(Convexity) Let $\phi$ be any psh function on the open subset of $\{1<|z_{i}|<\Lambda,i=1,\ldots p,|z_{k}|<1,k=p+1,\ldots n\}\subset(\mathbb{C}^{*})^{p}\times\mathbb{C}^{n-p}$. Then the function

$$\bar{\phi}(x_{1},\ldots x_{n})=\frac{1}{(2\pi)^{n}}\int_{D(1)^{n-p}}\prod_{p+1}^{n}\sqrt{-1}dz_{k}\wedge d\bar{z}_{k}\int_{T^{p}}\phi(e^{x_{1}+i\theta_{1}},\ldots e^{x_{p}+i\theta_{p}})d\theta_{1}\ldots d\theta_{p}$$

is convex.