Necessary and Sufficient Conditions for Schur Geometrical Convexity of the Four-Parameter Homogeneous Means

and Applied Analysis 3 where L x, y , I x, y denote logarithmic mean and identric exponential mean, respectively, G a, b √ ab. 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 . Until now, they have been perfectly solved by Chu and Zhang 7 , Wang and Zhang 8 , respectively. Recently, Chu and Xia also proved the same result as Wang and Zhang 9 . The Schur convexity of Sp,q a, b and Gp,q a, b on 0,∞ × 0,∞ and −∞, 0 × −∞, 0 with respect to p, q was investigated by Qi 10 and Sándor 11 , respectively. Now 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 12 . The Schur geometrical convexity was introduced by Zhang 13 . In 8, 14 , Wand and Zhang proved that Gp,q a, b is Schur geometrically convex Schur geometrically concave on 0,∞ × 0,∞ with respect to a, b if p q ≥ ≤ 0. Chu et al. 15 pointed out that this conclusion is also true for Sp,q a, b . Shi et al. 5, 16 , Li and Shi 6 , and Gu and Shi 17 also obtained similar results. 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 . Our main result is as follows. Theorem 1.2. For fixed p, 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. 2. Definitions and Lemmas Definition 2.1 see 18, 19 . Let x x1, x2, . . . , xn and y y1, y2, . . . , yn ∈ R n ≥ 2 . i x is said to by majorized by y in symbol x ≺ y if