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volume 4, issue 4, article 76, 2003.

Received 17 August, 2002;

accepted 3 June, 2003.

Communicated by:K.B. Stolarsky

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Journal of Inequalities in Pure and Applied Mathematics

CERTAIN BOUNDS FOR THE DIFFERENCES OF MEANS

PENG GAO

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.

EMail:penggao@umich.edu

2000c Victoria University ISSN (electronic): 1443-5756 089-02

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Certain Bounds for the Differences of Means

Peng Gao

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J. Ineq. Pure and Appl. Math. 4(4) Art. 76, 2003

http://jipam.vu.edu.au

Abstract

LetP_n,r(x)be the generalized weighted power means. We consider bounds for the differences of means in the following form:

max (Cu,v,β

x^2β−α₁ ,Cu,v,β

x^2β−α_n )

σ_n,w⁰_,β≥P_n,u^α −P_n,v^α α ≥min

(Cu,v,β

x^2β−α₁ ,Cu,v,β

x^2β−α_n )

σ_n,w,β.

Hereβ6= 0, σ_n,t,β(x) =P_n

i=1ω_i[x^β_i −P_n,t^β(x)]²andC_u,v,β=^u−v_2β2 . Some similar inequalities are also considered. The results are applied to inequalities of Ky Fan’s type.

2000 Mathematics Subject Classification:Primary 26D15, 26D20

Key words: Ky Fan’s inequality, Levinson’s inequality, Generalized weighted power means, Mean value theorem

The author thanks the referees for their many valuable comments and suggestions.

These greatly improved the presentation of the paper.

1. Introduction

LetP_n,r(x)be the generalized weighted power means:P_n,r(x) = (Pn

i=1ω_ix^r_i)¹^r, where ω_i > 0,1 ≤ i ≤ n with Pn

i=1ω_i = 1 andx = (x₁, x₂, . . . , x_n). Here P_n,0(x)denotes the limit ofP_n,r(x)asr →0⁺. In this paper, we always assume that0< x₁ ≤x₂ ≤ · · · ≤x_n. We write

σ_n,t,β(x) =

n

X

i=1

ω_ih

x^β_i −P_n,t^β (x)i2

and denoteσ_n,tasσ_n,t,1. We let

A_n(x) =P_n,1(x), G_n(x) =P_n,0(x), H_n(x) =Pn,−1(x)

and we shall writeP_n,rforP_n,r(x),A_nforA_n(x)and similarly for other means when there is no risk of confusion.

We consider upper and lower bounds for the differences of the generalized weighted means in the following forms (β 6= 0):

max

C_u,v,β

x^2β−α₁ ,C_u,v,β x^2β−αn

σ_n,w⁰_,β ≥ P_n,u^α −P_n,v^α (1.1) α

≥min

C_u,v,β

x^2β−α₁ , C_u,v,β x^2β−αn

σ_n,w,β, whereC_u,v,β = ^u−v_2β2. If we setx₁ =· · ·=xn−1 6=x_n, then we conclude from

xLim1→xn

P_n,u^α −P_n,v^α

ασ_n,w,β = u−v 2β²x^2β−αn

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that C_u,v,β is best possible. Here we define (P_n,u⁰ −P_n,v⁰ )/0 = ln(P_n,u/P_n,v), the limit of(P_n,u^α −P_n,v^α )/αasα→0.

In what follows we will refer to (1.1) as(u, v, α, β, w, w⁰). D.I. Cartwright and M.J. Field [8] first proved the case (1,0,1,1,1,1). H. Alzer [4] proved (1,0,1,1,1,0)and [5](1,0, α,1,1,1)withα ≤1. A.McD. Mercer [13] proved the right-hand side inequality with smaller constants for α = β = u = 1, v =−1, w=±1.

