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
volume 4, issue 4, article 76,
2003.
Received 17 August, 2002;
accepted 3 June, 2003.
Communicated by: K.B. Stolarsky
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
089-02
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Abstract
Let Pn,r (x) be the generalized weighted power means. We consider bounds for
the differences of means in the following form:
(
)
(
)
α − Pα
Pn,u
Cu,v,β Cu,v,β
C
C
n,v
u,v,β
u,v,β
max 2β−α , 2β−α σn,w0 ,β ≥
≥ min 2β−α , 2β−α σn,w,β .
α
x1
xn
x1
xn
P
β
. Some similar
Here β 6= 0, σn,t,β ( x) = ni=1 ωi [xβi − Pn,t
(x)]2 and Cu,v,β = u−v
2β 2
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.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Applications to Inequalities of Ky Fan’s Type . . . . . . . . . . . . . 14
4
A Few Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
References
Certain Bounds for the
Differences of Means
Peng Gao
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1.
Introduction
1
Pn
r r
Let Pn,r (x) be the generalized weighted
power
means:
P
(x)
=
(
ω
x
)
,
n,r
i
i
i=1
P
where ωi > 0, 1 ≤ i ≤ n with ni=1 ωi = 1 and x = (x1 , x2 , . . . , xn ). Here
Pn,0 (x) denotes the limit of Pn,r (x) as r → 0+ . In this paper, we always assume
that 0 < x1 ≤ x2 ≤ · · · ≤ xn . We write
n
h
i2
X
β
σn,t,β (x) =
ωi xβi − Pn,t
(x)
i=1
and denote σn,t as σn,t,1 .
We let
Certain Bounds for the
Differences of Means
Peng Gao
An (x) = Pn,1 (x), Gn (x) = Pn,0 (x), Hn (x) = Pn,−1 (x)
and we shall write Pn,r for Pn,r (x), An for An (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):
α
α
Pn,u
− Pn,v
Cu,v,β Cu,v,β
0
(1.1)
max
,
σ
≥
n,w ,β
α
x2β−α
x2β−α
n
1
Cu,v,β Cu,v,β
≥ min
, 2β−α σn,w,β ,
x2β−α
xn
1
where Cu,v,β =
u−v
.
2β 2
If we set x1 = · · · = xn−1 6= xn , then we conclude from
α
α
Pn,u
− Pn,v
u−v
=
x1 →xn ασn,w,β
2β 2 x2β−α
n
Lim
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0
0
that Cu,v,β is best possible. Here we define (Pn,u
− Pn,v
)/0 = ln(Pn,u /Pn,v ),
α
α
the limit of (Pn,u − Pn,v )/α as α → 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) with α ≤ 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 inequality, first published in the monograph Inequalities Q
by Beckenbach and Bellman
0
0
[7].(In this section, we set An = 1 − An , Gn = ni=1 (1 − xi )ωi . For general
definitions, see the beginning of Section 3.)
Theorem 1.1. For xi ∈ 0, 12 ,
(1.2)
A0n
G0n
≤
An
Gn
with equality holding if and only if x1 = · · · = xn .
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. Let 0 < a ≤ xi ≤ b < 1 (1 ≤ i ≤ n, n ≥ 2). For a 6= b we have
(1.3)
A0n
G0n
a
−
b
<
<
.
1−a
An − G n
1−b
Certain Bounds for the
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Peng Gao
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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]:
2
b
a 2
( 1−b
)
An ( 1−a )
A0n
An
≤ 0 ≤
.
Gn
Gn
Gn
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. If x1 6= xn , n ≥ 2, then
(1.4)
Gn − x1
x n − Gn
σn,1 > An − Gn >
σn,1 .
2x1 (An − x1 )
2xn (xn − An )
We generalize this in Section 2.
Certain Bounds for the
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Peng Gao
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2.
