Schur-convexity of the complete elementary symmetric function

Schur-convexity, Schur-geometric and Schur-harmonic convexity for a composite function of complete symmetric function

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.

Schur Convexity of Dual Form of the Complete Symmetric Function

In this paper, the Schur-convexity, the Schur-geometric-convexity and the Schur-harmonicconvexity of dual form of the complete symmetric function are investigated. As consequences, some new inequalities are established via majorilization theory. Mathematics subject classification (2010): 26B25, 05E05.

On Schur Convexity of Some Symmetric Functions

For x x1, x2, . . . , xn ∈ 0, 1 n and r ∈ {1, 2, . . . , n}, the symmetric function Fn x, r is defined as Fn x, r Fn x1, x2, . . . , xn; r ∑ 1≤i1<i2 ···<ir≤n ∏r j 1 1 xij / 1−xij , where i1, i2, . . . , in are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of Fn x, r are discussed. As consequences, several inequalities are est...

Schur–convexity, Schur Geometric and Schur Harmonic Convexities of Dual Form of a Class Symmetric Functions

By the properties of Schur-convex function, Schur geometrically convex function and Schur harmonically convex function, Schur-convexity, Schur geometric and Schur harmonic convexities of the dual form for a class of symmetric functions are simply proved. As an application, several inequalities are obtained, some of which extend the known ones. Mathematics subject classification (2010): 26D15, 0...

The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications