$ A = (a_ {ij}) _ {1 le i, j le n} $ is a matrix 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 $.And

$$ begin {equation}

begin {pmatrix}

y_1 \

vdots \

y_n \

{pmatrix}

= mathbf {A}

begin {pmatrix}

x_1 \

vdots \

x_n

{pmatrix}

end {equation} $$

$ y_i $ and $ x_i $ are not all negative. Prove: $ y_1 cdots y_n ge x_1 cdots x_n $

It can somehow be important to convexize the function.