The Main Theorem
0
Theorem 2.1. 1, rs , 1, γr , rt , tr , r 6= s, r 6= 0, γ 6= 0 holds for the following
three cases:
r
γ
0
1.
s
γ
≤
2.
r
γ
≥ 2, γr − 1 ≥
3.
r
γ
≤
s
γ
≤ 2, 1 ≥ γt , tγ ≥
s
γ
s
γ
≥
r
γ
− 1;
0
≥ γt , tγ ≥ 1;
0
≤ γt , tγ ≤ 1,
Certain Bounds for the
Differences of Means
with equality holding if and only if x1 = · · · = xn for all the cases.
Proof. Let γ = 1 and 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;
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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
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(2.1)
Dn (x) = An − Pn, rs −
n
r(r − s) X
2
−1
r
2xn
i=1
1
1
ωi xir − Pn,r t
2
.
r
We want to show that Dn ≥ 0 here. We can assume that x1 < x2 < · · · < xn
and use induction. The case n = 1 is clear, so assume that the inequality holds
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for n − 1 variables. Then
(2.2)
1 ∂Dn
·
=1−
ωn ∂xn
"
Pn, rs
xn
r1 #r−s
− (r − s) 1 −
Pn, rt
r1 !
+ S,
xn
where
1−t
n
S=
(2 − r)(r − s) X
2
2ωn xnr
−2
1
r
1
r
ωi xi − Pn, t
2
+ (r − s)
r
i=1
Pn,rt r
2−t
r
xn
1
1 Pn,r 1 − Pn,r t .
r
r
Certain Bounds for the
Differences of Means
Peng Gao
Thus, when s ≤ r ≤ 2, t ≤ 1, S ≥ 0.
Now by the mean value theorem
"
1−
r1 #r−s
s
Pn, r
xn
= (r − s)η r−s−1
1−
≥ (r − s) 1 −
r1 !
s
Pn, r
xn
1 !
Pn, rs
xn
r
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for r ≥ s ≥ r − 1 with
( min 1,
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Pn, rs
xn
1 )
r
( 1 )
Pn, rs r
≤ η ≤ max 1,
.
xn
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This implies
"
1−
Pn, rs
xn
r1 #r−s
− (r − s) 1 −
Pn, rt
r1 !
xn
"
≥ (r − s)
Pn, rt
xn
r1
−
Pn, rs
xn
r1 #
,
which is positive if s ≤ t.
n
Thus for s ≤ r ≤ 2, 1 ≥ t ≥ s ≥ r − 1, we have ∂D
≥ 0. By letting xn
∂xn
tend to xn−1 , we have Dn ≥ Dn−1 (with weights ω1 , . . . , ωn−2 , ωn−1 + ωn ) 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 if x1 = · · · = xn .
Now consider the left-hand side inequality of (1.1) and write
2
n
1
1
r(r − s) X
(2.3)
En (x) = An − Pn, rs −
ωi xir − P r t0 .
2
−1
n, r
r
2x1
i=1
n
Now ω11 ∂E
has an expression similar to (2.2) with xn ↔ x1 , ωn ↔ ω1 , t ↔
∂x1
0
n
t . It is then easy to see under the same condition, ∂E
≥ 0. Thus the left-hand
∂x1
side inequality of (1.1) holds by a similar induction process with the equality
holding if and only if x1 = · · · = xn .
Similarly, we can show Dn (x) ≤ 0, En (x) ≥ 0 for cases (2) and (3) with
equality holding if and only if x1 = · · · = xn for all the cases.
Now for an arbitrary γ, a change of variables y → y/γ for y = r, s, t, t0 in
the above cases leads to the desired conclusion.
Certain Bounds for the
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Peng Gao
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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 if x1 = · · · = xn for all the cases.
Proof. This follows from taking γ = 1 in Theorem 2.1 and another change of
variables: x1 → min{xr1 , xrn }, xn → max{xr1 , xrn } and xi = xri for 2 ≤ i ≤
n − 1 if n ≥ 3 and exchanging r and s for the case s > r.
