Journal of Inequalities in Pure and
Applied Mathematics
ON THE HOMOGENEOUS FUNCTIONS WITH TWO PARAMETERS
AND ITS MONOTONICITY
volume 6, issue 4, article 101,
2005.
ZHEN-HANG YANG
Zhejiang Electric Power Vocational Technical College
Hangzhou, Zhejiang, China, 311600.
Received 18 May, 2005;
accepted 04 August, 2005.
Communicated by: P.S. Bullen
EMail: yzhkm@163.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
155-05
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Abstract
Suppose f(x, y) is a positive homogeneous function defined on U(j R+ × R+ ),
h p pi 1
,b ) p−q
call Hf (a, b; p, q) = f(a
homogeneous function with two parameters.
f(aq ,bq )
If f(x, y) is 2nd differentiable, then the monotonicity in parameters p and q of
Hf (a, b; p, q) depend on the signs of I1 = (ln f)xy , for variable a and b depend
on the sign of I2a = [(ln f)x ln(y/x)]y and I2b = [(ln f)y ln(x/y)]x respectively.
As applications of these results, a serial of inequalities for arithmetic mean, geometric mean, exponential mean, logarithmic mean, power-Exponential mean
and exponential-geometric mean are deduced.
2000 Mathematics Subject Classification: Primary 26B35, 26E60; Secondary
26A48, 26D07
Key words: Homogeneous function with two parameters, f -mean with twoparameter, Monotonicity, Estimate for lower and upper bounds.
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
Zhen-Hang Yang
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Thanks for Mr. Zhang Zhihua
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Basic Concepts and Main Results . . . . . . . . . . . . . . . . . . . . . . . 5
3
Lemmas and Proofs of the Main Results . . . . . . . . . . . . . . . . . 11
4
Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
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1.
Introduction
The so-called two-parameter mean or extended mean values between two unequal positive numbers a and b were defined first by K.B. Stolarsky in [10] as
(1.1)
 1
q(ap −bp ) p−q


p 6= q, pq 6= 0

p(aq −bq )


1


p

ap −bp

p 6= 0, q = 0

 p(ln a−ln b)
q q 1q
.
E(a, b; p, q) =
a −b

p = 0, q 6= 0

q(ln
a−ln
b)


p


a ln a−bp ln b
1

−
p = q 6= 0
exp

p
p
a −b
p


 √
ab
p=q=0
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
Zhen-Hang Yang
Title Page
The monotonicity of E(a, b; p, q) has been researched by E. B. Leach and
M. C. Sholander in [4], and others also in [9, 8, 7, 6, 5, 11, 14, 15, 17] using
different ideas and simpler methods.
As the generalized power-mean, C. Gini obtained a similar two-parameter
type mean in [1]. That is:

