On the Ky Fan inequality

On the Ky Fan Inequality This is the Published version of the following publication Dragomir, Sever S and Scarmozzino, Fernandos P (2001) On the Ky Fan Inequality. RGMIA research report collection, 4 (1). The publisher’s official version can be found at Note that access to this version may require subscription. Downloaded from VU Research Repository https://vuir.vu.edu.au/17371/ ON THE KY FAN INEQUALITY S.S. DRAGOMIR AND F.P. SCARMOZZINO Abstract. Some inequalities related to the Ky Fan and C.-L. Wang inequalities for weighted arithmetic and geometric means are given. 1. Introduction In 1961, E.F. Beckenbach and R. Bellman published in their well known book “Inequalities” the following “unpublished result due to Ky Fan” [2, p. 5] (see also [1, p. 150]). Theorem 1. If 0 < xi ≤ 12 , (i = 1, . . . , n) ; then: , n " n # n1 , n n Y Y X X (1.1) ≤ (1 − xi ) xi (1 − xi ) xi i=1 i=1 i=1 i=1 with equality only if x1 = · · · = xn . A generalisation of Ky Fan’s inequality for weighted means was proved by C.-L. Wang in 1980, [9]. Theorem 2. If 0 < xi ≤ 12 , (i = 1, . . . , n) , then An (p̄, x̄) Gn (p̄, x̄) ≥ , An (p̄, 1 − x̄) Gn (p̄, 1 − x̄) Pn Pn where pi > 0 (i = 1, . . . , n) with pi = 1 and An (p̄, x̄) := i=1Q i=1 pi xi is the n pi weighted arithmetic mean, Gn (p̄, x̄) := i=1 xi is the weighted geometric mean. The equality holds in (1.2) iff x1 = · · · = xn . For a survey on related results of Ky Fan’s inequality, see [1] by H. Alzer. For different refinements and generalisations, see [4] – [8]. (1.2) 2. The Results The following result holds. 1 Theorem Pn 3. Assume that 0 < m ≤ xi ≤ M ≤ 2 , (i = 1, . . . , n), pi > 0 (i = 1, . . . , n) , with i=1 pi = 1, then we have the inequalities: (2.1)  M2  m2   An (p̄, x̄) (1−M )2 An (p̄, x̄) (1−m)2 An (p̄, x̄) An (p̄, 1 − x̄) ≥ ≥ ≥ ≥ 1. Gn (p̄, x̄) Gn (p̄, x̄) Gn (p̄, 1 − x̄) Gn (p̄, x̄) The equality will hold in all inequalities iff x1 = · · · = xn . Date: January 11, 2001. 1991 Mathematics Subject Classification. Primary 26D15, 26D10. Key words and phrases. Ky Fan’s Inequality, Weighted arithmetic and geometric means. 1 2 S.S. DRAGOMIR AND F.P. SCARMOZZINO An (p̄,x̄) ≥1 Proof. The first and the last inequality in (2.1) follow by the fact that G n (p̄,x̄)  1 (by the weighted arithmetic mean - geometric mean inequality), m ∈ 0, 2 and  M ∈ 0, 21 .
