The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications

For x = (x(1), x(2), ... , x(n)) is an element of R-+(n), the second dual form of the Hamy symmetric function is defined by H-n** (x, r) = H-n** (x(1), x(2), ... , x(n); r) = Pi(1 <= i1<i2<...<ir <= n) (Sigma(r)(j=1) x(ij))(1/r), where r is an element of {1, 2, ... , n} and i(1), i(2), ... , i(n) are positive integers. In this paper, we prove that H-n* (x, r) is Schur concave, and Schur multiplicatively and harmonic convex in R-+(n). Some applications in inequalities and reliability theory are presented. (C) 2011 Elsevier Inc. All rights reserved.