Uniform limits of Green potentials in the unit disc (Q803330)

In a previous work [Proc. Am. Math. Soc. 93, 567-568 (1985; Zbl 0561.31003)] the author proved the following: Theorem A. If \(G_{\mu}\) is the Green potential of a measure \(\mu\) satisfying \(\int_{D}(1-| z|)d\mu (z)<\infty\) then \((1)\quad \liminf_{r\to 1^-}(1-r)M_{\infty}(G_{\mu},r)=0,\) where \(M_{\infty}(G_{\mu},r)=\sup_{| z| =r}G_{\mu}(z).\) This paper proves a variation of (1) where the weight function is \((1- r)^{\alpha}\), \(0\leq \alpha \leq 1\). Then it provides sufficient conditions that guarantee that the limit in (1) is zero, rather than just the limit inferior.