Certain Bounds for the
Differences of Means
Peng Gao
We remark here since σn,t0 = σn,t + (2An − Pn,t − Pn,t0 )(Pn,t − Pn,t0 ), we
have σn,1 ≤ σn,t for t 6= 1 and σn,t ≤ σn,t0 for t0 ≤ t ≤ 1, σn,t ≥ σn,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. For l − r ≥ t − s ≥ 0, l 6= t, xi ∈ [a, b], a > 0,
(P r − P r )/r (r − s) 1
(r − s) 1
n,r
n,s
(2.4)
(l − t) al−r ≥ (P l − P l )/l ≥ (l − t) bl−r .
n,t
n,l
Except for the trivial cases r = s or (l, t) = (r, s), the equality holds if and only
if x1 = · · · = xn , where we define 0/0 = xr−l
for any i.
i
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Proof. This is a generalization of a result A.McD. Mercer [12]. We may assume
that x1 = a, xn = b and consider
r
r
D(x) = Pn,r
− Pn,s
−
r(r − s)
l
l
(Pn,l
− Pn,t
),
l(l − t)xl−r
n
r(r − s)
l
l
(Pn,l
− Pn,t
).
l(l − t)xl−r
1
We will show that Dn · En ≤ 0. Suppose r − s ≥ 0 here; the case r − s ≤ 0 is
similar. We have

l−t 
"
r−s
r−s # r−s
1−r
∂Dn
Pn,s
r−s
Pn,t
xn
 + S,
−
·
=1−
1−
rωn ∂xn
xn
l−t
xn
r
r
E(x) = Pn,r
− Pn,s
−
Certain Bounds for the
Differences of Means
Peng Gao
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where
Contents
(r − s)(l − r) l
l
S=
(Pn,l − Pn,t
) ≥ 0.
l(l − t)xl−2
ω
n
n
Now by the mean value theorem
"
1−
where
Pn,t
xn
Pn,t
xn
l−t
r−s # r−s
l − t l−t−r+s
=
η
r−s
1−
Pn,t
xn
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< η < 1 and
∂Dn
x1−r
n
≥1−
rωn ∂xn
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Pn,s
xn
r−s
−
1−
Pn,t
xn
r−s !
≥0
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since t ≥ s.
x1−r
∂En
1
Similarly, we have rω
≥ 0 and by a similar induction process as the
1 ∂x1
one in the proof of Theorem 2.1, we have Dn · En ≤ 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
1
2
(Pn,2
− G2n ) ≥ An − Hn ≥
(P 2 − G2n )
2x1
2xn n,2
Certain Bounds for the
Differences of Means
and the right-hand side inequality above gives a refinement of a result of A.McD.
Mercer [13].
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Proposition 2.4. For r > s, α > β,
(2.6)
xβ−α
1
≥
β−α
Pn,s
Peng Gao
β
β
(Pn,r
− Pn,s
)/β
β−α
≥ Pn,r
≥ xβ−α
≥
n
α
α
(Pn,r − Pn,s )/α
with equality holding if and only if x1 = · · · = xn , where we define 0/0 = xβ−α
i
for any i.
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Proof. By the mean value theorem,
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β
β
α β/α
α β/α
Pn,r
− Pn,s
= (Pn,r
)
− (Pn,s
)
where Pn,s < η < Pn,r and (2.6) follows.
β
α
α
= η β−α (Pn,r
− Pn,s
),
α
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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
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. If x1 6= xn , n ≥ 2, then for 1 > s ≥ 0
(2.7)
1−s
Pn,s
− x1−s
1
2x1−s
1 (An − x1 )
σn,1 > An − Pn,s >
1−s
x1−s
− Pn,s
n
σn,1 .
2x1−s
n (xn − An )
Proof. We prove the right-hand inequality; the left-hand side inequality is similar. Let
1−s
x1−s
− Pn,s
n
Dn (x) = (xn − An )(An − Pn,s ) −
σn,1 .
