The existing various notions of generalized convexity are very useful, in particular in optimal control theory (Cannarsa and Sinestrari, 2004) and optimization (for more information and references see Rolewicz, 2005). Therefore, convenient conditions which guarantee generalized convexity are very useful. As we know from the classical theory of convex functions, midconvexity and local upper boun...

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of the Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity for a composite function of the complete symmetric function.