The problem of evaluating the norms of the strong Fejer sums with the power p for orthogonal polynomials has been considered. It has been proved that the norms of the strong Fejer sums have been limited for 1≤p≤2. The estimations have contained the concrete values of constants. The results have been refined for Legendre polynomials. The following inequality A_n(p,x) ≤ √ 1/2 ln n + 1 has been obtained for Legendre polynomials at p=1, x=1.