Some properties of a generalized Hamy symmetric function and its applications

This paper is concerned with the generalized Hamy symmetric function ∑ n ( x , r ; f ) = ∑ 1 ⩽ i 1 < i 2 < ⋯ < i r ⩽ n f ( ∏ j = 1 r x i j 1 r ) , where f is a positive function defined in a subinterval of ( 0 , + ∞ ) . Some properties, including Schur-convexity, geometric Schur-convexity and harmonic Schur-convexity are investigated. As applications, several inequalities are obtained, some of which extend the known ones.