There is a close relationship between (1.1) and the following Ky Fan inequal- ity, first published in the monograph Inequalities by Beckenbach and Bellman [7].(In this section, we set A⁰_n = 1−A_n, G⁰_n = Qn

i=1(1−x_i)^ωⁱ. For general definitions, see the beginning of Section3.)

Theorem 1.1. Forx_i ∈ 0,¹₂

,

(1.2) A⁰_n

G⁰_n ≤ A_n G_n with equality holding if and only ifx₁ =· · ·=x_n.

P. Mercer [15] observed that the validity of(1,0,1,1,1,1)leads to the following refinement of the additive Ky Fan inequality:

Theorem 1.2. Let0< a≤x_i ≤b <1 (1≤i≤n, n≥2). Fora6=bwe have

(1.3) a

1−a < A⁰_n−G⁰_n A_n−G_n < b

1−b.

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Thus, by a result of P. Gao [9], it yields the following refinement of Ky Fan’s inequality, first proved by Alzer [6]:

A_n G_n

(1−a^a )²

≤ A⁰_n G⁰_n ≤

A_n G_n

(1−b^b )² .

For an account of Ky Fan’s inequality, we refer the reader to the survey article [2] and the references therein.

The additive Ky Fan’s inequality for generalized weighted means is a con- sequence of (1.1). Since it does not always hold (see [9]), it follows that (1.1) does not hold for arbitrary(u, v, α, β, w, w⁰).

Our main result is a theorem that shows the validity of (1.1) for someα, β, u, v, w, w⁰. We apply it in Section3to obtain further refinements and general- izations of inequalities of Ky Fan’s type.

One can obtain further refinements of (1.1). Recently, A.McD. Mercer proved the following theorem [14]:

Theorem 1.3. Ifx₁ 6=x_n, n≥2, then (1.4) G_n−x₁

2x₁(A_n−x₁)σ_n,1 > A_n−G_n> x_n−G_n

2x_n(x_n−A_n)σ_n,1. We generalize this in Section2.

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2. The Main Theorem

Theorem 2.1. 1,^s_r,1,^γ_r,^t_r,^t_r⁰

, r 6= s, r 6= 0, γ 6= 0 holds for the following three cases:

1. ^s_γ ≤ _γ^r ≤2,1≥ _γ^t,^t_γ⁰ ≥ _γ^s ≥ _γ^r −1;

2. ^r_γ ≥2,_γ^r −1≥ _γ^s ≥ _γ^t,^t_γ⁰ ≥1;

3. ^r_γ ≤ _γ^s ≤ _γ^t,^t_γ⁰ ≤1,

with equality holding if and only ifx₁ =· · ·=x_nfor all the cases.

Proof. Let γ = 1and r 6= s. We will show that (1.1) holds for the following three cases:

1. s≤r≤2,1≥t, t⁰ ≥s≥r−1;

2. r≥2, r−1≥s≥t, t⁰ ≥1;

3. r≤s≤t, t⁰ ≤1.

For case (1), consider the right-hand side inequality of (1.1) and let (2.1) D_n(x) = A_n−P_n,^s

r −r(r−s) 2x

2 r−1 n

n

X

i=1

ω_i x

1 r

i −P

1 r

n,_r^t

2

.

We want to show thatD_n ≥0here. We can assume thatx₁ < x₂ <· · ·< x_n and use induction. The case n = 1is clear, so assume that the inequality holds

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forn−1variables. Then (2.2) 1

ω_n ·∂D_n

∂x_n = 1−

"

P_n,^s

r

x_n

¹_r#^r−s

−(r−s) 1− P_n,^t

r

x_n ¹r!

+S, where

S = (2−r)(r−s) 2ω_nx

2 r−2 n

n

X

i=1

ω_i x

1 r

i −P

1 r

n,_r^t

²

+ (r−s) P

1−t r

n,_r^t

x

2−t

nr

P

1 r

n,¹_r −P

1 r

n,^t_r

.

Thus, whens≤r ≤2, t≤1,S ≥0.

