Schur m -Power Convexity of a Class of Multiplicatively Convex Functions and Applications

Abstract EN

We investigate the conditions under which the symmetric functions F n , k ( x , r ) = ∏ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n f ( ∑ j = 1 k x i j r ) 1 / r , k = 1,2 , … , n, are Schur m -power convex for x ∈ R + + n and r > 0 .

As a consequence, we prove that these functions are Schurgeometrically convex and Schur harmonically convex, whichgeneralizes some known results.

By applying the theory ofmajorization, several inequalities involving the p th power mean andthe arithmetic, the geometric, or the harmonic means are presented.