Let $A=(a_{ij})_{1 \le i, j \le n}$ be a matrix such that $\sum_\limits{i=1}^{n} a_{ij}=1$ for every $j$, and $\sum_\limits{j=1}^n a_{ij} = 1$ for every $i$, and $a_{ij} \ge 0$. Let
$$\begin{equation}
  \begin{pmatrix}
  y_1 \\
  \vdots \\
  y_n \\
  \end{pmatrix}
  =\mathbf{A}
  \begin{pmatrix}
  x_1 \\
  \vdots \\
  x_n
  \end{pmatrix}
  \end{equation}$$
with non-negative $y_i$ and $x_i$. Prove that  $y_1 \cdots y_n \ge x_1 \cdots x_n$.

It may somehow matter to convex function.