+ α ln t with α ∈ R. We We define the function f : (0, 1) → R, f (t) = ln 1−t t have 1 α f ′ (t) = − + , t ∈ (0, 1) , t (1 − t) t # " α 1 − 2t 1 1 − 2t ′′ f (t) = 2 − t2 = t2 2 − α , t ∈ (0, 1) . [t (1 − t)] (1 − t) 1−2t ′ If we consider the function g : (0, 1) → R, g (t) = (1−t) 2 , then g (t) = showing that the function g is monotonically strictly decreasing on (0, 1). Consequently for t ∈ (m, M ), we have 1 − 2m 1 − 2M (2.2) 2 = g (M ) ≤ g (t) ≤ g (m) = 2. (1 − M ) (1 − m) 2t(t−1) , (t−1)4 Using (2.2) we observe that the function f is strictly convex on (m, M ) if α ≤ 1−2M . (1−M )2 Applying Jensen’s discrete inequality for the function f : (m, M ) → R, f (t) =
1−2M ln 1−t + α ln t, with α ≤ (1−M , we deduce t )2 !     n n n X X X 1 − xi pi ln pi xi pi f (xi ) ≥ f + α ln xi = xi i=1 i=1 i=1 ! Pn   n X 1 − i=1 pi xi Pn pi xi , + α ln = ln i=1 pi xi i=1 which is equivalent to     An (p̄, 1 − x̄) Gn (p̄, 1 − x̄) ln + α ln Gn (p̄, x̄) ≥ ln + α ln An (p̄, x̄) Gn (p̄, x̄) An (p̄, x̄) or, moreover, to   α   Gn (p̄, x̄) An (p̄, 1 − x̄) Gn (p̄, 1 − x̄) ln ≥ ln , An (p̄, x̄) An (p̄, x̄) Gn (p̄, x̄) that is,  (2.3) α−1 ≥ 1−2M (1−M )2 −1 Gn (p̄, x̄) An (p̄, x̄) An (p̄, 1 − x̄) . Gn (p̄, 1 − x̄) Now, we observe that the inequality (2.3) is the best possible if α is maximal, i.e., 1−2M , getting α = (1−M )2  Gn (p̄, x̄) An (p̄, x̄)  ≥ An (p̄, 1 − x̄) , Gn (p̄, 1 − x̄) which is clearly equivalent to the second inequality in (2.1). The third inequality is produced in a similar fashion, using the function h (t) =
1−2m which is strictly convex on (m, M ) if β ≥ (1−m) β ln t − ln 1−t 2. t The case of equality follows by the fact that in Jensen’s inequality for strictly convex functions, the equality holds iff x1 = · · · = xn . KY FAN INEQUALITY 3 We omit the details. Remark 1. Since Wang’s inequality (1.2) is equivalent to: An (p̄, x̄) An (p̄, 1 − x̄) ≥ , Gn (p̄, x̄) Gn (p̄, 1 − x̄) (2.4) then the first part of (2.1) may be seen as a refinement of Wang’s result while the second part  m2  An (p̄, x̄) (1−m)2 An (p̄, 1 − x̄) ≥ Gn (p̄, 1 − x̄) Gn (p̄, x̄) (2.5) can be considered a counterpart of (1.2). Now, let us recall the Lah-Ribarić inequality for convex functions (see for example [3, p. 140]). PnIf f : [a, b] ⊂ R → R is convex on [a, b], xi ∈ [a, b], pi ≥ 0 (i = 1, . . . , n) and i=1 pi = 1, then (2.6) n X pi f (xi ) ≤ i=1 b− Pn i=1 pi xi b−a · f (a) + Pn pi xi − a · f (b) . b−a i=1 Now, we can state and prove the following inequality related to the Ky Fan result. Theorem 4. Assume that 0 < m ≤ xi ≤ M ≤ Pn p = 1, then we have the inequalities: i=1 i (2.7)  1−m 2 m m( 1−m ) n (p̄,x̄)   M −A M −m 1−M 2 m M ( 1−m ) 1 2, pi > 0 (i = 1, . . . , n) with (p̄,x̄)−m  AnM −m 2 m · Gn (p̄, x̄)( 1−m ) ≤ Gn (p̄, 1 − x̄) (p̄,x̄)−m n (p̄,x̄)    M −A  AnM M −m −m 2 M 1−M 1−m ≤ Gn (p̄, x̄)( 1−M ) . 2 2 M M m( 1−M ) M ( 1−M ) Proof. 3, we know that the function f : (m, M ) ⊂  From the proof of