Now by the mean value theorem 1−

"

P_n,^s

r

x_n

¹_r#^r−s

= (r−s)η^r−s−1 1− P_n,^s

r

x_n ¹_r!

≥(r−s) 1− P_n,^s

r

x_n ¹_r!

forr≥s≥r−1with min

( 1,

P_n,^s

r

x_n ¹_r)

≤η≤max (

1, P_n,^s

r

x_n ¹_r)

.

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This implies

1−

"

P_n,^s

r

x_n

¹_r#^r−s

−(r−s) 1− P_n,^t

r

x_n ¹r!

≥(r−s)

"

P_n,^t

r

x_n

1 r

− P_n,^s

r

x_n ¹_r#

, which is positive ifs≤t.

Thus fors ≤ r ≤ 2, 1 ≥ t ≥ s ≥ r−1, we have ^∂D_∂xⁿ

n ≥ 0. By lettingx_n tend tox_n−1, we haveD_n ≥D_n−1(with weightsω₁, . . . , ω_n−2, ω_n−1 +ω_n) and thus the right-hand side inequality of (1.1) holds by induction. It is also easy to see that equality holds if and only ifx₁ =· · ·=x_n.

Now consider the left-hand side inequality of (1.1) and write (2.3) E_n(x) =A_n−P_n,^s

r − r(r−s) 2x

2 r−1 1

n

X

i=1

ω_i

x

1 r

i −P

1 r

n,^t_r⁰

2

.

Now _ω¹

1

∂En

∂x1 has an expression similar to (2.2) withx_n ↔x₁, ω_n ↔ω₁, t↔ t⁰. It is then easy to see under the same condition, ^∂E_∂xⁿ

1 ≥0. Thus the left-hand side inequality of (1.1) holds by a similar induction process with the equality holding if and only ifx₁ =· · ·=x_n.

Similarly, we can showD_n(x) ≤ 0, E_n(x) ≥ 0for cases (2) and (3) with equality holding if and only ifx₁ =· · ·=x_nfor all the cases.

Now for an arbitraryγ, a change of variablesy → y/γ fory = r, s, t, t⁰ in the above cases leads to the desired conclusion.

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In what follows our results often include the casesr = 0 ors = 0 and we will leave the proofs of these special cases to the reader since they are similar to what we give in the paper.

Corollary 2.2. Forr > s,min{1, r−1} ≤s≤max{1, r−1}andmin{1, s} ≤ t, t⁰ ≤ max{1, s}, (r, s, r,1, t, t⁰)holds. Fors ≤ r ≤ t, t⁰ ≤ 1, (r, s, s,1, t, t⁰) holds, with equality holding if and only ifx₁ =· · ·=x_nfor all the cases.

Proof. This follows from takingγ = 1in Theorem2.1 and another change of variables: x₁ → min{x^r₁, x^r_n}, x_n → max{x^r₁, x^r_n} and x_i = x^r_i for 2 ≤ i ≤ n−1ifn≥3and exchangingrandsfor the cases > r.

We remark here sinceσ_n,t⁰ = σ_n,t+ (2A_n−P_n,t−P_n,t⁰)(P_n,t−P_n,t⁰), we have σ_n,1 ≤ σ_n,t for t 6= 1 and σ_n,t ≤ σ_n,t⁰ fort⁰ ≤ t ≤ 1, σ_n,t ≥ σ_n,t⁰ for t ≥ t⁰ ≥ 1. Thus the optimal choices for the set {t, t⁰} will be{1, s} for the case(r, s, r,1, t, t⁰)and{1, r}for the case(r, s, s,1, t, t⁰).

Our next two propositions give relations between differences of means with different powers:

Proposition 2.3. Forl−r≥t−s≥0, l6=t, x_i ∈[a, b], a >0,

(2.4)

(r−s) (l−t)

1 a^l−r ≥

(P_n,r^r −P_n,s^r )/r (P_n,l^l −P_n,t^l )/l

≥

(r−s) (l−t)

1 b^l−r.