2x1−s
n
We show by induction that Dn ≥ 0. We have
1−s s 1 − s Pn,s
xn
∂Dn
= (1 − ωn )(An − Pn,s ) −
1−
ωn σn,1
∂xn
2xn
xn
Pn,s
1−s
≥ (1 − ωn ) An − Pn,s −
σn,1 ≥ 0,
2xn
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 have Dn ≥ 0. Since not all the xi ’s
are equal, we get the desired result.
Corollary 2.6. For 1 > s ≥ 0,
1 − s Pn,s
1−s
(2.8)
·
σn,1 ≥ An − Pn,s ≥
σn,s ,
2x1
An
2xn
with equality holding if and only if x1 = · · · = xn .
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Peng Gao
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P 1−s −x1−s
n,s
1
≤
Proof. By Theorem 2.5, we only need to show 2x1−s
(An −x1 )
1
is easily verified by using the mean value theorem.
1−s Pn,s
2x1 An
and this
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3.
Applications to Inequalities of Ky Fan’s Type
Let f (x, y) be a real function. We regard y as an implicit function defined by
f (x, y) = 0 and for y = (y1 , . . . , yn ), let f (xi , yi ) = 0, 1 ≤ i ≤ n. We write
0
0
0
0
0
0
0
Pn,r
= Pn,r (y) with An = Pn,1
, Gn = Pn,0
, Hn = Pn,−1
. Furthermore, we
write x1 = a > 0 and xn = b so that xi ∈ [a, b] with yi ∈ [a0 , b0 ], a0 > 0 and
require that fx0 , fy0 exist for xi ∈ [a, b], yi ∈ [a0 , b0 ].
To simplify expressions, we define:
α
α
Pn,r
(y) − Pn,s
(y)
α
α
Pn,r (x) − Pn,s (x)
. Pn,r (x)
(y)
ln
and, in order to include the case of
with ∆r,s,0 = ln PPn,r
Pn,s (x)
n,s (y)
equality for various inequalities in our discussion, we define 0/0 = 1 from 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 some α > 0, r > s, β = 1 and t = t0 = 1 , write
0
σn,1 (y) = σn,1
, apply (1.1) to sequences x, y and then take their quotients to
get
0
0
aσn,1
bσn,1
≤
∆
≤
.
r,s,α
b0 σn,1
a0 σn,1
P
P
0
Since σn,1
= ni=1 wi ( nk=1 wk (yi − yk ))2 , the mean value theorem yields
(3.1)
∆r,s,α =
yi − yk = −
fx0
(ξ, y(ξ))(xi − xk )
fy0
Certain Bounds for the
Differences of Means
Peng Gao
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for some ξ ∈ (a, b). Thus
0 2
0 2
fx f 0
min 0 σn,1 ≤ σn,1 ≤ max x0 σn,1 ,
a≤x≤b fy
a≤x≤b fy
which implies
0 2
0 2
fx fx b
a
.
min
≤
∆
≤
max
r,s,α
b0 a≤x≤b fy0 a0 a≤x≤b fy0 Certain Bounds for the
Differences of Means
We next apply the above argument to a special case.
p
Peng Gao
p
Corollary 3.1. Let f (x, y) = cx + dy − 1, 0 < c ≤ d, p ≥ 1, xi ∈ [0, (c +
1
d)− p ]. For s ∈ [0, 2] and α = max{s, 1} we have
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∆1,s,α ≤ 1
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(3.2)
with equality holding if and only if x1 = · · · = xn .
Proof. This follows from Corollary 2.2 by the appropriate choice of r and s.
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From now on we will concentrate on the case f (x, y) = x+y−1. Extensions
to the case of general functions f (x, y) are left to the reader.
Corollary 3.2. Let f (x, y) = x + y − 1, 0 < a < b < 1 and xi ∈ [a, b]
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(i = 1, . . . , n), n ≥ 2. Then for r > s, min{1, r − 1} ≤ s ≤ max{1, r − 1}
(
2−r 2−r )
a
b
(3.3)
max
,
1−b
1−a
> ∆r,s,r
(
2−r 2−r )
b
a
> min
,
.