Theorem 1−2M 0, 21 → R, f (t) = ln 1−t + ln t is strictly convex on (m, M ). Now, if we 2 t (1−M ) apply the Lah-Ribarić inequality for f as above, a = m and b = M , we get: # "   n X 1 − 2M 1 − xi + pi ln 2 ln xi xi (1 − M ) i=1 Pn Pn n X pi xi − m M − i=1 pi xi = pi f (xi ) ≤ f (m) + i=1 f (M ) M −m M −m i=1 # "  Pn  M − i=1 pi xi 1−m 1 − 2M = ln + 2 ln m M −m m (1 − M ) # "  Pn  1 − 2M 1−M i=1 pi xi − m + ln + 2 ln M , M −m M (1 − M ) 4 S.S. DRAGOMIR AND F.P. SCARMOZZINO which is equivalent to  1 − 2M Gn (p̄, 1 − x̄) + 2 ln Gn (p̄, x̄) Gn (p̄, x̄) (1 − M )     1−2M M − An (p̄, x̄) 1−m 2 (1−M ) ≤ ln + ln (m) M −m m     1−2M An (p̄, x̄) − m 1−M 2 (1−M ) , + ln + ln (M ) M −m M ln that is,  1−2M Gn (p̄, 1 − x̄) · [Gn (p̄, x̄)] (1−M )2 Gn (p̄, x̄)  o  M −An (p̄,x̄)  o  An (p̄,x̄)−m n n M −m M −m 1−2M 1−2M 2 −1 2 −1 (1−M ) (1−M ) ≤ (1 − m) m · (1 − M ) M from which we obtain the second inequality in (2.7). To prove the first inequality, we apply the Lah-Ribarić inequality for the function
1−t 1−2m which is strictly convex on (m, M ). ln t − ln h : (m, M ) → R, h (t) = (1−m) 2 t We omit the details. Finally, let us recall Dragomir-Ionescu’s inequality for differentiable convex functions (see [7]) ! n n X X (2.8) pi f (xi ) − f pi xi 0 ≤ i=1 ≤ n X i=1 i=1 pi xi f ′ (xi ) − n X i=1 pi xi n X pi f ′ (xi ) i=1 provided f : (a, b) ⊆PR → R is differentiable convex on (a, b), xi ∈ (a, b) and pi > 0 n (i = 1, . . . , n) with i=1 pi = 1. If f is strictly convex on (a, b), then the equality holds in (2.8) iff x1 = · · · = xn , we may state the following result. Theorem 5. With the assumptions of Theorem 4, we have      1 1 (2.9) − An p̄, exp An (p̄, x̄) An p̄, x̄ (1 − x̄) 1 − x̄ "   #   1−2M 1 − 2M 1 An (p̄, x̄) (1−M )2 × 1 − An (p̄, x̄) An p̄, × 2 x̄ Gn (p̄, x̄) (1 − M )     Gn (p̄, 1 − x̄) An (p̄, 1 − x̄) ≥ Gn (p̄, x̄) An (p̄, x̄)      1 1 − An p̄, ≥ exp An (p̄, x̄) An p̄, x̄ (1 − x̄) 1 − x̄ " #   1−2m   An (p̄, x̄) (1−m)2 1 1 − 2m × , 1 − An (p̄, x̄) An p̄, × 2 x̄ Gn (p̄, x̄) (1 − m)   where x̄1 denotes the vector x11 , . . . , x1n , ȳ · z̄ := (y1 z1 , . . . , zn yn ), and x̄ ∈ Rn , x̄ > 0̄ (i.e., xi > 0 for any i ∈ {1, . . . , n}), ȳ, z̄ ∈ Rn . KY FAN INEQUALITY 5 
