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.
$$y_i=\sum_j a_{ij} x_j\geqslant \prod_j x_j^{a_{ij} }$$ by Jensen inequality for logarithm. Now take the product over $i=1,2,\dots,n$.
This is a special case of the so called Schur's majorization inequality. Here are the details. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bx}{\boldsymbol{x}}$ $\newcommand{\by}{\boldsymbol{y}}$
Given $\bx\in\bR^n$ we denote by $\bar{\bx}$ the vector obtained from $\bx$ by rearranging its coordinates in decreasing order. We say that $\bx$ dominates $\by$ and we write this $\bx\succ\by$ if
$$\sum_{i=1}^k \bar{x}_i\geq \sum_{i=1}^k \bar{y}_i,\;\;\forall k=1,\dotsc, n-1, $$
$$\sum_{i=1}^n \bar{x}_i= \sum_{i=1}^n \bar{y}_i.$$
The symmetric group $S_n$ acts on $\bR^n$ by permuting the coordinates of a vector. For $\bx\in\bR^n$ we denote by $S_n\cdot\bx$ the orbit of $\bx$ with respect to this action, i.e., the set of vectors that can be obtained from $\bx$ by permuting its coordinates. $\DeclareMathOperator{\conv}{conv}$
For a set $S\subset \bR^n$ we denote by $\conv(S)$ its convex hull.
The next nontrivial theorem characterizes the dominance relation. For a nice presentation of this theorem and Schur's majorization inequality I refer to Chapter 13 of
J.Michael Steele: The Cauchy-Schwarz Master Class, Cambridge University Press, 2004.
Theorem. Let $\bx,\by\in\bR^n$. The following statements are equivalent.
Fix an interval $(a,b)\subset \bR$. A symmetric $C^1$-function $f:(a,b)^n\to\bR$ is called Schur convex if $\newcommand{\pa}{\partial}$
$$ (x_j-x_k)\left( \frac{\pa f}{\pa x_j}(\bx)-\frac{\pa f}{\pa x_k}(\bx)\right)\geq 0,$$
for any $\bx\in (a,b)^n$ and any $j,k=1,\dotsc, n$.
The Schur majorization inequality states that
$$ \bx\succ \by \implies f(\bx)\geq f(\by), $$
for any Schur convex function $f:(a,b)^n\to\bR$ and any $\bx,\by\in(a,b)^n$.
The function
$$ p:[0,\infty)^n\to\bR,\;\;p(\bx)=-x_1\cdots x_n $$
is Schur convex. The inequality $p(\bx)\geq p(\by)$ is the inequality that interests you.
By the equivalence of conditions (ii) and (iv) in Theorem A.3 on page 14, the condition $y=Ax$ for $y=[y,\dots,y_n]^T$ and $x=[x,\dots,x_n]^T$ is equivalent to the condition that $\sum_1^n g(y_i)\ge\sum_1^n g(x_i)$ for all continuous concave functions $g$. Now take $g=\ln$ to get the desired inequality $y_1 \cdots y_n \ge x_1 \cdots x_n$.