1
ap +bp p−q

p 6= q

q +bq

a

p
p ln b
(1.2)
G(a, b; p, q) =
exp( a lnapa+b
) p = q 6= 0 .
p
+b


√


ab
p=q=0
Recently, the sufficient and necessary conditions comparing the two-parameter
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mean with the Gini mean were put forward by using the so-called concept of
“strong inequalities” ([3]).
From the above two-parameter type means, we find that their forms are both
1
p p p−q
f (a ,b )
, where f (x, y) is a homogeneous function of x and y.
f (aq ,bq )
The main aim of this paper is to establish the concept of “two-parameter homogeneous functions”, and study the monotonicity of functions in the form of
1
p p p−q
f (a ,b )
. As applications of the main results, we will deduce three inequalf (aq ,bq )
ity chains which contain the arithmetic, geometric, exponential, logarithmic,
power-exponential and exponential-geometric means, prove an upper bound for
the Stolarsky mean in [12], and present two estimated expressions for the exponential mean.
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
Zhen-Hang Yang
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2.
Basic Concepts and Main Results
Definition 2.1. Assume that f : U(j R+ × R+ ) → R+ is a homogeneous
function of variable x and y, and is continuous and exists first order partial
derivative, (a, b) ∈ R+ × R+ with a 6= b, (p, q) ∈ R × R. If (1, 1) ∈
/ U, then we
define
1
f (ap , bp ) p−q
(2.1)
Hf (a, b; p, q) =
(p 6= q, pq 6= 0),
f (aq , bq )
(2.2)
Hf (a, b; p, p) = limHf (a, b; p, q) = Gf,p (a, b)
(p = q 6= 0),
q→p
where
(2.3)
1
p
Zhen-Hang Yang
p
p
Gf,p (a, b) = Gf (a , b ), Gf (x, y)
xfx (x, y) ln x + yfy (x, y) ln y
= exp
,
f (x, y)
fx (x, y) and fy (x, y) denote partial derivative to 1st and 2nd variable of f (x, y)
respectively.
If (1, 1) ∈ U, then define further
1
f (ap , bp ) p
(2.4) Hf (a, b; p, 0) =
(p 6= 0, q = 0),
f (1, 1)
1
f (aq , bq ) q
(2.5) Hf (a, b; 0, q) =
(p = 0, q 6= 0),
f (1, 1)
(2.6)
Hf (a, b; 0, 0) = limHf (a, b; p, 0) = a
p→0
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
fx (1,1)
f (1,1)
b
fy (1,1)
f (1,1)
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(p = q = 0).
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From Lemma 3.1, Hf (a, b; p, q) is still a homogeneous function of positive
numbers a and b. We call it a homogeneous function for positive numbers a
and b with two parameters p and q, and call it a two-parameter homogeneous
function for short. To avoid confusion, we also denote it by Hf (p, q) or Hf (a, b)
or Hf .
If f (x, y) is a positive 1-order homogeneous mean function defined on R+ ×
R+ , then call Hf (a, b; p, q) the two-parameter f -mean of positive numbers a
and b.
Remark 1. If f (x, y) is a positive 1-order homogeneous function defined on
R+ × R+ , and is continuous and exists 1st order partial derivative, and satisfies
f (x, y) = f (y, x), then
√
Gf,0 (a, b) = Hf (a, b; 0, 0) = ab.
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
Zhen-Hang Yang
Title Page
In fact, by (2.3), we have
fx (1, 1) ln a + fy (1, 1) ln b
Gf,0 (a, b) = exp
= Hf (a, b; 0, 0).
f (1, 1)
Since f (x, y) is a positive 1-order homogeneous function, from (3.1) of Lemma
3.2, we obtain
(2.7)
1 · fx (1, 1) 1 · fy (1, 1)
+
= 1.
f (1, 1)
f (1, 1)
If f (x, y) = f (y, x), then fx (x, y) = fy (y, x), so we have
(2.8)
fx (1, 1) = fy (1, 1).
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By (2.7) and (2.8), we get
fx (1, 1)
fy (1, 1)
1
=
= ,
f (1, 1)
f (1, 1)
2
√
thereby Gf,0 = ab.
Thus it can be√seen that despite the form of f (x, y) we always have Hf (a, b; 0, 0)
= Gf,0 (a, b) = ab, so long as f (x, y) is a positive 1-order homogeneous symmetric function defined on R+ × R+ .
x−y
ln x−ln y
Example 2.1. In Definition 2.1, let f (x, y) = L(x, y) =
(x, y > 0, x 6=
y), we get (1.1), i.e.
 1

q(ap −bp ) p−q

p 6= q, pq 6= 0

q
q
p(a −b )




1

p p
p

p 6= 0, q = 0

 L (a , b )
1
(2.9)
HL (a, b; p, q) =
,
q (aq , bq )
L
p = 0, q 6= 0






GL,p (a, b)
p = q 6= 0





G(a, b)
p=q=0
On the Homogeneous
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and Its Monotonicity
Zhen-Hang Yang
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where
1
GL,p (a, b) = Ep (a, b) = E p (ap , bp ) = Ep ,
1
a a−b
√
a
−1
E(a, b) = e
,
G(a,
b)
=
ab.
bb
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Remark 2. That
−1
E(a, b) = e
aa
bb
1
a−b
(a, b > 0 with a 6= b)
is called the exponential mean of unequal positive numbers a and b, and is also
called the identical mean and denoted by I(a, b). To avoid confusion, we adopt
our terms and notations in what follows.
Example 2.2. In Definition 2.1, let f (x, y) = A(x, y) = x+y
(x, y > 0, x 6= y),
2
we get (1.2), i.e.

