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In this paper, the relationship between r -convexity and Schur-convexity is investigated first. Moreover, the r -convexity and Schur-convexity of a class of functions are studied. As applications, some new inequalities on Minkowski's inequality are established. [ABSTRACT FROM AUTHOR]

The main aim of the article is to prove that the symmetric function Φ n ( x , r ) = ∏ i 1 + i 2 + ⋯ + i n = r ( x 1 i 1 + x 2 i 2 + ⋯ + x n i n ) is Schur geometrically convex for x ∈ R + + n and fixed r ∈ N + = { 1 , 2 , ⋯ } , where i 1 , i 2 , ⋯ , i n are non-negative integers. Further, we obtain Φ n ( x , r ) is also Schur m -power convex for m ≤ 0 . As applications, a Klamkin–Newman type inequality is derived. Finally, we list a counter example to illustrate Φ n ( x , r ) is neither Schur convex nor Schur concave. [ABSTRACT FROM AUTHOR]

The decision theorem of the Schur harmonic convexity for the compositions involving Schur harmonically convex functions is established and used to determine the Schur harmonic convexity of some symmetric functions. [ABSTRACT FROM AUTHOR]

The decision theorem of the Schur geometric convexity for the compositions involving Schur geometrically convex functions is established and used to determine the Schur geometric convexity of some symmetric functions. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish some integral inequalities of Hermite–Hadamard type involving the product of an s -convex function and a symmetric function and apply these new established inequalities to construct inequalities for special means. [ABSTRACT FROM AUTHOR]

We prove inequalities for mixed generalized means; from these inequalities we obtain analogs or generalizations of inequalities due to Ness, to Carlson, Meany, and Nelson, to Godunova, and to Marcus and Minc, etc. [ABSTRACT FROM AUTHOR]