On sums and products of distinct numbers

Let A be a set of k complex numbers, and let A+ (respectively, A×) be the set of sums (resp. products) of distinct elements of A. LetgC(k)=minA⊂C,|A|=k {|A+|+|A×|}.Ruzsa posed the question whether gC(k) grows faster than any power of k. In this note we give an affirmative answer to this question.