1
ap +bp p−q


p 6= q
q
q

 a +b
(2.10)
HA (a, b; p, q) =
GA,p (a, b) p = q 6= 0 ,



 G(a, b)
p=q=0
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
Zhen-Hang Yang
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1
p
p
a
a+b
p
b
a+b
where GA,p (a, b) = Zp (a, b) = Z (a , b ) = Zp . Z(a, b) = a b
is called
the power-exponential mean between positive numbers a and b.
1
−1 xx x−y
Example 2.3. In Definition 2.1, let f (x, y) = E(x, y) = e
(x, y >
yy
0, x 6= y), then
 1
p p

 E(aq ,bq ) p−q p 6= q

 E(a ,b )
(2.11)
HE (a, b; p, q) =
GE,p (a, b)




G(a, b)
p = q 6= 0 ,
p=q=0
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G2
1
where GE,p (a, b) = Yp (a, b) = Y p (ap , bp ) = Yp . Y (a, b) = Ee1− L2 is called
the exponential-geometric mean between positive numbers a and b, where E =
E(a, b), L = L(a, b), G = G(a, b).
Example 2.4. In Definition 2.1, let f (x, y) = D(x, y) = |x − y| (x, y > 0, x 6=
y), then
 1
 ap −bp p−q
p 6= q, pq 6= 0
aq −bq
,
(2.12)
HD (a, b; p, q) =
 G (a, b) p = q 6= 0
D,p
1
1
1
where GD,p (a, b) = GD,p = e p E p (ap , bp ) = e p Ep .
In order to avoid confusion, we rename HL (a, b; p, q) (or E(a, b; p, q)) and
HA (a, b; p, q) (or G(a, b; p, q)) as the two-parameter logarithmic mean and twoparameter arithmetic mean respectively. In the same way, we call HE (a, b; p, q)
in Example 2.3 the two-parameter exponential mean.
In Example 2.4, since D(x, y) = |x − y| is not a certain mean between
positive numbers x and y, but one absolute value function of difference of two
positive numbers, we call HD (a, b; p, q) a two-parameter homogeneous function
of difference.
It is obvious that the conception of two-parameter homogeneous functions
has greatly developed the extension of the concept of two-parameter means.
For monotonicity of two-parameter homogeneous functions Hf (a, b; p, q),
we have the following main results.
Theorem 2.1. Let f (x, y) be a positive n−order homogenous function defined
on U(j R+ ×R+ ), and be second order differentiable. If I1 = (ln f )xy > (<)0,
On the Homogeneous
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Zhen-Hang Yang
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then Hf (p, q) is strictly increasing (decreasing) in both p and q on (−∞, 0) ∪
(0, +∞).
Corollary 2.2.
1. HL (p, q), HA (p, q), HE (p, q) are strictly increasing both p and q on
(−∞, +∞),
2. HD (p, q) is strictly decreasing both p and q on (−∞, 0) ∪ (0, +∞).
Theorem 2.3. Let f (x, y) be a positive 1-order homogeneous function defined
on U(j R+ × R+ ), and be second order differentiable.
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
1. If I2a = [(ln f )x ln(y/x)]y > (<)0, then Hf (a, b) is strictly increasing
(decreasing) in a.
Zhen-Hang Yang
2. If I2b = [(ln f )y ln(x/y)]x > (<)0, then Hf (a, b) is strictly increasing
(decreasing) in b.
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Corollary 2.4. HL (a, b), HD (a, b) is strictly increasing in both a and b.
