Schur-Convexity for a Class of Symmetric Functions and Its Applications

@inproceedings{Xia2009SchurConvexityFA,
  title={Schur-Convexity for a Class of Symmetric Functions and Its Applications},
  author={Wei-feng Xia and Yu-Ming Chu},
  year={2009}
}
For x x1, x2, . . . , xn ∈ R , the symmetric function φn x, r is defined by φn x, r φn x1, x2, . . . , xn; r ∏ 1≤i1<i2 ···<ir≤n ∑r j 1 xij / 1 xij 1/r , where r 1, 2, . . . , n and i1, i2, . . . , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn x, r are discussed. As applications, some inequalities are established by use of the theory of majorization.