Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications
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=11−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., number=10, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Wei-feng, Xia, year=2009, month=Oct., pages=1306–1318