2018
Том 70
№ 9

# Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

Wei-feng Xia

Abstract

For $x = (x_1, x_2, …, x_n) ∈ (0, 1 ]^n$ and $r ∈ \{ 1, 2, … , n\}$, a symmetric function $F_n(x, r)$ is defined by the relation $$F_n(x,r) = F_n(x_1, x_2, …, x_n; r) = ∑_{1 ⩽ i_1 < i_2…i_r ⩽n } ∏^r_{j=1}\frac{1−x_{i_j}}{x_{i_j}},$$ where $i_1 , i_2 , ... , i_n$ are positive integers. This paper deals with the Schur convexity and Schur multiplicative convexity of $F_n(x, r)$. As applications, some inequalities are established by using the theory of majorization.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 10, pp 1541-1555.

Citation Example: Wei-feng Xia Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications // Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1306-1318.

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