On Logarithmic Convexity for Ky-Fan Inequality
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Research Article On Logarithmic Convexity for Ky-Fan Inequality ˇ c´ 1, 2 Matloob Anwar1 and J. Pecari 1 2
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54660, Pakistan Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
Correspondence should be addressed to Matloob Anwar, matloob [email protected] 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 related results. Copyright q 2008 M. Anwar and J. Peˇcari´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 x1 , x2 , . . . , xn and p1 , p2 , . . . , pn be real numbers such that xi ∈ 0, 1/2, pi > 0 with Pn n A be the weighted geometric mean and arithmetic mean, respectively, i1 pi . Let Gn and n p defined by Gn ni1 xi i 1/Pn , and An 1/Pn ni1 pi xi x. In particular, consider the above mentioned means Gn ni1 1 − xi pi 1/Pn , and An 1/Pn ni1 pi 1 − xi . Then the wellknown Ky-Fan inequality is Gn An ≤ . Gn An
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, then for 0 < xi < a, pi > 0, n 1 p i f xi − f Pn i1
n 1 p i xi Pn i1
≤
n 1 pi f 2a − xi − f Pn i1
In 3, the second author proved the following result.
n 1 pi 2a − xi . Pn i1
1.2
2
Journal of Inequalities and Applications
Theorem 1.2. Let f be a real-valued 3-convex function on 0, 2a and xi 1 ≤ i ≤ n n points on 0, 2a, then n 1 p i f xi − f Pn i1
n 1 p i xi Pn i1
n 1 p i f a xi − f ≤ Pn i1
n 1 p i a xi . Pn i1
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 ⎧ xs ⎪ ⎪ , s / 0, 1, 2, ⎪ ⎪ ss − 1s − 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 log x, s 0, ϕs x 2 ⎪ ⎪ ⎪ ⎪ −xlog x, s 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 x2 log x, s 2. 2
2.1
Then φs x xs−3 , that is, ϕs x is 3-convex for x > 0. Theorem 2.2. Define the function ξs
n 1 pi ϕs 2a − xi − ϕs xi − ϕs 2a − x ϕs x Pn i1
2.2
for xi , pi as in 1.2. Then 1 for all s, t ∈ I ⊆ R, 2 , ξs ξt ≥ ξr2 ξst/2
2.3
that is, ξs is log convex in the Jensen sense; 2 ξs is continuous on I ⊆ R, it is also log convex, that is, for r < s < t, ξst−r ≤ ξrt−s ξts−r
2.4
a Gn An 1 , ξ0 ln 2 Gn Aan
2.5
with
where Gan ni1 2a − xi pi 1/Pn , Aan 1/Pn ni1 pi 2a − xi .
M. Anwar and J. Peˇcari´c
3
Proof. 1 Let us consider the function fx, u, v, r, s, t fx u2 ϕs x 2uvϕr x v2 ϕt x,
2.6
where r s t/2, u, v, r, s, t are reals. 2 f x uxs/2−3/2 vxt/2−3/2 ≥ 0
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