On the Ky Fan Inequality

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ON THE KY FAN INEQUALITY

S.S. DRAGOMIR AND F.P. SCARMOZZINO

Abstract. Some inequalities related to the Ky Fan and C.-L. Wang inequal-ities 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In 1961, E.F. Beckenbach and R. Bellman published in their well known book “Inequalities” the following “unpublished result due to Ky Fan” [2, p. 5] (see also [1, p. 150]).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:

(1.1)

" n

Y

i=1

xi

, n

Y

i=1

(1−xi)

#n1

≤

n

X

i=1

xi

, n

X

i=1

(1−xi)

with equality only ifx1=· · ·=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

(1.2) An(¯p,x¯)

An(¯p,1−¯x)

≥ Gn(¯p,x¯) Gn(¯p,1−x¯)

,

where pi > 0 (i= 1, . . . , n) with Pn_i=1pi = 1 and An(¯p,x¯) := Pn_i=1pixi is the

weighted arithmetic mean,Gn(¯p,x¯) :=Qn_i=1xpii is the weighted geometric mean.

The equality holds in (1.2) iffx1=· · ·=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].

2. The Results

The following result holds.

Theorem 3. Assume that0< m≤xi≤M ≤1₂,(i= 1, . . . , n),pi >0 (i= 1, . . . , n),

withPn

i=1pi= 1,then we have the inequalities:

(2.1) An(¯p,x¯)

Gn(¯p,x¯)

≥

_A

n(¯p,x¯)

Gn(¯p,x¯)

M

2 (1−M)2

≥An(¯p,1−x¯) Gn(¯p,1−x¯)

≥

_A

n(¯p,x¯)

Gn(¯p,x¯)

m

2 (1−m)2

≥1.

The equality will hold in all inequalities iffx1=· · ·=xn.

Date: January 11, 2001.

1991Mathematics Subject Classification. Primary 26D15, 26D10.

Key words and phrases. Ky Fan’s Inequality, Weighted arithmetic and geometric means.

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Proof. The first and the last inequality in (2.1) follow by the fact that An( ¯p,¯x)

Gn( ¯p,¯x) ≥1

(by the weighted arithmetic mean - geometric mean inequality), m ∈ 0,1₂ and

M ∈ 0,1₂

.

We define the function f : (0,1)→R, f(t) = ln 1−tt

+αlnt with α∈R. We

have

f0(t) =− 1 t(1−t)+

α

t, t∈(0,1),

f00(t) = 1−2t [t(1−t)]2 −

α t2 =

1

t2

"

1−2t

(1−t)2 −α

#

, t∈(0,1).

If we consider the function g : (0,1) → R, g(t) = _(1−t)1−2t2, then g0(t) =

2t(t−1) (t−1)4,

showing that the functiongis monotonically strictly decreasing on (0,1). Consequently fort∈(m, M), we have

(2.2) 1−2M

(1−M)2 =g(M)≤g(t)≤g(m) =

1−2m

(1−m)2.

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 functionf : (m, M)→R, f(t) =

ln 1−t_t +αlnt, withα≤_(1−M1−2M₎2, we deduce

n

X

i=1

pi

ln

₁₋_x

i

xi

+αlnxi

= n

X

i=1

pif(xi)≥f n

X

i=1

pixi

!

= ln

₁₋Pn

i=1pixi

Pn

i=1pixi

+αln n

X

i=1

pixi

!

,

which is equivalent to

ln

_G

n(¯p,1−¯x)

Gn(¯p,x¯)

+αlnGn(¯p,x¯)≥ln

_A

n(¯p,1−x¯)

An(¯p,x¯)

+αlnAn(¯p,x¯)

or, moreover, to

ln

_G

n(¯p,x¯)

An(¯p,x¯)

α

≥ln

_A

n(¯p,1−¯x)

An(¯p,x¯)

_G

n(¯p,1−x¯)

Gn(¯p,x¯)

,

that is,

(2.3)

_G

n(¯p,x¯)

An(¯p,x¯)

α−1

≥ An(¯p,1−x¯) Gn(¯p,1−x¯)

.

