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For x=(x₁,x₂,…,x_n)∈R₊ⁿ, the second dual form of the Hamy symmetric function is defined by where r∈{1,2,…,n} and i₁,i₂,…,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.