1−b
1−a
For s < r ≤ 1,
(
(3.4)
max
Certain Bounds for the
Differences of Means
Peng Gao
2−s 2−s )
b
a
,
1−b
1−a
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> ∆r,s,s
(
2−s 2−s )
b
a
,
.
> min
1−b
1−a
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Proof. Apply Corollary 2.2 to sequences x, y with t = t0 = 1 and take their
quotients, by noticing σn,1 (x) = σn,1 (y).
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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]:
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(3.5)
Gn
Hn
2
a
( 1−a
)
≤
G0n
≤
Hn0
Gn
Hn
2
b
( 1−b
)
.
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We can use Corollary 2.6 to get further refinements of inequalities of Ky
Fan’s type. Since σn,s = σn,1 + (An − Pn,s )2 , we can rewrite the right-hand side
inequality in (2.8) as
1−s
1−s
(3.6) (Pn,1 (x) − Pn,s (x)) 1 −
(Pn,1 (x) − Pn,s (x)) ≥
σn,1 .
2b
2b
Apply (2.8) to y and taking the quotient with (3.6), we get
0
0
0
bσn,1
Pn,s
Pn,1 (y) − Pn,s (y)
b Pn,s
≤ 0
= 0 0 .
a σn,1 A0n
a An
(Pn,1 (x) − Pn,s (x)) 1 − 1−s
(Pn,1 (x) − Pn,s (x))
2b
Certain Bounds for the
Differences of Means
Similarly,
Peng Gao
1−s
(Pn,1 (y)
2a0
(Pn,1 (y) − Pn,s (y)) 1 −
Pn,1 (x) − Pn,s (x)
− Pn,s (y))
≥
a An
.
b0 Pn,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 < 1 and xi ∈ [a, b] (i = 1, . . . , n), n ≥ 2. Then
for α ≤ 1, 0 ≤ s < 1
2−α 0
2−α
Pn,s
b
a
An
(3.7)
B > ∆1,s,α >
A,
0
1−b
An
1−a
Pn,s
where
−1
1−s
A= 1−
(Pn,1 (y) − Pn,s (y))
,
2a0
1−s
B =1−
(Pn,1 (x) − Pn,s (x)).
2b
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We note here when α = 1, s = 0, b ≤ 21 , the left-hand side inequality of
(3.7) yields
A0n − G0n
b G0n 0
<
(A + Gn )
An − G n
1 − b A0n n
(3.8)
a refinement of the following two results of H. Alzer [1]: A0n /G0n ≤ (1 −
Gn )/(1 − An ), which is equivalent to (A0n − G0n )/(An − Gn ) < G0n /A0n and [3]:
A0n − G0n ≤ (An − Gn )(A0n + Gn ).
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. Let J(x) be the smallest closed interval that contains all of xi and
let y ∈ J(x) and f (x), g(x) ∈ C 2 (J(x)) be two twice continuously differentiable functions. Then
Pn
P
ωi f (xi ) − f (y) − ( ni=1 ωi xi − y)f 0 (y)
f 00 (ξ)
i=1
P
P
(3.9)
= 00
n
n
0
g (ξ)
i=1 ωi g(xi ) − g(y) − (
i=1 ωi xi − y)g (y)
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
wi f (txi + (1 − t)y) − f (y) − t(A − y)f 0 (y)
i=1
and consider W (t) = (Qf )(t) − K(Qg)(t), where K is the left-hand side expression in (3.9). The lemma then follows by the same argument as in [12].
Certain Bounds for the
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Peng Gao
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By taking g(x) = x2 , y = Pn,t in the lemma, we get:
Corollary 3.5. Let f (x) ∈ C 2 [a, b] with m = min f 00 (x), M = max f 00 (x).
a≤x≤b
a≤x≤b
Then
n
n
X
X
M
(3.10)
σn,t ≥
ωi f (xi ) − f
2
i=1
!
ωi xi − (An − Pn,t )f 0 (Pn,t ) ≥
i=1
m
σn,t .
2
Moreover, if f 000 (x) exists for x ∈ [a, b] with f 000 (x) > 0 or f 000 (x) < 0 for
x ∈ [a, b] then the equality holds if and only if x1 = · · · = xn .
