SCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN

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1 SCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN HUAN-NAN SHI Abstract. The monotonicity and the Schur-convexity with parameters s, t) in R 2 for fixed x, y) and the Schur-convexity and the Schur-geometrically convexity with variables x, y) in R 2 ++ for fixed s, t) of Gini mean Gr, s; x, y) are discussed. Some new inequalities are obtained. 1. Introduction Throughout the paper we assume that the set of the real number, the nonnegative real number, the nonpositive real number and the positive real number by R, R +, R andr ++ respectively. let s, t) R 2, x, y) R The Gini mean of x, y) is defined in [1] and [2, p. 44] as x s + y s ) 1/s r) Gr, s; x, y) = x r + y r, r s, x s ln x + y s ) ln y exp x r + y r, r = s. Clearly, G0, 1; x, y) is the harmonic mean, G0, 0; x, y) is the geometric mean, G1, 0; x, y) is the arithmetic mean. In 1988, Zs.Páles proved the following result on the comparison of the Gini means. Theorem A [3]). Let u, v, t, w R. Then the comparison inequality Gu, v; x, y) Gt, w; x, y) 1) is valid if and if u + v t + w, and i) min{u, v} min{t, w}, if 0 min{u, v, t, w}, ii) ku, v) kt, w), if min{u, v, t, w} < 0 < max{u, v, t, w}, iii) max{u, v} max{t, w}, if max{u, v, t, w} 0, here { x y x y kx, y) =, x y signx), x = y Mathematics Subject Classification. Primary 26D15, 26A51. Key words and phrases. Gini mean, Stolarsky mean, Schur-convexity, Schur-geometric concavity, inequality. The author was supported in part by the Scientific Research Common Program of Beijing Municipal Commission of Education KM ). This paper was typeset using AMS-LATEX. 1

2 2 H.-N. SHI The extended mean or Stolarsky mean) of x, y) is defined in [2, p. 43] as r s ys x s ) 1/s r) y r x r, rsr s)x y) 0, 1 r y r x r ) 1/r, rx y) 0; ln y ln x Er, s; x, y) = ) 1 x x r 1/x r y r ), rx y) 0; e 1/r y yr xy, x y; x, x = y. The Schur-convexities of the extended mean Er, s; x, y) with r, s) and x, y) were presented in [4, 5] as follows. Theorem B [4]). For fixed x, y) with x > 0, y > 0 and x y, the extended mean Er, s; x, y) are Schur-concave on [0, + ) [0, + ) and Schur-convex on, 0], 0] with r, s). Theorem C [5]). For fixed r, s) R 2, 1) if 2 < 2r < s or 2 2s r, then the extended mean values Er, s; x, y) is Schur-convex with x, y) 0, ) 0, ), 2) if r, s) {r < s 2r, 0 < r 1} {s < r 2s, 0 < s 1} {0 < s < r 1} {0 < r < s 1} {s 2r < 0} {r 2s < 0}, then the extended mean Er, s; x, y) is Schur-concave with x, y) 0, ) 0, ). For more information on the extended mean values and Gini mean, please refer to [3, 4, 5, 6, 7, 8, 9, 10] and the references therein. In this paper, the monotonicity and the Schur-convexity with parameters s, t) in R 2 for fixed x, y) and the Schur-convexity and the Schur-geometrically convexity with variables x, y) in R 2 ++ for fixed s, t) of Gini mean Gr, s; x, y) are discussed. Moreover, some new inequalities are obtained. 2. Definitions and Lemmas We need the following definitions and lemmas. Definition 1 [11, 12]). Let x = x 1,..., x n ) and y = y 1,..., y n ) R n. 1) x is said to be majorized by y in symbols x y) if k i=1 x [i] k i=1 y [i] for k = 1, 2,..., n 1 and n i=1 x i = n i=1 y i, where x [1] x [n] and y [1] y [n] are rearrangements of x and y in a descending order. 2) x y means x i y i for all i = 1, 2,..., n. let Ω R n, ϕ: Ω R is said to be increasing if x y implies ϕx) ϕy). ϕ is said to be decreasing if and only if ϕ is increasing. 3) Ω R n is called a convex set if αx 1 + βy 1,..., αx n + βy n ) Ω for any x and y Ω, where α and β [0, 1] with α + β = 1. 4) let Ω R n, ϕ: Ω R is said to be a Schur-convex function on Ω if x y on Ω implies ϕ x) ϕ y). ϕ is said to be a Schur-concave function on Ω if and only if ϕ is Schur-convex function. Definition 2 [13, 14]). Let x = x 1,..., x n ) and y = y 1,..., y n ) R n ++.

