Abstract:

In this paper, we study the Schur convexity and Schur geometrical convexity
of identric and logarithmic means in n variables. We prove the following results: (1) The identric mean is Schur concave and S-geometrically convex; (2) The first-logarithmic mean is Schur concave; (3) The second-logarithmic mean is Schur convex. At last we give two open problems.

Key words: Convex functions, Schur convex functions, Geometrically convex functions,
Schur geometrically convex functions