I have a question in Demailly's book(Complex Analytic and Differential Geometry in Page 43):
Let $\theta \in C^{\infty}(R)$ be a nonnegative function with support in [-1,1] such that $\int_{\mathbb{R}}\theta(h)dh=1$ and $\int_{\mathbb{R}}h\theta(h)dh=0$ . For arbitrary $\eta=(\eta_1,\eta_2,...,\eta_p)\in \mathbb{R_+}^p $,the function \begin{equation} M_{\eta}(t_1,t_2,...t_p)=\int_{\mathbb{R}^p}max\{t_1+h_1,t_2+h_2,...t_p+h_p\}\prod_{1\leq i \leq p}\theta(h_i/\eta_i)/\eta_idh_1...dh_p \end{equation} is non decreasing in all variables, smooth and convex on $\mathbb{R}^p$.
I know that the function is non decreasing and smooth. However,I'm failed in proving the convexity of this function.I have tried to calculate the Hessian Matrix of it, but I'm not sure that the Hessian Matrix is semi-positive definite.