The results are applied to inequalities of Ky Fan’s type

http://jipam.vu.edu.au/

Volume 4, Issue 4, Article 76, 2003

CERTAIN BOUNDS FOR THE DIFFERENCES OF MEANS

PENG GAO

DEPARTMENT OFMATHEMATICS, UNIVERSITY OFMICHIGAN, ANNARBOR, MI 48109, USA.

penggao@umich.edu

Received 17 August, 2002; accepted 3 June, 2003 Communicated by K.B. Stolarsky

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

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 α ≥min

(C_u,v,β x^2β−α₁ ,C_u,v,β

x^2β−αn

) σ_n,w,β. Hereβ6= 0, σ_n,t,β(x) =Pn

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.

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

2000 Mathematics Subject Classification. Primary 26D15, 26D20.

1. INTRODUCTION

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

i=1ω_ix^r_i)^1r, where ωi >0,1≤i ≤nwithPn

i=1ωi = 1andx= (x1, x2, . . . , xn). HerePn,0(x)denotes the limit ofPn,r(x)asr→0⁺. In this paper, we always assume that0< x1 ≤x2 ≤ · · · ≤xn. 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) =P_n,−1(x)

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

ISSN (electronic): 1443-5756

c 2003 Victoria University. All rights reserved.

The author thanks the referees for their many valuable comments and suggestions. These greatly improved the presentation of the paper.

089-02

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

(1.1) 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

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

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 forα =β =u= 1, v =−1, w=±1.

There is a close relationship between (1.1) and the following Ky Fan inequality, first pub- lished 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 Section 3.) 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.

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 consequence 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 some α, β, u, v, w, w⁰. We apply it in Section 3 to obtain further refinements and generalizations 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=xn, 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 Section 2.

2. THEMAIN THEOREM

Theorem 2.1. 1,^s_r,1,^γ_r,_r^t,^t_r⁰

, r6=s, r6= 0, γ 6= 0holds for the following three cases:

(1) _γ^s ≤ ^r_γ ≤2,1≥ _γ^t,^t_γ⁰ ≥ ^s_γ ≥ _γ^r −1;

(2) _γ^r ≥2,_γ^r −1≥ _γ^s ≥ _γ^t,^t_γ⁰ ≥1;

(3) _γ^r ≤ _γ^s ≤ _γ^t,^t_γ⁰ ≤1,

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

Proof. Letγ = 1andr 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,^t_r

² .

We want to show that D_n ≥ 0 here. We can assume that x₁ < x₂ < · · · < x_n and use induction. The casen = 1is clear, so assume that the inequality holds 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,^t_r

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)

. 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 ¹r

− 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 toxn−1, we haveD_n≥Dn−1(with weightsω₁, . . . , ωn−2, ωn−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 =· · ·=xn.

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.

In what follows our results often include the cases r = 0 or s = 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. For r > s, min{1, r −1} ≤ s ≤ max{1, r− 1} and min{1, s} ≤ t, t⁰ ≤ max{1, s}, (r, s, r,1, t, t⁰) holds. For s ≤ 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 Theorem 2.1 and another change of variables: x₁ → min{x^r₁, x^r_n}, xn → max{x^r₁, x^r_n}andxi = x^r_i for2 ≤ i ≤ n−1ifn ≥ 3and exchangingr

andsfor 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 haveσ_n,1 ≤σ_n,tfor t 6= 1andσ_n,t ≤σ_n,t⁰ fort⁰ ≤t ≤ 1, σ_n,t ≥ σ_n,t⁰ fort ≥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.

Proof. This is a generalization of a result A.McD. Mercer [12]. We may assume that x1 = 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 xn

r−s#_r−s^l−t

= l−t

r−sη^l−t−r+s 1− P_n,t

xn

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

sincet ≥s.

Similarly, we have ^x

1−r 1

rω1

∂En

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

By takingl= 2, t= 0, r = 1, s=−1in 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,r and (2.6) follows.

We apply (2.6) to the case (1,0,1,1,1,1) to see that (1,0, α,1,1,1) holds with α ≤ 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 similar. 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 2x_n σ_n,1

≥0,

where the last inequality holds by Theorem 2.1. By an induction process similar to the one in the proof of Theorem 2.1, we haveD_n ≥0. Since not all thex_i’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 =· · ·=x_n.

Proof. By Theorem 2.5, 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.