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3.
Lemmas and Proofs of the Main Results
For proving the main results in this article, we need some properties of homogeneous functions in [16]. For convenience, we quote them as follows.
Lemma 3.1. Let f (x, y), g(x, y) be n, m-order homogenous functions over Ω
respectively, then f · g, f /g (g 6= 0) are n + m, n − m-order homogenous
functions over Ω respectively.
If for a certain p with (xp , y p ) ∈ Ω, and f p (x, y) exists, then f (xp , y p ), f p (x, y)
are both np-order homogeneous functions over Ω.
Lemma 3.2. Let f (x, y) be a n-order homogeneous function over Ω, and fx , fy
both exist, then fx , fy are both (n − 1)-order homogeneous functions over Ω,
furthermore we have
(3.1)
xfx + yfy = nf.
On the Homogeneous
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and Its Monotonicity
Zhen-Hang Yang
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In particular, when n = 1 and f (x, y) is second order differentiable over Ω,
then
(3.2)
(3.3)
(3.4)
xfx + yfy = f,
xfxx + yfxy = 0,
xfxy + yfyy = 0.
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Lemma 3.3. Let f (x, y) be a positive n−order homogenous function defined
on U(j R+ × R+ ), and be second order differentiable. Set
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T (t) = ln f (at , bt ), where x = at , y = bt , a, b > 0,
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then
∂ 2 ln f (x, y)
= (ln f )xy .
T (t) = −xyI1 (ln b − ln a) , where I1 =
∂x∂y
00
2
Proof. Since f (x, y) is a positive n-order homogeneous function, from (3.1),
we can obtain x(ln f )x + y(ln f )y = n or x(ln f )x = n − y(ln f )y , y(ln f )y =
n − x(ln f )x , so
at fx (at , bt ) ln a + bt fy (at , bt ) ln b
f (at , bt )
xfx (x, y) ln a + yfy (x, y) ln b
=
f (x, y)
= x(ln f )x ln a + y(ln f )y ln b.
T 0 (t) =
(3.5)
(3.6)
(3.7)
Hence
On the Homogeneous
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and Its Monotonicity
Zhen-Hang Yang
Title Page
T 00 (t) =
0
0
∂T (t) dx ∂T (t) dy
+
∂x dt
∂y dt
Contents
t
= [y(ln f )y (ln b − ln a) + n ln a]x a ln a
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= y(ln f )yx (ln b − ln a)x ln a + x(ln f )xy (ln a − ln b)y ln b
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= −xy(ln f )xy (ln b − ln a)2
= −xyI1 (ln b − ln a)2 .
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Lemma 3.4. Let f (x, y) be a positive 1-order homogeneous function defined
on U(j R+ × R+ ), and be second order differentiable. Set
S(t) =
txfx (x, y)
, where x = at , y = bt a, b > 0,
f (x, y)
then
S 0 (t) = xyI2a , where I2a = [(ln f )x ln(y/x)]y .
Proof.
d xfx (x, y)
xfx (x, y)
S (t) =
+t
f (x, y)
dt f (x, y)
∂(x(ln f )x ) dx ∂(x(ln f )x ) dy
= x(ln f )x + t
+
∂x
dt
∂y
dt
∂(x(ln f )x ) t
∂(x(ln f )x ) t
= x(ln f )x + t
a ln a +
b ln b
∂x
∂y
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
0
= x(ln f )x + t [x(x(ln f )x )x ln a + y(x(ln f )x )y ln b] .
Zhen-Hang Yang
