We investigate the conditions under which the symmetric functionsFn,k(x,r)=∏1≤i1<i2<⋯<ik≤n ‍f(∑j=1k‍xijr)1/r, k=1,2,…,n,are Schurm-power convex forx∈R++nandr>0. As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving thepth power mean and the arithmetic, the geometric, or the harmonic means are presented.