Abstract
The judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions are given. As their application, some analytic inequalities are established.
Highlights
1 Introduction Throughout this paper, R denotes the set of real numbers, x = (x, x, . . . , xn) denotes ntuple (n-dimensional real vectors), the set of vectors can be written as
Theorem A Let A ⊂ Rk be a symmetric convex set, and let φ be a Schur-convex function defined on A with the property that for each fixed x, . . . , xk, φ(z, x, . . . , xk) is convex in z on {z : (z, x, . . . , xk) ∈ A}
Lemma [, p. ] Let ⊂ Rn+ be a symmetric geometrically convex set with a nonempty interior
Summary
Theorem A Let A ⊂ Rk be a symmetric convex set, and let φ be a Schur-convex function defined on A with the property that for each fixed x , . Theorem Let A ⊂ Rk be a symmetric geometrically convex set, and let φ be a Schur geometrically convex (concave) function defined on A with the property that for each fixed x , . Theorem Let A ⊂ Rk be a symmetric harmonically convex set, and let φ be a Schur harmonically convex (concave) function defined on A with the property that for each fixed x , . ] Let ⊂ Rn+ be a symmetric geometrically convex set with a nonempty interior. Lemma [ , ] Let ⊂ Rn+ be a symmetric harmonically convex set with a nonempty interior. Lemma [ ] Let I ⊂ R+ be an open subinterval, and let φ : I → R+ be differentiable. (i) φ is GA-convex (concave) if and only if xφ (x) is increasing (decreasing). (ii) φ is HA-convex (concave) if and only if x φ (x) is increasing (decreasing)
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