Except for the trivial casesr =sor(l, t) = (r, s), the equality holds if and only ifx₁ =· · ·=x_n, where we define0/0 = x^r−l_i for anyi.

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Proof. This is a generalization of a result A.McD. Mercer [12]. We may assume thatx1 =a, xn=band consider

D(x) = P_n,r^r −P_n,s^r − r(r−s)

l(l−t)x^l−r_n (P_n,l^l −P_n,t^l ), E(x) =P_n,r^r −P_n,s^r − r(r−s)

l(l−t)x^l−r₁ (P_n,l^l −P_n,t^l ).

We will show thatD_n·E_n ≤0. Supposer−s≥0here; the caser−s≤0is similar. We have

x^1−r_n rω_n · ∂D_n

∂x_n = 1− P_n,s

x_n ^r−s

− r−s l−t



1−

"

P_n,t x_n

^r−s#_r−s^l−t

+S, where

S= (r−s)(l−r)

l(l−t)x^l−2_n ω_n(P_n,l^l −P_n,t^l )≥0.

Now by the mean value theorem 1−

"

P_n,t x_n

r−s#_r−s^l−t

= l−t

r−sη^l−t−r+s 1− P_n,t

x_n

r−s! ,

where ^P_x^n,t

n < η <1and x^1−r_n

rω_n

∂D_n

∂x_n ≥1− P_n,s

x_n r−s

− 1− P_n,t

x_n

r−s!

≥0

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sincet≥s.

Similarly, we have ^x

1−r 1

rω1

∂En

∂x1 ≥ 0 and by a similar induction process as the one in the proof of Theorem 2.1, we have D_n·E_n ≤ 0. This completes the proof.

By taking l = 2, t = 0, r = 1, s = −1 in the proposition, we get the following inequality:

(2.5) 1

2x₁(P_n,2² −G²_n)≥A_n−H_n≥ 1

2x_n(P_n,2² −G²_n)

and the right-hand side inequality above gives a refinement of a result of A.McD.

Mercer [13].

Proposition 2.4. Forr > s, α > β,

(2.6) x^β−α₁ ≥P_n,s^β−α ≥ (P_n,r^β −P_n,s^β )/β

(P_n,r^α −P_n,s^α )/α ≥P_n,r^β−α ≥x^β−α_n

with equality holding if and only ifx₁ =· · ·=x_n, where we define0/0 = x^β−α_i for anyi.

Proof. By the mean value theorem,

P_n,r^β −P_n,s^β = (P_n,r^α )^β/α−(P_n,s^α )^β/α = β

αη^β−α(P_n,r^α −P_n,s^α ), whereP_n,s < η < P_n,rand (2.6) follows.

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We apply (2.6) to the case (1,0,1,1,1,1)to see that (1,0, α,1,1,1) holds withα ≤ 1, a result of Alzer [5]. We end this section with a generalization of (1.4) and leave the formulation of similar refinements to the reader.

Theorem 2.5. Ifx₁ 6=x_n, n≥2, then for1> s≥0 (2.7) P_n,s^1−s−x^1−s₁

2x^1−s₁ (A_n−x₁)σ_n,1 > A_n−P_n,s > x^1−s_n −P_n,s^1−s 2x^1−s_n (x_n−A_n)σ_n,1.

Proof. We prove the right-hand inequality; the left-hand side inequality is sim- ilar. Let

D_n(x) = (x_n−A_n)(A_n−P_n,s)− x^1−s_n −P_n,s^1−s 2x^1−s_n σ_n,1. We show by induction thatD_n≥0. We have

∂D_n

∂x_n = (1−ω_n)(A_n−P_n,s)−1−s 2x_n

P_n,s x_n

1−s 1−

x_n P_n,s

s

ω_n

σ_n,1

≥(1−ω_n)

A_n−P_n,s− 1−s 2xn

σ_n,1

≥0,

where the last inequality holds by Theorem2.1. By an induction process similar to the one in the proof of Theorem2.1, we have Dn ≥ 0. Since not all thexi’s are equal, we get the desired result.

