space ${²}_{\alpha }$.

We first show that the Gaussian integral means of $f:ℂ\to ℂ$ (with respect to the area measure ${\mathrm{e}}^{-{\alpha \left|z\right|}^2}\mathrm{d}A\left(z\right)$) is a convex function of $r$ on $(0,\infty )$ when $\alpha \le 0$. We then prove that the weighted integral means $A_{\alpha ,\beta }(f,r)$ and $L_{\alpha ,\beta }(f,r)$ of the mixed area and the mixed length of $f\left(r𝔻\right)$ and $\partial f\left(r𝔻\right)$, respectively, also have the property of convexity in the case of $\alpha \le 0$. Finally, we show with examples that the range $\alpha \le 0$ is the best possible.