Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 870950, 4 pages doi:10.1155/2008/870950 Research Article On Logarithmic Convexity for Ky-Fan Inequality Matloob Anwar 1 and J. Pe ˇ cari ´ c 1, 2 1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54660, Pakistan 2 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia Correspondence should be addressed to Matloob Anwar, matloob t@yahoo.com Received 19 November 2007; Accepted 14 February 2008 Recommended by Sever Dragomir We give an improvement and a reversion of the well-known Ky-Fan inequality as well as some re- lated results. Copyright q 2008 M. Anwar and J. Pe ˇ cari ´ c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and preliminaries Let x 1 ,x 2 , ,x n and p 1 ,p 2 , ,p n be real numbers such that x i ∈ 0, 1/2,p i > 0withP n   n i1 p i .LetG n and A n be the weighted geometric mean and arithmetic mean, respectively, defined by G n   n i1 x p i i  1/P n ,andA n 1/P n   n i1 p i x i  x. In particular, consider the above- mentioned means G  n   n i1 1 − x i  p i  1/P n ,andA  n 1/P n   n i1 p i 1 − x i . Then the well- known Ky-Fan inequality is G n G  n ≤ A n A  n . 1.1 It is well known that Ky-Fan inequality can be obtained from the Levinson inequality 1, see also 2, page 71. Theorem 1.1. Let f be a real-valued 3-convex function on 0, 2a,thenfor0 <x i <a, p i > 0, 1 P n n  i1 p i f  x i  − f  1 P n n  i1 p i x i  ≤ 1 P n n  i1 p i f  2a − x i  − f  1 P n n  i1 p i  2a − x i   . 1.2 In 3, the second author proved the following result. 2 Journal of Inequalities and Applications Theorem 1.2. Let f be a real-valued 3-convex function on 0, 2a and x i 1 ≤ i ≤ n n points on 0, 2a,then 1 P n n  i1 p i f  x i  − f  1 P n n  i1 p i x i  ≤ 1 P n n  i1 p i f  a  x i  − f  1 P n n  i1 p i  a  x i   . 1.3 In this paper, we will give an improvement and reversion of Ky-Fan inequality as well as some related results. 2. Main results Lemma 2.1. Define the function ϕ s x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x s ss − 1s − 2 ,s /  0, 1, 2, 1 2 log x, s  0, −xlog x, s  1, 1 2 x 2 log x, s  2. 2.1 Then φ  s xx s−3 , that is, ϕ s x is 3-convex for x>0. Theorem 2.2. Define the function ξ s  1 P n n  i1 p i  ϕ s  2a − x i  − ϕ s  x i  − ϕ s 2a − x ϕ s x2.2 for x i ,p i as in 1.2.Then 1 for all s, t ∈ I ⊆ R, ξ s ξ t ≥ ξ 2 r  ξ 2 st/2 , 2.3 that is, ξ s is log convex in the Jensen sense; 2 ξ s is continuous on I ⊆ R,itisalsolog convex, that is, for r<s<t, ξ t−r s ≤ ξ t−s r ξ s−r t 2.4 with ξ 0  1 2 ln  G a n A n G n A a n  , 2.5 where G a n   n i1 2a − x i  p i  1/P n , A a n 1/P n   n i1 p i 2a − x i . M. Anwar and J. Pe ˇ cari ´ c3 Proof. 1 Let us consider the function fx, u, v, r,s, tfxu 2 ϕ s x2uvϕ r xv 2 ϕ t x, 2.6 where r s  t/2, u, v, r, s, t are reals. f  x  ux s/2−3/2  vx t/2−3/2  2 ≥ 0 2.7 for x>0. This implies that f is 3-convex. Therefore, by 1.2,wehaveu 2 ξ s 2uvξ r v 2 ξ t ≥ 0, that is, ξ s ξ t ≥ ξ 2 r  ξ 2 st/2 . 2.8 This follows that ξ s is log convex in the Jensen sense. 