Key words Complete symmetric function, Schur multiplicative convexity, Schur harmonic convexity, theory of majorization MSC (2010) 26B25 This paper investigates the Schur multiplicative and harmonic convexities of the complete symmetric function Fn (x, r) = i 1 + i 2 + ···+ i n = r x i 1 1 x i 2 2 . . . x i n n and the function ϕn (x, r) = F n (x ,r ) F n (x ,r −1) , where i1 , i2 , . . . , in are nonnegative integers and r ≥ 1. As applications, some analytic inequalities are established by use of the theory of majorization.