3 SCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN 3 1) Ω R n ++ is called a geometrically convex set if x α 1 y β 1,..., xα ny β n) Ω for any x and y Ω, where α and β [0, 1] with α + β = 1. 2) let Ω R n ++, ϕ: Ω R + is said to be a Schur-geometrically convex function on Ω if ln x 1,..., ln x n ) ln y 1,..., ln y n ) on Ω implies ϕ x) ϕ y). ϕ is said to be a Schur-geometrically concave function on Ω if and only if ϕ is Schur-geometrically convex function. Lemma 1 [11, p. 38]). A function ϕx) is increasing if and only if ϕx) 0 for x Ω, where Ω R n is an open set, ϕ : Ω R is differentiable, and ϕx) ϕx) =,..., ϕx) ) R n. x 1 x n Lemma 2 [11, p. 58]). Let Ω R n is symmetric and has a nonempty interior set. Ω 0 is the interior of Ω. ϕ : Ω R is continuous on Ω and differentiable in Ω 0. Then ϕ is the Schur convexschur concave)function, if and only if ϕ is symmetric on Ω and ϕ x 1 x 2 ) ϕ ) 0 0) 2) x 1 x 2 holds for any x = x 1, x 2,, x n ) Ω 0. Lemma 3 [13, p. 108]). Let Ω R n ++ is a symmetric and has a nonempty interior geometrically convex set. Ω 0 is the interior of Ω. ϕ : Ω R + is continuous on Ω and differentiable in Ω 0. If ϕ is symmetric on Ω and ) ϕ ϕ ln x 1 ln x 2 ) x 1 x 2 0 0) 3) x 1 x 2 holds for any x = x 1, x 2,, x n ) Ω 0, then ϕ is the Schur-geometrically convex Schur-geometrically concave) function. Lemma 4 [15, 5]). Let f be a continuous function. The arithmetic mean of function f defined by 1 y ft) dt, x y, F x, y) = x y x 4) fx), x = y. Then 1) function F x, y) is increasing decreasing) on I 2 if and only if f is an increasing decreasing) function on I. 2) F x, y) is Schur-convex Schur-concave) on I 2 if and only if fx)is convex concave) on I Lemma 5 [16]). Let a b, ut) = tb + 1 t)a, vt) = ta + 1 t)b and let 1/2 t 2 t 1 1. Then ut 2 ), vt 2 )) ut 1 ), vt 1 )) a, b). Lemma 6 [17]). Let l, t, p, q R ++, p > q and p + q 3l + t). Assume also that 1/3 l/t 3 or q l + t. Then Gl, t; x, y) p/q) 1/p q) Ep, q; x, y).

