A polynomial is called sums of squares (SOS) if it can be written as a sum of squares of polynomials. Clearly any such polynomial is nonnegative. Although Hilbert could show that the converse is in general false, Hurwitz could certify that the geometry mean is dominated by the arithmetic mean using sum of squares decompositions. In this talk we will present results on the relation between symmetric mean inequalities and sums of squares. In particular we will show that such inequalities of degree 4 can be always verified using sums of squares.