Abstract
The necessary and sufficient conditions for Schur geometrical convexity of the four‐parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means.
Highlights
Introduction and Main ResultLet p, q ∈ R and a, b > 0
The necessary and sufficient conditions for Schur geometrical convexity of the four-parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means
The Schur convexity of Sp,q a, b and Gp,q a, b on 0, ∞ × 0, ∞ with respect to a, b was investigated by Qi et al 4, Shi et al 5, Li and Shi 6, and Chu and Zhang 7
Summary
Q / 0, p 0, p q / 0, p q 0, Abstract and Applied Analysis and Sp,q a, a a see 1 , where. The Schur convexity of Sp,q a, b and Gp,q a, b on 0, ∞ × 0, ∞ with respect to a, b was investigated by Qi et al 4 , Shi et al 5 , Li and Shi 6 , and Chu and Zhang 7. Schur convexity of a four-parameter homogeneous means family containing Stolarsky and Gini means on −∞, ∞ × −∞, ∞ with respect to p, q has been perfectly solved by Yang. The purpose of this paper is to present the necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means with respect to a, b. Q , r, s ∈ R×R the four-parameter homogeneous means F p, q; r, s; a, b are Schur geometrically convex (Schur geometrically concave) on 0, ∞ × 0, ∞ with respect to a, b if and only if p q r s > < 0
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