Now, we observe that the inequality (2.3) is the best possible ifαis maximal, i.e.,

α= 1−2M

(1−M)2, getting

_G

n(¯p,x¯)

An(¯p,x¯)

₍₁1₋−_M2M₎₂−1

≥ An(¯p,1−x¯) Gn(¯p,1−x¯)

,

which is clearly equivalent to the second inequality in (2.1).

The third inequality is produced in a similar fashion, using the functionh(t) =

βlnt−ln 1−t_t

which is strictly convex on (m, M) ifβ ≥ 1−2m (1−m)2.

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We omit the details.

Remark 1. Since Wang’s inequality (1.2) is equivalent to:

(2.4) An(¯p,x¯)

Gn(¯p,x¯)

≥ An(¯p,1−x¯) Gn(¯p,1−x¯)

,

then the first part of (2.1) may be seen as a refinement of Wang’s result while the second part

(2.5) An(¯p,1−x¯)

Gn(¯p,1−x¯)

≥

_A

n(¯p,x¯)

Gn(¯p,x¯)

m

2 (1−m)2

can be considered a counterpart of (1.2).

Now, let us recall the Lah-Ribari´c inequality for convex functions (see for example [3, p. 140]).

If f : [a, b] ⊂ R→R is convex on [a, b], xi ∈ [a, b], pi ≥ 0 (i= 1, . . . , n) and

Pn

i=1pi= 1, then

(2.6)

n

X

i=1

pif(xi)≤

b−Pn

i=1pixi

b−a ·f(a) +

Pn

i=1pixi−a

b−a ·f(b).

Now, we can state and prove the following inequality related to the Ky Fan result.

Theorem 4. Assume that 0 < m ≤ xi ≤ M ≤ ₂1, pi > 0 (i= 1, . . . , n) with

Pn

i=1pi= 1, then we have the inequalities:

₁

−m

m(1−mm)

2

M−MAn−m( ¯p,x¯) ₁

−M

M(1−mm)

2

AnM( ¯p,−¯xm)−m

·Gn(¯p,x¯)(

m

1−m)

2

(2.7)

≤ Gn(¯p,1−x¯)

≤

₁₋_m

m(1−MM)

2

M−MAn−( ¯mp,x¯) ₁₋_M

M(1−MM)

2

AnM( ¯p,−x¯m)−m

Gn(¯p,x¯)(

M

1−M)

2 .

Proof. From the proof of Theorem 3, we know that the function f : (m, M) ⊂

0,1₂→R,f(t) = ln 1−tt

+_(1−M1−2M₎2 lnt is strictly convex on (m, M). Now, if we

apply the Lah-Ribari´c inequality forf as above,a=mandb=M, we get:

n

X

i=1

pi

"

ln

₁₋_x

i

xi

+ 1−2M (1−M)2lnxi

#

= n

X

i=1

pif(xi)≤

M −Pn

i=1pixi

M−m f(m) +

Pn

i=1pixi−m

M −m f(M)

= M −

Pn

i=1pixi

M−m

"

ln

₁₋_m

m

+ 1−2M (1−M)2lnm

#

+

Pn

i=1pixi−m

M−m

"

ln

1−M M

+ 1−2M (1−M)2lnM

#

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ln

_G

n(¯p,1−x¯)

Gn(¯p,x¯)

+ 1−2M

(1−M)2lnGn(¯p,x¯)

≤ M −An(¯p,x¯)

M−m

ln

₁

−m m

+ ln (m)

1−2M

(1−M)2

+An(¯p,¯x)−m

M−m

ln

₁₋_M

M

+ ln (M)

1−2M

(1−M)2

,

that is,

Gn(¯p,1−x¯)

Gn(¯p,x¯)

·[Gn(¯p,x¯)]

1−2M

(1−M)2

≤

(1−m)m n ₁₋₂_M

(1−M)2−1

oM−MAn−( ¯mp,x¯)

·

(1−M)M n ₁₋₂_M

(1−M)2−1

oAnM( ¯p,−x¯m)−m

from which we obtain the second inequality in (2.7).

To prove the first inequality, we apply the Lah-Ribari´c inequality for the function

h: (m, M)→_R,h(t) = 1−2m

(1−m)2lnt−ln 1−t_t

which is strictly convex on (m, M). We omit the details.

Finally, let us recall Dragomir-Ionescu’s inequality for differentiable convex func-tions (see [7])

0 ≤

n

X

i=1

pif(xi)−f n

X

i=1

pixi

!