1−2M + (1−M Proof. Since the function f : (m, M ) ⊂ 0, 12 → R, f (t) = ln 1−t ln t t )2 is strictly convex on (m, M ), by (2.8) we may state that "  # Pn    n X 1 − i=1 pi xi 1 − xi 1 − 2M P − ln pi ln ln x + i n 2 xi (1 − M ) i=1 pi xi i=1 ! n X 1 − 2M − pi xi 2 ln (1 − M ) i=1 ! n n n n n X X X X X pi f ′ (xi ) pi xi pi xi f ′ (xi ) − = pi xi ≤ pi f (xi ) − f i=1 i=1 = n X pi xi " 1 − 2M 2 · i=1 i=1 1 1 − xi xi (1 − xi ) i=1 # (1 − M ) # " n X 1 1 1 − 2M , − − pi xi pi 2 · xi (1 − xi ) (1 − M ) xi i=1 i=1 i=1 n X which is equivalent to     An (p̄, 1 − x̄) 1 − 2M Gn (p̄, 1 − x̄) ln G (p̄, x̄) − ln + ln n 2 Gn (p̄, x̄) An (p̄, x̄) (1 − M ) 1 − 2M − 2 ln An (p̄, x̄) (1 − M )   1 − 2M 1 ≤ 2 − An p̄, 1 − x̄ (1 − M ) "  #   1 1 1 − 2M , − An p̄, −An (p̄, x̄) × 2 An p̄, x̄ x̄ (1 − x̄) (1 − M ) which is equivalent to     Gn (p̄, 1 − x̄) An (p̄, 1 − x̄) ln Gn (p̄, x̄) An (p̄, x̄)    1−2M2   1 An (p̄, x̄) (1−M ) 1 − 2M p̄, 1 − A (p̄, x̄) A ≤ ln + n n 2 Gn (p̄, x̄) x̄ (1 − M )     1 1 +An (p̄, x̄) An p̄, − An p̄, x̄ (1 − x̄) 1 − x̄ ( "  1−2M #   An (p̄, x̄) (1−M )2 1 − 2M 1 = ln · exp 1 − An (p̄, x̄) An p̄, 2 Gn (p̄, x̄) x̄ (1 − M )      1 1 − An p̄, , × exp An (p̄, x̄) An p̄, x̄ (1 − x̄) 1 − x̄ hence the first inequality in (2.9). The second inequality follows by (2.8) applied for the strictly convex function
1−2m 1−t , t ∈ (m, M ). h (t) = (1−m) 2 ln t − ln t We omit the details. 6 S.S. DRAGOMIR AND F.P. SCARMOZZINO References [1] H. ALZER, The inequality of Ky Fan and related results, Acta Applicandae Mathematicae, An International Survey Journal on Applying Mathematics and Mathematical Applications, 38 (1995), 305-354. [2] E. F. BECKENBACH and R. BELLMAN, Inequalities, Springer-Verlag, Berlin, 1961. [3] P.S. BULLEN, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics, 97, Addison Wesley Longman, 1998. [4] S.S. DRAGOMIR, Some refinements of Ky Fan’s inequality, J. Math. Anal. Appl., 163 (1992), 317-321. [5] S.S. DRAGOMIR, A further improvement of Jensen’s inequality, Tamkang J. of Math., 25(1) (1994), 29-36. [6] S.S. DRAGOMIR, A new improvement of Jensen’s inequality, Indian J. Pure Appl. Math., 26(10) (1995), 959-968. [7] S.S. DRAGOMIR and N.M. IONESCU, Some converse of Jensen’s inequality and applications, Anal. Num. Theor. Approx., 23 (1994), 71-78. [8] S.S. DRAGOMIR and D.M. MILOŠEVIĆ, A sequence of mappings connected with Jensen’s inequality and applications, Matematiki Vesnik, 44 (1992), 113-121. [9] C.L. WANG, A Ky Fan inequality of the complementary A. − G. type and its variants, J. Math. Anal. Appl., 73 (1980), 501-505. School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC 8001, Victoria, Australia. E-mail address: sever@matilda.vu.edu.au URL: http://rgmia.vu.edu.au/SSDragomirWeb.html E-mail address: fps@matilda.vu.edu.au URL: http://cams.vu.edu.au/staff/fernandos.html