Corollary 2.6. For1> s≥0,

(2.8) 1−s

2x₁ · P_n,s

A_n σ_n,1 ≥A_n−P_n,s ≥ 1−s 2x_n σ_n,s, with equality holding if and only ifx1 =· · ·=xn.

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Proof. By Theorem2.5, we only need to show ^P

1−s n,s −x^1−s₁

2x^1−s₁ (An−x1) ≤ ^1−s_2x

1

Pn,s

An and this is easily verified by using the mean value theorem.

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3. Applications to Inequalities of Ky Fan’s Type

Let f(x, y) be a real function. We regardy as an implicit function defined by f(x, y) = 0and fory = (y1, . . . , yn), letf(xi, yi) = 0,1 ≤ i ≤ n. We write P_n,r⁰ = P_n,r(y) with A⁰_n = P_n,1⁰ , G⁰_n = P_n,0⁰ , H_n⁰ = P_n,−1⁰ . Furthermore, we write x₁ = a > 0andx_n = b so thatx_i ∈ [a, b]withy_i ∈ [a⁰, b⁰], a⁰ > 0 and require thatf_x⁰, f_y⁰ exist forxi ∈[a, b],yi ∈[a⁰, b⁰].

To simplify expressions, we define:

(3.1) ∆_r,s,α = P_n,r^α (y)−P_n,s^α (y) P_n,r^α (x)−P_n,s^α (x) with ∆r,s,0 =

ln^P_P^n,r^(y)

n,s(y)

.

ln^P_P^n,r^(x)

n,s(x)

and, in order to include the case of equality for various inequalities in our discussion, we define0/0 = 1from now on.

In this section, we apply our results above to inequalities of Ky Fan’s type.

Let f(x, y) be any function satisfying the conditions in the first paragraph of this section. We now show how to get inequalities of Ky Fan’s type in general.

Suppose (1.1) holds for someα > 0, r > s, β = 1andt = t⁰ = 1, write σ_n,1(y) = σ⁰_n,1, apply (1.1) to sequences x,y and then take their quotients to get

aσ_n,1⁰

b⁰σ_n,1 ≤∆_r,s,α ≤ bσ⁰_n,1 a⁰σ_n,1. Sinceσ_n,1⁰ =Pn

i=1w_i(Pn

k=1w_k(y_i−y_k))², the mean value theorem yields y_i−y_k =−f_x⁰

f_y⁰(ξ, y(ξ))(x_i−x_k)

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for someξ ∈(a, b). Thus

a≤x≤bmin

f_x⁰ f_y⁰

2

σ_n,1 ≤σ_n,1⁰ ≤ max

a≤x≤b

f_x⁰ f_y⁰

2

σ_n,1, which implies

a b⁰ min

a≤x≤b

f_x⁰ f_y⁰

2

≤∆_r,s,α ≤ b a⁰ max

a≤x≤b

f_x⁰ f_y⁰

2

. We next apply the above argument to a special case.

Corollary 3.1. Let f(x, y) = cx^p +dy^p −1,0< c ≤ d, p ≥ 1, x_i ∈ [0,(c+ d)⁻¹^p]. Fors∈[0,2]andα= max{s,1}we have

(3.2) ∆_1,s,α ≤1

with equality holding if and only ifx₁ =· · ·=x_n.

Proof. This follows from Corollary2.2by the appropriate choice ofrands.

From now on we will concentrate on the casef(x, y) = x+y−1. Extensions to the case of general functionsf(x, y)are left to the reader.