2 Note that ξ s is continuous at all points s  0,s 1, and s  2 since ξ 0  lim s→0 ξ s  1 2 ln  G a n A n G n A a n  , ξ 1  lim s→1 ξ s  1 P n n  i1 p i  x i ln x i −  2a − x i  ln  2a − x i  2a − x ln2a − x − x ln x, ξ 2  lim s→2 ξ s  1 2  1 P n n  i1 p i  2a − x i  2 ln  2a − x i  − x 2 i ln x i  − 2a − x 2 ln2a − xx 2 ln x  . 2.9 Since ξ s is a continuous and convex in Jensen sense, it is log convex. That is, t − r ln ξ s ≤ t − s ln ξ r s − r ln ξ t , 2.10 which completes the proof. Corollary 2.3. For x i , p i as in 1.2, 1 < exp  2ξ 4 3 ξ −3 4  ≤ G a n A n G n A a n ≤ exp  2ξ 3/4 −1 ξ 1/4 3  . 2.11 Proof. Setting s  0,r −1, and t  3inTheorem 1.2,wegetξ 4 0 ≤ ξ 3 −1 ξ 3 or ξ 0 ≤ ξ 3/4 −1 ξ 1/4 3 . 2.12 Again setting s  3,r 0, and t  4inTheorem 1.2,wegetξ 4 3 ≤ ξ 0 ξ 3 4 or ξ 0 ≥ ξ 4 3 ξ −3 4 . 2.13 Combining both inequalities 2.12, 2.13,weget ξ 4 3 ξ −3 4 ≤ ξ 0 ≤ ξ 3/4 −1 ξ 1/4 3 . 2.14 4 Journal of Inequalities and Applications Also we have ξ s positive for s>2; therefore, we have 0 <ξ 4 3 ξ −3 4 ≤ ξ 0 ≤ ξ 3/4 −1 ξ 1/4 3 . 2.15 Applying exponentional function, we get 1 < exp  2ξ 4 3 ξ −3 4  ≤ G a n A n G n A a n ≤ exp  2ξ 3/4 −1 ξ 1/4 3  . 2.16 Remark 2.4. In Corollary 2.3, putting 2a  1 we get an improvement of Ky-Fan inequality. Theorem 2.5. Define the function ρ s  1 P n n  i1 p i  ϕ s  a  x i  − ϕ s  x i  − ϕ s a  x ϕ s x, 2.17 for x i ,p i ,aas for Theorem 1.1.Then 1 for all s, t ∈ I ⊆ R, ρ s ρ t ≥ ρ 2 r  ρ 2 st/2 , 2.18 that is, ρ s is log convex in the Jensen sense; 2 ρ s is continuous on I ⊆ R,itisalsolog convex. That is for r<s<t, ρ t−r s ≤ ρ t−s r ρ s−r t 2.19 with ρ 0  1 2 ln   G n A n G n  A n  , 2.20 where  G n   n i1 a  x i  p i  1/P n ,  A n 1/P n   n i1 p i a  x i . Proof. The proof is similar to the proof of Theorem 2.2. Remark 2.6. Let us note that similar results for difference of power means were recently ob- tained by Simic in 4. References 1 N. Levinson, “Generalization of an inequality of Ky-Fan,” Journal of Mathematical Analysis and Applica- tions, vol. 8, no. 1, pp. 133–134, 1964. 2 J. Pe ˇ cari ´ c, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992. 3 J. Pe ˇ cari ´ c, “An inequality for 3-convex functions,” Journal of Mathematical Analysis and Applications, vol. 90, no. 1, pp. 213–218, 1982. 4 S. Simic, “On logarithmic convexity for differences of power means,” Journal of Inequalities and Applica- tions, vol. 2007, Article ID 37359, 8 pages, 2007. . Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 870950, 4 pages doi:10.1155/2008/870950 Research Article On Logarithmic Convexity for Ky-Fan Inequality Matloob. Boston, Mass, USA, 1992. 3 J. Pe ˇ cari ´ c, “An inequality for 3-convex functions,” Journal of Mathematical Analysis and Applications, vol. 90, no. 1, pp. 213–218, 1982. 4 S. Simic, On logarithmic. the second author proved the following result. 2 Journal of Inequalities and Applications Theorem 1.2. Let f be a real-valued 3-convex function on 0, 2a and x i 1 ≤ i ≤ n n points on 0,