4 4 H.-N. SHI 3. Main results and their proofs In the following, we are in a position to state our main results and give proofs of them. Theorem 1. For fixed x, y) R 2 ++, 1) Gr, s; x, y) is increasing in R 2, 2) Gr, s; x, y) is Schur-concave in R 2 +, 3) Gr, s; x, y) is Schur-convex in R 2 withr, s). Proof. Let gt) = xt ln x + y t ln y x t + y t. It is easy to check that ln Gr, s; x, y) = 1 r s s r gt) dt, r s, see the proof of Lemma 3.1 of in [18]). By computing, we get g t) = x t y t x t + y t ) 2 ln x ln y)2, g t) = xt y t ln x ln y) 2 x t y t ) 4 ln x ln y)x 2t y 2t ). 1) Since g t) 0, gt) is increasing in R. By 1) in lemma 4, it follows that ln Gr, s; x, y) is increasing in R 2 with r, s), and so does Gr, s; x, y). 2) When r, s) R 2 +, ln u and u 2t are increasing in R + with u, hence ln x ln y)x 2t y 2t ) 0, moreover g t) 0. That is, gt) is concave in R +. By 2) in lemma 4, it follows that { ln Gr, s; x, y), r s, F r, s) = gt), r = s is Schur-concave in R 2 + with r, s), and so does Gr, s; x, y) by Corollary 6.14 in [11, p. 64]. 3) When r, s) R 2, ln u is increasing and u 2t is decreasing in R + with u, hence ln x ln y)x 2t y 2t ) 0, moreover g t) 0. That is, gt) is convex in R. By 2) in lemma 4, it follows that F r, s) is Schur-convex in R 2 with r, s), and so does Gr, s; x, y). The proof is complete. Remark 1. Let 0 min{u, v, t, w}. If u, v) t, w), it follows that u+v = t+w and min{u, v} min{t, w}. By the Theorem A, we have Gu, v; x, y) Gt, w; x, y), but by 2) in the Theorem 1, we have Gu, v; x, y) Gt, w; x, y). In fact, 1) is wrong for the cases i), for example, taking u = 2, v = 1, t = 3, w = 0, then 1) is reduces to it is equivalent to x 2 + y 2 x + y x 3 + y 3 2 ) 1 3, x 2 + y 2 ) 3 3xyx 4 + y 4 ) + 2x 3 y 3. Taking x > 0 and let y 0 +, it following that x 6 0, this leads to a contradiction with x > 0.

5 SCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN 5 In fact, 3) of Theorem 1 gives a sufficient condition such that the reversed inequality of 1) is valid, i.e. max{u, v, t, w} 0, max{u, v} max{t, w}, u + v = t + w. Theorem 2. If r, s) Ω 1 = {r s 1} {s r 1} R 2, then Gr, s; x, y) is the Schur-convex with x, y) in R Proof. Let ϕx, y) = xs + y s x r + y r. When r s, for fixed x, y) R2, we have ϕ x = sxs 1 x r + y r ) rx r 1 x s + y s ) x r + y r ) 2, ϕ = sys 1 x r + y r ) ry r 1 x s + y s ) x r + y r ) 2. ϕ x ϕ = sxr + y r )x s 1 y r 1 ) rx s + y s )x r 1 y r 1 ) x r + y r ) 2 and then = sxr 1 y r 1 ) x r + y r ) = sxr 1 y r 1 ) x r + y r ) G x y) x G ) = x y s r = sx y)xr 1 y r 1 ) s r)x r + y r ) and [ s 1 r 1 r 1)xs 1 y s 1 ) s 1)x r 1 y r 1 ) r s xs + y s ] x r + y r [ s 1 r 1 Es r r 1, s 1; x, y) r ] s Gs r r, s; x, y), ϕ x ϕ ) ϕ 1 s r 1 x, y) [ s 1 r 1 Es r r 1, s 1; x, y) r s Gs r r, s; x, y) In lemma 6, taking l = r, t = s, p = r 1, q = s 1, we have l > 0, t > 0, p > 0, q > 0 r > 1, s > 1 p > q r > s 3s r > s > 1 p + q 3l + t) r + s 1 1/3 l/t 3 s/3 r 3s l > 0, t > 0, p > 0, q > 0 p > q p + q 3l + t) q l + t r > 1, s > 1 r > s r + s 1 r 1 r > s > 1. Hence, when r > s > 1, we have ) 1 r 1 r s Gr, s; x, y) Er 1, s 1; x, y), s 1 i.e. G s r r, s; x, y) s 1 r 1 Es r r 1, s 1; x, y). 5) ] ϕ 1 s r 1 x, y)

