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 ) ∈ 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 ) = ∏ 1 ≤ i 1 < i 2 < ⋯ < i r ≤ n ( ∑ j = 1 r x i j ) 1 r , where r ∈ { 1 , 2 , … , n } and i 1 , i 2 , … , i n are positive integers. s 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.