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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