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 = · · · = xn
is not satisfied, for example, for f (x) = x2 , we have the following identity:
!2
!2
n
n
n
n
X
X
X
X
ωi x2i −
ωi xi
=
ωi xi −
ωk xk .
i=1
i=1
i=1
Corollary 3.6. Let xi ∈ (0, a]. If f 000 (x) ≥ 0 in (0, 2a), then
!
n
n
X
X
(3.11)
ωi f (xi ) − f
ωi xi
≤
i=1
ωi f (2a − xi ) − f
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i=1
n
X
Peng Gao
k=1
Corollary 3.5 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]:
i=1
Certain Bounds for the
Differences of Means
n
X
i=1
!
ωi (2a − xi ) .
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If f 000 (x) > 0 on (0, 2a) then equality holds if and only if x1 = · · · = xn .
Proof. Take t = 1 in (3.10) and apply Corollary 3.5 to (x1 , . . . , xn ) and (2a −
x1 , . . . , 2a − xn ). Since f 000 (x) ≥ 0 in (0, 2a), it follows that max f 00 (x) ≤
0≤x≤a
min f 00 (x) and the corollary is proved.
a≤x≤2a
Now we establish an inequality relating different ∆r,s,α ’s:
Corollary 3.7. For l − r ≥ t − s ≥ 0, l 6= t, r 6= s, (l, t) 6= (r, s), xi ∈ [a, b],
yi ∈ [a, b], n ≥ 2,
(3.12)
Certain Bounds for the
Differences of Means
l−r ∆r,s,r a l−r
b
>
> .
a0
∆l,t,l b0
Proof. Apply (2.4) to both x and y and take their quotients.
Peng Gao
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For another proof of inequality (3.5), use this corollary with l = 1, t =
0, s = −1 and r = 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):
r−s r−s
Pn,r − Pn,s
(4.1)
max
σn,t0 ≥
2−r ,
2−r
1−r
2xn
Pn,r
2x1
r−s r−s
≥ min
, 2−r σn,t .
2xn
2x2−r
1
The conjecture presented here has been reformulated (one can compare it
with the original one in [13]), since here (r − s)/2 is the best possible constant
by the same argument as above.
Note when r = 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)
2
Pn,2
−
4x1
G2n
≥ An − G n ≥
2
Pn,2
−
4xn
G2n
.
He also pointed out that the above inequality is not comparable to either of
the inequalities in (1.1) with α = β = u = 1, v = 0, t = t0 = 0, 1. We note that
(4.2) can be obtained from (1.1) by averaging the case α = β = u = t = t0 =
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Differences of Means
Peng Gao
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1, v = 0 with the following trivial bound:
A2 − G2n
A2n − G2n
≥ An − G n ≥ n
.
2x1
2xn
Thus the incomparability of (4.2) and (4.1) with r = 1, s = 0, t = 1 reflects the
2
fact that Pn,2
− A2n and A2n − G2n are in general not comparable.
We also note when replacing Cu,v,β by a smaller constant, that we sometimes
get a trivial bound. For example, for s ≤ 12 , the following inequality holds:
An − Pn,s ≥
n
n
2
1 X
1 X 1/2
ωk xk − A1/2
≥
ωk (xk − An )2 .
n
2 k=1
8xn k=1
1/2
1/2
The first inequality is equivalent to Pn,1/2 An
apply the mean value theorem to
1/2
xk
−
A1/2
n
2
=
≥ Pn,s . For the second, simply
2
1
1 −1/2
ξk (xk − An ) ≥
(xk − An )2 ,
2
4xn
with ξk in between xk and An .
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Peng Gao
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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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Peng Gao
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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 inequalities 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
An − Gn , Rocky Mountain J. Math., 31 (2001), 553–560.
[15] P. MERCER, A note on Alzer’s refinement of an additive Ky Fan inequality, Math. Inequal. Appl., 3 (2000), 147–148.
[16] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.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.
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