6 6 H.-N. SHI When r > s > 1, we have s r < 0 and )x y)x r 1 y r 1 ) 0. Combining G with 5), it follows that x y) x G 0. By lemma 2, Gr, s; x, y) is the Schur-convex with x, y) in R Since Gr, s; x, y) is symmetric with r, s), when s > r > 1, Gr, s; x, y) is also the Schur-convex with x, y) in R When r = s, let ψx, y) = xs ln x + y s ln y x r + y r where By computing, x s 1 hx, y) y s 1 kx, y) = xs ln x + y s ln y x s + y s. Then ψ x = xs 1 hx, y) x s + y s ) 2, ψ = ys 1 kx, y) x s + y s ) 2, hx, y) = s ln x + 1)x s + y s ) sx s ln x + y s ln y), kx, y) = s ln y + 1)x s + y s ) sx s ln x + y s ln y). = x s + y s ) [ x s 1 s ln x + 1) y s 1 s ln y + 1) ] sx s ln x + y s ln y)x s 1 y s 1 ) = s s 1 y s 1 x + y)ln x ln y) + x s 1 y s 1 )x s + y s ), and then, G x y) x G ) ψ = x y) x ψ ) e ψx,y) = sxs 1 y s 1 x + y)x y)ln x ln y) + x y)x s 1 y s 1 )x s + y s ) x s + y s ) 2 e ψx,y). Since ln t and t s 1 are increasing in R + with t for s 1, therefore )x y)ln x ln y) 0 and x y)x s 1 y s 1 G ) 0, moreover x y) x G 0. That is, when r = s 1, Gr, s; x, y) is the Schur-convex with x, y) in R The proof is complete. Theorem 3. If r, s) Ω 2 = {r s 0} {s r 0} R 2, then Gr, s; x, y) is the Schur-geometrically convex with x, y) in R Proof. Let ϕx, y) = xs + y s and then x r + y r. When r s, for fixed x, y) R2, we have x ϕ x = sxs x r + y r ) rx r x s + y s ) x r + y r ) 2, y ϕ = sys x r + y r ) ry r x s + y s ) x r + y r ) 2. x ϕ x y ϕ = sxr + y r )x s y r ) rx s + y s )x r y r ) x r + y r ) 2 = sxr y r ) x r + y r [ s r rxs y s ) sx r y r ) r s xs + y s x r + y r = sxr y r ) x r + y r [ s r Es r r, s; x, y) r s Gs r r, s; x, y) ] ],

7 SCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN 7 ln x ln y) x G x y G and = sln x ln y)xr y r ) s r)x r + y r ) ) ln x ln y = x ϕ s r x y ϕ ) ϕ 1 s r 1 x, y) [ s r Es r r, s; x, y) r s Gs r r, s; x, y)] ϕ 1 s r 1 x, y) In lemma 6, taking l = p = r, t = q = s, we have l > 0, t > 0, p > 0, q > 0 r > 0, s > 0 p > q r > s p + q 3l + t) r + s 1 1/3 l/t 3 s/3 r 3s l > 0, t > 0, p > 0, q > 0 p > q p + q 3l + t) q l + t Hence, when r > s > 0, we have i.e. Gr, s; x, y) r > 0, s > 0 r > s r + s 1 r 0 r s) 1 r s Er, s; x, y), 3s r > s > 0 r > s > 0. G s r r, s; x, y) s r Es r r, s; x, y). 6) When r > s > 0, we have s r < 0, and since ln t and t r are increasing in R + with t, therefore ln x ) ln y)x r y r ) 0. Combining with 6), it follows that ln x ln y) x G x y G 0. By lemma 3, Gr, s; x, y) is the Schur-geometrically convex with x, y) in R Since Gr, s; x, y) is symmetric with r, s), when s > r > 0, Gr, s; x, y) is also the Schur-geometrically convex with x, y) in R When r = s, we have ψ x = xs 1 hx, y) x s + y s ) 2, ψ = xs 1 kx, y) x s + y s ) 2, where hx, y), kx, y) and ψx, y) are same as in Theorem 2. By computing, and then, x s hx, y) y s kx, y) = s s y s x + y)ln x ln y) + x s y s )x s + y s ), ln x ln y) x G x y G ) = ln x ln y) x ψ x y ψ ) e ψx,y) = sxs y s x + y)ln x ln y) 2 + ln x ln y)x s y s )x s + y s ) x s + y s ) 2 e ψx,y). Since when s 0, ln t and t s are ) increasing in R +, ln x ln y)x s y s ) 0, moreover ln x ln y) x G x y G 0. That is, when r = s 0, Gr, s; x, y) is the Schur-geometrically convex with x, y) in R The proof is complete.