Corollary 3.2. Let f(x, y) = x +y − 1, 0 < a < b < 1 and x_i ∈ [a, b]

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(i= 1, . . . , n), n≥2. Then forr > s,min{1, r−1} ≤s≤max{1, r−1}

max (

b 1−b

^2−r ,

a 1−a

^2−r) (3.3)

>∆_r,s,r

>min (

b 1−b

2−r

, a

1−a

2−r) . Fors < r ≤1,

max (

b 1−b

2−s

, a

1−a

2−s) (3.4)

>∆_r,s,s

>min (

b 1−b

^2−s ,

a 1−a

^2−s) .

Proof. Apply Corollary 2.2 to sequencesx,y with t = t⁰ = 1 and take their quotients, by noticingσ_n,1(x) =σ_n,1(y).

As a special case of the above corollary, by taking r = 0, s = −1, we get the following refinement of the Wang-Wang inequality [17]:

(3.5)

G_n Hn

(1−a^a )²

≤ G⁰_n H_n⁰ ≤

G_n Hn

(1−b^b )² .

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We can use Corollary 2.6 to get further refinements of inequalities of Ky Fan’s type. Sinceσn,s =σn,1+ (An−Pn,s)², we can rewrite the right-hand side inequality in (2.8) as

(3.6) (P_n,1(x)−P_n,s(x))

1− 1−s

2b (P_n,1(x)−P_n,s(x))

≥ 1−s 2b σ_n,1. Apply (2.8) toyand taking the quotient with (3.6), we get

Pn,1(y)−Pn,s(y)

(P_n,1(x)−P_n,s(x)) 1− ^1−s_2b (P_n,1(x)−P_n,s(x)) ≤ bσ⁰_n,1 a⁰σ_n,1

P_n,s⁰ A⁰_n = b

a⁰ P_n,s⁰

A⁰_n . Similarly,

(P_n,1(y)−P_n,s(y)) 1−^1−s_2a0(P_n,1(y)−P_n,s(y)) P_n,1(x)−P_n,s(x) ≥ a

b⁰ A_n P_n,s.

Combining these with a result in [9], we obtain the following refinement of Ky Fan’s inequality:

Corollary 3.3. Let0< a < b <1andx_i ∈ [a, b] (i= 1, . . . , n), n≥ 2. Then forα≤1,0≤s <1

(3.7)

b 1−b

^2−α P_n,s⁰

A⁰_n B >∆_1,s,α >

a 1−a

^2−α A_n Pn,s

A, where

A=

1−1−s

2a⁰ (Pn,1(y)−Pn,s(y)) −1

, B = 1−1−s

2b (P_n,1(x)−P_n,s(x)).

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We note here whenα = 1, s = 0, b ≤ ¹₂, the left-hand side inequality of (3.7) yields

(3.8) A⁰_n−G⁰_n

A_n−G_n < b 1−b

G⁰_n

A⁰_n(A⁰_n+Gn)

a refinement of the following two results of H. Alzer [1]: A⁰_n/G⁰_n ≤ (1− G_n)/(1−A_n), which is equivalent to(A⁰_n−G⁰_n)/(A_n−G_n)< G⁰_n/A⁰_nand [3]:

A⁰_n−G⁰_n≤(A_n−G_n)(A⁰_n+G_n).

Next, we give a result related to Levinson’s generalization of Ky Fan’s inequality. We first generalize a lemma of A.McD. Mercer [12].

Lemma 3.4. LetJ(x)be the smallest closed interval that contains all ofx_i and let y ∈ J(x) and f(x), g(x) ∈ C²(J(x)) be two twice continuously differen- tiable functions. Then

(3.9)

Pn

i=1ω_if(x_i)−f(y)−(Pn

i=1ω_ix_i−y)f⁰(y) Pn

i=1ω_ig(x_i)−g(y)−(Pn

i=1ω_ix_i−y)g⁰(y) = f⁰⁰(ξ) g⁰⁰(ξ)

for some ξ ∈ J(x), provided that the denominator of the left-hand side is nonzero.

