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Hindawi Publishing CorporationJournal of Inequalities and Applications Volume 2008, Article ID 870950, 4 pages doi:10.1155/2008/870950 Research Article On Logarithmic Convexity for Ky-Fa

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 ˇcari ´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 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, , x n andp1, p2, , p n be real numbers such that x i ∈ 0, 1/2, p i > 0 with P n 

n

i1 p i Let G n and A n be the weighted geometric mean and arithmetic mean, respectively, defined byG n n

i1 x p i

i 1/P n, andA n  1/P nn

i1 p i x i  x In particular, consider the

above-mentioned means G

n  n i1 1 − x ip i1/P n, and A

n  1/P nn i1 p i 1 − x i Then the well-known Ky-Fan inequality is

G n

G

A

It is well known that Ky-Fan inequality can be obtained from the Levinson inequality1, see also2, page 71

Theorem 1.1 Let f be a real-valued 3-convex function on 0, 2a, then for 0 < x i < a, p i > 0,

1

P n

n



i1

p i fx i

− f

 1

P n

n



i1

p i x i



≤ 1

P n

n



i1

p i f2a − x i

− f

 1

P n

n



i1

p i

2a − x i

In3, 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



i1

p i fx i− f

 1

P n

n



i1

p i x i



≤ 1

P n

n



i1

p i fa  x i− f

 1

P n

n



i1

p ia  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

ss − 1s − 2 , s / 0, 1, 2,

1

2logx, s  0,

−xlog x, s  1,

1

2x2logx, 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 P1

n

n



i1

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 in1.2 Then

1 for all s, t ∈ I ⊆ R,

ξ s ξ t ≥ ξ2

r  ξ2

that is, ξ s is log convex in the Jensen sense;

2 ξ s is continuous on I ⊆ R, it is also log convex, that is, for r < s < t,

ξ t−r

r ξ s−r

with

ξ0 1

2ln

a

n A n

G n A a



where G a

n n i1 2a − x ip i1/P n , A a

n  1/P nn i1 p i 2a − x i .

M Anwar and J Peˇcari´c 3

Proof. 1 Let us consider the function

fx, u, v, r, s, t  fx  u2ϕ s x  2uvϕ r x  v2ϕ t x, 2.6 wherer  s  t/2, u, v, r, s, t are reals.

fx ux s/2−3/2  vx t/2−3/22

forx > 0 This implies that f is 3-convex Therefore, by 1.2, we have u2ξ s 2uvξ r v2ξ t≥ 0, that is,

ξ s ξ t ≥ ξ2

r  ξ2

This follows thatξ sis log convex in the Jensen sense

2 Note that ξ sis continuous at all pointss  0, s  1, and s  2 since

ξ0 lim

s→0 ξ s 1

2ln

a

n A n

G n A a



,

ξ1 lim

s→1 ξ s P1

n

n



i1

p i

x ilnx i−2a − x i

ln

2a − x i

 2a − x ln2a − x − x ln x,

ξ2 lim

s→2 ξ s 1

2

 1

P n

n



i1

p i

2a − x i2

ln

2a − x i

− x2

i lnx i

− 2a − x2ln2a − x  x2lnx



.

2.9 Sinceξ sis 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 in1.2,

1< exp2ξ4

3ξ−3 4



≤ G G a n A n

n A a

n ≤ exp2ξ3/4

−1 ξ1/4

3



Proof Setting s  0, r  −1, and t  3 inTheorem 1.2, we getξ4

0 ≤ ξ3

−1ξ3or

ξ0 ≤ ξ3/4

−1 ξ1/4

Again settings  3, r  0, and t  4 inTheorem 1.2, we getξ4

3 ≤ ξ0ξ3

4or

ξ0≥ ξ4

3ξ−3

Combining both inequalities2.12, 2.13, we get

ξ4

3ξ−3

4 ≤ ξ0≤ ξ3/4

−1 ξ1/4

4 Journal of Inequalities and Applications Also we haveξ s positive fors > 2; therefore, we have

0< ξ4

3ξ−3

4 ≤ ξ0≤ ξ3/4

−1 ξ1/4

Applying exponentional function, we get

1< exp2ξ4

3ξ−3 4



≤ G G a n A n

n A a ≤ exp2ξ3/4

−1 ξ1/4

3



Remark 2.4 InCorollary 2.3, putting 2a  1 we get an improvement of Ky-Fan inequality.

Theorem 2.5 Define the function

ρ s  P1

n

n



i1

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 , a as for Theorem 1.1 Then

1 for all s, t ∈ I ⊆ R,

ρ s ρ t ≥ ρ2

r  ρ2

that is, ρ s is log convex in the Jensen sense;

2 ρ s is continuous on I ⊆ R, it is also log convex That is for r < s < t,

ρ t−r

with

ρ0 1

2ln

n A n

G n An



where  G n n i1 a  x ip i1/P n ,  A n  1/P nn i1 p i a  x i .

Proof The proof is similar to the proof ofTheorem 2.2

Remark 2.6 Let us note that similar results for difference of power means were recently ob-tained by Simic in4

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 Peˇcari´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 Peˇcari´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.