Maple 2018 does it by

    restart; 
     A := log((1+x[1]/(1+x[1]+(1/2)*x[2]+(1/4)*x[3]))*(1+x[2]/(1+x[2]+
    (1/2)*x[3]+(1/4)*x[1]))*(1+x[3]/(1+x[3]+(1/2)*x[2]+(1/4)*x[3]))):
    H := VectorCalculus:-Hessian(A, [x[1], x[2], x[3]]):
    LinearAlgebra:-IsDefinite(H, query = 'negative_semidefinite')
    assuming x[1]>=0,x[1]<=1,x[2]>=0,x[2]<=1,x[3]>=0,x[3]<=1;

> true

Addition. Maple cracks the Wolfgang's modification too, but only for concrete values of $n$:

    restart; n := 25: xx := [seq(x[j], j = 1 .. n)]:
    A := log(mul(1+x[k]/(1+add(q^j*x[k+j], j = 0 .. n-1)), k = 1 .. n)):
    H := VectorCalculus:-Hessian(A, xx):
    LinearAlgebra:-IsDefinite(H, query = 'negative_semidefinite')
    assuming seq(x[s]>=0,s=1..n),seq(x[s]<=1,s=1..n),q>0,q<1

> true