Proof. The proof is very similar to the one given in [12]. Write

(Qf)(t) =

n

X

i=1

w_if(tx_i+ (1−t)y)−f(y)−t(A−y)f⁰(y)

and considerW(t) = (Qf)(t)−K(Qg)(t),whereK is the left-hand side expression in (3.9). The lemma then follows by the same argument as in [12].

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By takingg(x) = x², y =P_n,tin the lemma, we get:

Corollary 3.5. Let f(x) ∈ C²[a, b]withm = min

a≤x≤bf⁰⁰(x), M = max

a≤x≤bf⁰⁰(x).

Then (3.10) M

2 σ_n,t ≥

n

X

i=1

ω_if(x_i)−f

n

X

i=1

ω_ix_i

!

−(A_n−P_n,t)f⁰(P_n,t)≥ m 2σ_n,t. Moreover, if f⁰⁰⁰(x) exists for x ∈ [a, b] with f⁰⁰⁰(x) > 0 or f⁰⁰⁰(x) < 0 for x∈[a, b]then the equality holds if and only ifx₁ =· · ·=x_n.

The caset = 1in the above corollary was treated by A.McD. Mercer [11].

Note for an arbitraryf(x), equality can hold even if the conditionx₁ =· · ·=x_n is not satisfied, for example, forf(x) =x², we have the following identity:

n

X

i=1

ω_ix²_i −

n

X

i=1

ω_ix_i

!2

=

n

X

i=1

ω_i x_i−

n

X

k=1

ω_kx_k

!2

.

Corollary3.5 can be regarded as a refinement of Jensen’s inequality and it leads to the following well-known Levinson’s inequality for 3-convex functions [10]:

Corollary 3.6. Letx_i ∈(0, a]. Iff⁰⁰⁰(x)≥0in(0,2a), then (3.11)

n

X

i=1

ω_if(x_i)−f

n

X

i=1

ω_ix_i

!

≤

n

X

i=1

ω_if(2a−x_i)−f

n

X

i=1

ω_i(2a−x_i)

! .

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Iff⁰⁰⁰(x)>0on(0,2a)then equality holds if and only ifx₁ =· · ·=x_n. Proof. Taket = 1in (3.10) and apply Corollary 3.5to(x₁, . . . , x_n)and(2a− x₁, . . . ,2a−x_n). Since f⁰⁰⁰(x) ≥ 0 in (0,2a), it follows that max

0≤x≤af⁰⁰(x) ≤

a≤x≤2amin f⁰⁰(x)and the corollary is proved.

Now we establish an inequality relating different∆_r,s,α’s:

Corollary 3.7. Forl−r ≥ t−s≥ 0, l 6=t, r 6= s,(l, t) 6= (r, s), x_i ∈[a, b], y_i ∈[a, b], n≥2,

(3.12)

b a⁰

^l−r

>

∆_r,s,r

∆_l,t,l

>a b⁰

l−r

. Proof. Apply (2.4) to bothxandyand take their quotients.

For another proof of inequality (3.5), use this corollary with l = 1, t = 0, s=−1andr= 0.

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4. A Few Comments

A variant of (1.1) is the following conjecture by A.McD. Mercer [13] (r >

s, t, t⁰ =r, s):

max

r−s

2x^2−r₁ , r−s 2x^2−r_n

σn,t⁰ ≥ P_n,r−P_n,s P_n,r^1−r (4.1)

≥min

r−s

2x^2−r₁ , r−s 2x^2−r_n

σn,t.

The conjecture presented here has been reformulated (one can compare it with the original one in [13]), since here(r−s)/2is the best possible constant by the same argument as above.

Note whenr = 1, (4.1) coincides with (1.1) and thus the conjecture in general is false.

There are many other kinds of expressions for the bounds of the difference between the arithmetic and geometric means. See Chapter II of the book Clas- sical and New Inequalities in Analysis [16].

