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.



Question about an inequality described by matrices
$$y\_i=\sum\_j a\_{ij} x\_j\geqslant \prod\_j x\_j^{a\_{ij} }$$
by Jensen inequality for logarithm. Now take the product over $i=1,2,\dots,n$.