(2.8)

≤

n

X

i=1

pixif0(xi)− n

X

i=1

pixi n

X

i=1

pif0(xi)

providedf : (a, b)⊆R→Ris differentiable convex on (a, b),xi∈(a, b) andpi>0 (i= 1, . . . , n) withPn

i=1pi= 1.

Iff is strictly convex on (a, b), then the equality holds in (2.8) iffx1=· · ·=xn, we may state the following result.

Theorem 5. With the assumptions of Theorem 4, we have

exp

An(¯p,x¯)An

¯

p, 1

¯

x(1−x¯)

−An

¯

p, 1

1−x¯

(2.9)

×

"

1−2M

(1−M)2

1−An(¯p,x¯)An

¯

p,1

¯

x

#

×

An(¯p,x¯)

Gn(¯p,x¯)

₍₁1−₋_M2M₎₂

≥

_G

n(¯p,1−x¯)

Gn(¯p,x¯)

An(¯p,1−x¯)

An(¯p,x¯)

≥ exp

An(¯p,x¯)An

¯

p, 1

¯

x(1−x¯)

−An

¯

p, 1

1−x¯

×

"

1−2m

(1−m)2

1−An(¯p,x¯)An

¯

p,1

¯

x

#

×

An(¯p,x¯)

Gn(¯p,x¯)

₍₁1−₋_m2m₎₂

,

where _x1_¯ denotes the vector _x1 1, . . . ,

1 xn

, y¯·z¯ := (y1z1, . . . , znyn), and ¯x∈ Rn,

¯

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Proof. Since the function f : (m, M)⊂ 0,₂1→ R, f(t) = ln 1−tt

+_(1−M1−2M₎2lnt

is strictly convex on (m, M), by (2.8) we may state that

n X i=1 pi " ln

1−xi

xi

+ 1−2M (1−M)2lnxi

#

−ln

₁

−Pn

i=1pixi

Pn

i=1pixi

− 1−2M

(1−M)2ln n

X

i=1

pixi

!

= n

X

i=1

pif(xi)−f n

X

i=1

pixi

!

≤

n

X

i=1

pixif0(xi)− n

X

i=1

pixi n

X

i=1

pif0(xi)

= n

X

i=1

pixi

"

1−2M

(1−M)2 · 1

xi

− 1

xi(1−xi)

#

−

n

X

i=1

pixi n

X

i=1

pi

"

1−2M

(1−M)2· 1

xi

− 1

xi(1−xi)

#

,

ln

_G

n(¯p,1−¯x)

Gn(¯p,x¯)

+ 1−2M

(1−M)2lnGn(¯p,x¯)−ln

_A

n(¯p,1−x¯)

An(¯p,x¯)

− 1−2M

(1−M)2lnAn(¯p,x¯)

≤ 1−2M

(1−M)2 −An

¯

p, 1

1−x¯

−An(¯p,x¯)×

"

1−2M

(1−M)2An

¯ p,1 ¯ x

−An

¯

p, 1

¯

x(1−x¯)

#

,

ln Gn(¯p,1−x¯)

Gn(¯p,x¯)

An(¯p,1−¯x)

An(¯p,x¯)

≤ ln

_A

n(¯p,x¯)

Gn(¯p,x¯)

₍₁1−₋2_MM₎₂

+ 1−2M (1−M)2

1−An(¯p,x¯)An

¯ p,1 ¯ x

+An(¯p,x¯)An

¯

p, 1

¯

x(1−x¯)

−An

¯

p, 1

1−x¯

= ln

(

_A

n(¯p,¯x)

Gn(¯p,x¯)

₍₁1−₋2_MM₎₂

·exp

"

1−2M

(1−M)2

1−An(¯p,¯x)An

¯ p,1 ¯ x # ×exp

An(¯p,x¯)An

¯

p, 1

¯

x(1−x¯)

−An

¯

p, 1

1−x¯

,

hence the first inequality in (2.9).

The second inequality follows by (2.8) applied for the strictly convex function

h(t) = _(1−m)1−2m2lnt−ln

1−t t

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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. MILOˇSEVI ´C, 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.

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URL:http://rgmia.vu.edu.au/SSDragomirWeb.html

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