In [12], A.McD. Mercer showed (4.2) P_n,2² −G²_n

4x1

≥A_n−G_n≥ P_n,2² −G²_n 4xn

.

He also pointed out that the above inequality is not comparable to either of the inequalities in (1.1) withα=β =u= 1, v = 0, t=t⁰ = 0,1. We note that (4.2) can be obtained from (1.1) by averaging the caseα =β =u = t =t⁰ =

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1, v = 0with the following trivial bound:

A²_n−G²_n

2x₁ ≥An−Gn≥ A²_n−G²_n 2x_n .

Thus the incomparability of (4.2) and (4.1) withr= 1, s= 0, t = 1reflects the fact thatP_n,2² −A²_nandA²_n−G²_nare in general not comparable.

We also note when replacingC_u,v,β by a smaller constant, that we sometimes get a trivial bound. For example, fors≤ ¹₂, the following inequality holds:

An−Pn,s ≥ 1 2

n

X

k=1

ωk

x^1/2_k −A^1/2_n 2

≥ 1 8x_n

n

X

k=1

ωk(xk−An)².

The first inequality is equivalent to P_n,1/2^1/2 A^1/2n ≥ P_n,s. For the second, simply apply the mean value theorem to

x^1/2_k −A^1/2_n 2

= 1

2ξ_k^−1/2(x_k−A_n) 2

≥ 1

4x_n(x_k−A_n)², withξ_k in betweenx_kandA_n.

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References

[1] H. ALZER, Refinements of Ky Fan’s inequality, Proc. Amer. Math. Soc., 117 (1993), 159–165.

[2] H. ALZER, The inequality of Ky Fan and related results, Acta Appl. Math., 38 (1995), 305–354.

[3] H. ALZER, On Ky Fan’s inequality and its additive analogue, J. Math.

Anal. Appl., 204 (1996), 291–297.

[4] H. ALZER, A new refinement of the arithmetic mean–geometric mean inequality, Rocky Mountain J. Math., 27(3) (1997), 663–667.

[5] H. ALZER, On an additive analogue of Ky Fan’s inequality, Indag.

Math.(N.S.), 8 (1997), 1–6.

[6] H. ALZER, Some inequalities for arithmetic and geometric means, Proc.

Roy. Soc. Edinburgh Sect. A, 129 (1999), 221–228.

[7] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, Berlin-Göttingen-Heidelberg 1961.

[8] D.I. CARTWRIGHT AND M.J. FIELD, A refinement of the arithmetic mean-geometric mean inequality, Proc. Amer. Math. Soc., 71 (1978), 36–

38.

[9] P. GAO, A generalization of Ky Fan’s inequality, Int. J. Math. Math. Sci., 28 (2001), 419–425.

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[10] N. LEVINSON, Generalization of an inequality of Ky Fan, J. Math. Anal.

Appl., 8 (1964), 133–134.

[11] A.McD. MERCER, An "error term" for the Ky Fan inequality, J. Math.

Anal. Appl., 220 (1998), 774–777.

[12] A.McD. MERCER, Some new inequalities involving elementary mean values, J. Math. Anal. Appl., 229 (1999), 677–681.

[13] A.McD. MERCER, Bounds for A-G, A-H, G-H, and a family of inequal- ities of Ky Fan’s type, using a general method, J. Math. Anal. Appl., 243 (2000), 163–173.

[14] A.McD. MERCER, Improved upper and lower bounds for the difference A_n−G_n, Rocky Mountain J. Math., 31 (2001), 553–560.

[15] P. MERCER, A note on Alzer’s refinement of an additive Ky Fan inequal- ity, Math. Inequal. Appl., 3 (2000), 147–148.

[16] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers Group, Dordrecht, 1993.

[17] P.F. WANG AND W.L. WANG, A class of inequalities for the symmetric functions, Acta Math. Sinica (in Chinese), 27 (1984), 485–497.