3. APPLICATIONS TOINEQUALITIES OFKY FAN’S TYPE

Letf(x, y)be a real function. We regardyas an implicit function defined byf(x, y) = 0and fory = (y1, . . . , yn), letf(xi, yi) = 0,1 ≤ i ≤ n. We write P_n,r⁰ = Pn,r(y)withA⁰_n = P_n,1⁰ , G⁰_n=P_n,0⁰ , H_n⁰ =P_n,−1⁰ . Furthermore, we writex1 =a >0andxn=bso thatxi ∈ [a, b]with y_i ∈[a⁰, b⁰], a⁰ >0and require thatf_x⁰, f_y⁰ exist forx_i ∈[a, b],y_i ∈[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, writeσ_n,1(y) = σ_n,1⁰ , apply (1.1) to sequencesx,yand then take their quotients to get

aσ⁰_n,1

b⁰σ_n,1 ≤∆_r,s,α ≤ bσ_n,1⁰ 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) for someξ∈(a, b). Thus

a≤x≤bmin

f_x⁰ f_y⁰

2

σ_n,1 ≤σ⁰_n,1 ≤ 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)^−p1]. For s∈[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 Corollary 2.2 by 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. Letf(x, y) =x+y−1,0< a < b < 1andx_i ∈ [a, b] (i = 1, . . . , n), n ≥2.

Then forr > s,min{1, r−1} ≤s ≤max{1, r−1}

(3.3) max (

b 1−b

^2−r ,

a 1−a

^2−r)

>∆_r,s,r >min (

b 1−b

^2−r ,

a 1−a

^2−r) . Fors < r ≤1,

(3.4) max (

b 1−b

2−s

, a

1−a

2−s)

>∆_r,s,s >min (

b 1−b

2−s

, a

1−a

2−s) . Proof. Apply Corollary 2.2 to sequences x,y with t = t⁰ = 1 and take their quotients, by

noticingσn,1(x) = σn,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 H_n

(1−a^a )²

≤ G⁰_n H_n⁰ ≤

G_n H_n

(1−b^b )² .

We can use Corollary 2.6 to get further refinements of inequalities of Ky Fan’s type. Since σ_n,s =σ_n,1+ (A_n−P_n,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

P_n,1(y)−P_n,s(y)

(Pn,1(x)−Pn,s(x)) 1− ^1−s_2b (Pn,1(x)−Pn,s(x)) ≤ bσ_n,1⁰ a⁰σn,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. Let 0 < a < b < 1and xi ∈ [a, b] (i = 1, . . . , n), n ≥ 2. Then for α ≤ 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 P_n,sA, 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)).

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+G_n)

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 of x_i and lety ∈ J(x) andf(x), g(x)∈C²(J(x))be two twice continuously differentiable functions. Then

(3.9)

Pn

i=1ωif(xi)−f(y)−(Pn

i=1ωixi−y)f⁰(y) Pn

i=1ω_ig(x_i)−g(y)−(Pn

i=1ω_ix_i−y)g⁰(y) = f⁰⁰(ξ) g⁰⁰(ξ) for someξ∈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].

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

Corollary 3.5. Letf(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, iff⁰⁰⁰(x)exists forx ∈ [a, b]with f⁰⁰⁰(x) > 0orf⁰⁰⁰(x) < 0forx ∈ [a, b]then the equality holds if and only ifx₁ =· · ·=x_n.

The case t = 1 in the above corollary was treated by A.McD. Mercer [11]. Note for an arbitrary f(x), equality can hold even if the condition x₁ = · · · = 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

.

Corollary 3.5 can be regarded as a refinement of Jensen’s inequality and it leads to the fol- lowing 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)

! . Iff⁰⁰⁰(x)>0on(0,2a)then equality holds if and only ifx₁ =· · ·=x_n.

Proof. Taket= 1in (3.10) and apply Corollary 3.5 to(x₁, . . . , x_n)and(2a−x₁, . . . ,2a−x_n).

Since f⁰⁰⁰(x) ≥ 0 in (0,2a), it follows that max

0≤x≤af⁰⁰(x) ≤ min

a≤x≤2af⁰⁰(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 withl = 1, t= 0, s=−1andr= 0.

4. A FEWCOMMENTS

A variant of (1.1) is the following conjecture by A.McD. Mercer [13] (r > s, t, t⁰ =r, s):

(4.1) max

r−s

2x^2−r₁ , r−s 2x^2−r_n

σ_n,t⁰ ≥ P_n,r−P_n,s

P_n,r^1−r ≥min

r−s

2x^2−r₁ , r−s 2x^2−r_n

σ_n,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 Classical 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 4x_n .

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 = 1, v = 0with the following trivial bound:

A²_n−G²_n

2x₁ ≥A_n−G_n ≥ A²_n−G²_n 2x_n .

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

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

A_n−P_n,s ≥ 1 2

n

X

k=1

ω_k

x^1/2_k −A^1/2_n 2

≥ 1 8x_n

n

X

k=1

ω_k(x_k−A_n)².

The first inequality is equivalent toP_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^−1/2_k (xk−An) 2

≥ 1

4x_n(xk−An)², withξ_kin betweenx_kandA_n.

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