8 8 H.-N. SHI 4. Applications Theorem 4. Let ut) = ts + 1 t)r, vt) = tr + 1 t)s, and let 1 2 t 2 t 1 1. If s r 0, then for fixed x, y) R 2 ++, r + s G 2, r + s ) 2 ; x, y G ut 2 ), vt 2 ); x, y) G ut 1 ), vt 1 ); x, y) Gr, s; x, y) Gr + s, 0; x, y), 7) and if s r 0, then inequalities in 7) are all reversed. Proof. From lemma 5, we have r + s 2, r + s ) ut 2 ), vt 2 )) ut 1 ), vt 1 )) s, r). 2 and it is clear that r, s) r + s, 0), therefore, by Theorem 1, it follows that 7) are holds. The proof is complete. According to Theorem 2, The following Theorem 5 can be proved in a similar way as shown Theorem 4. Theorem 5. Let x, y) R 2 ++, ut) = ty + 1 t)x, vt) = tx + 1 t)y, and let 1 2 t 2 t 1 1. If r, s) Ω 1 = {r s 1} {s r 1} R 2, then G r, s; x + y, x + y ) G r, s; ut 2 ), vt 2 )) 2 2 G r, s; ut 1 ), vt 1 )) Gr, s; x, y) Gr, s; x + y, 0). 8) Theorem 6. Let x, y) R If r, s) Ω 2 = {r s 0} {s r 0} R 2, then G r, s; xy, xy) Gr, s; x, y). 9) Proof. Since ln xy, ln xy) ln x, ln y), by Theorem 3, it follows that 9) is holds. The proof is complete. References [1] C. Gini, Di una formula compresive delle medie, METRON ), 3-22 [2] Ji-chang Kuang, Applied Inequalities Chang yong bu deng shi )3nd ed. Shandong Press of science and technology, Jinan, China, Chinese). [3] Edward Neuman and Zsolt, Ples, On comparison of Stdlarsky and Gini means, J. Math. Anal. Appl., ), [4] F. Qi, Schur-convexity of the extended mean values, RGMIA Res. Rep. Coll ), no. 4, Art. 4, Available online at [5] Huan-nan Shi, Shan-he Wu and Feng Qi, An alternative note on the Schur-convexity of the extended mean values, Mathematical Inequalities and Applications 2006, 9 2): MR a:26016) [6] Zsolt, Ples, Inequalities for differences of powers, J. Math. Anal. Appl ), [7] Zsolt, Ples, Inequalities for sums of powers, J. Math. Anal. Appl ), [8] József Sándor, Anote on the Gini means, Genneral Mathematics, Vol. 12 No ), [9] C. E. M. Pearce, J. Pečarić and J. Sándor, Ageneralzation of Pólya s inequality to Stolarsky and Gini means, Mathematical Inequalities & Applications, Vol. 1 No ), [10] Peter Czinder and Zsolt Ples, An extension of the Gini and Stdlarsky means,j.inequl. Pure and Appl. Math. Vol ) no. 2, Art. 42; Available online at jipam.vu.edu.an/ [11] Bo-ying. Wang. Foundations of Majorization Inequalities, Beijing Normal Univ. Press, Beijing, China, Chinese)

9 SCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN 9 [12] A. M. Marshall and I. Olkin. Inequalities:theory of majorization and its application. New York :Academies Press, [13] Xiao-ming. Zhang. Geometrically Convex Functions. Hefei: An hui University Press, 2004.Chinese) [14] Niculescu, Convexity According to the Geometric Mean, Mathematical Inequalities & Applications, 2000, 32): [15] Elezovic N. and Pecaric J., A note on Schur-convex fuctions,rocky Mountain J. Math. 2000, 30 3): [16] Da-mao Li, Chun Gu and Huan-nan Shi, Schur Convexity of the Power-Type Generalization of Heronian Mean, Mathematics in practice and theorychinese), 2006, 369): [17] Peter A. Hasto, Monotonicity property of ratios of symmetric homogeneous means, J. Ineq.Pure Appl.Math.Volume 3, Issue 5, Article :1-23 [18] Czindeer, Pter; Ples, Zsolt, An extension of the Hermite-Hadamard inequality and an application for Gini and Stolarsky means, J. Inequal. Pure Appl. Math. 5, No. 2, Paper No. 42, 8 p., electronic only 2004). Zbl H.-N. Shi) Department of Electronic information, Teacher s College, Beijing Union University, Beijing City, , P.R.China address: shihuannan@yahoo.com.cn, sfthuannan@buu.com.cn