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(x,y)
By Lemma 3.1, that x(ln f )x = xff x(x,y)
is a 0-order homogeneous function,
from (3.1) of Lemma 3.2, we obtain
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x [x(ln f )x ]x + y [x(ln f )x ]y = 0 or x [x(ln f )x ]x = −y [x(ln f )x ]y ,
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hence
S 0 (t) = x(ln f )x + ty [x(ln f )x ]y (ln b − ln a)
= x(ln f )x + txy(ln f )xy (ln b − ln a)
= x(ln f )x + xy(ln f )xy (ln bt − ln at )
= x(ln f )x + xy(ln f )xy (ln y − ln x)
= xy y −1 (ln f )x + (ln f )xy ln(y/x)
= xy [(ln f )x ln(y/x)]y = xyI2a .
Based on the above lemmas, then next we will go on proving the main results
in this paper.
Proof of Theorem 2.1. Since Hf (p, q) is symmetric with respect to p and q, we
only need to prove the monotonicity for p of ln Hf .
1) When p 6= q,
1
f (ap , bp )
T (p) − T (q)
ln
=
,
p − q f (aq , bq )
p−q
(p − q)T 0 (p) − T (p) + T (q)
∂ ln Hf
=
.
∂p
(p − q)2
ln Hf =
Set g(p) = (p − q)T 0 (p) − T (p) + T (q), then g(q) = 0, g 0 (p) = (p − q)T 00 (p),
and then exist ξ = q + θ(p − q) with θ ∈ (0, 1) by Mean-value Theorem, such
that
∂ ln Hf
g(p) − g(q)
g 0 (ξ)
(ξ − q)T 00 (ξ)
=
=
=
= (1 − θ)T 00 (ξ).
2
∂p
(p − q)
p−q
p−q
On the Homogeneous
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and Its Monotonicity
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By Lemma 3.3, T 00 (ξ) = −xyI1 (ln b − ln a)2 , x = aξ , y = bξ . Obviously, when
∂ ln H
I1 < (>)0, we get ∂p f > (<)0.
2) When p = q, from (2.2) and (3.6),
1
ln Hf = ln Gfp (ap , bp ) =
xfx (x, y) ln x + yfy (x, y) ln y
= T 0 (p),
f (x, y)
∂ ln Hf
= T 00 (p) = −xyI1 (ln b − ln a)2 .
∂p
∂ ln Hf
∂p
when I1 < (>)0, we get
> (<)0.
Combining 1) with 2), the proof is completed.
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
Zhen-Hang Yang
Proof of Corollary 2.2. It follows from Theorem 2.1 that the monotonicity of
Hf (p, q) depends on the sign of I1 = (ln f )xy .
1) For f (x, y) = L(x, y),
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1
1
I1 = (ln f )xy =
−
2
(x − y)
xy(ln x − ln y)2
1
√ 2
2
(
xy)
−
L
(x,
y)
.
=
xy(x − y)2
√
By the well-known inequality L(x, y) > xy ([13]), we have I1 < 0.
2) For f (x, y) = A(x, y),
I1 = (ln f )xy = −
1
< 0.
(x + y)2
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3) For f (x, y) = E(x, y),
1
[2(x − y) − (x + y)(ln x − ln y)]
(x − y)3
x+y
2(ln x − ln y)
=
L(x, y) −
.
(x − y)3
2
I1 = (ln f )xy =
By the well-known inequality L(x, y) <
4) For f (x, y) = D(x, y),
I1 = (ln f )xy =
x+y
2
([13]), we have I1 < 0.
1
> 0.
(x − y)2
Applying mechanically Theorem 2.1, we immediately obtain Corollary 2.2.
Proof of Theorem 2.3.
1) Since
p−1
∂ ln Hf
1
pa fx (ap , bp ) qaq−1 fx (aq , bq )
S(p) − S(q)
=
−
=
,
p
p
q
q
∂a
p−q
f (a , b )
f (a , b )
a(p − q)
by the Mean-value Theorem, there exists ξ = q + θ(p − q) with θ ∈ (0, 1), such
that
∂ ln Hf
S(p) − S(q)
=
= a−1 S 0 (ξ).
∂a
a(p − q)
From Lemma 3.4, S 0 (ξ) = xyI2a , where x = aξ , y = bξ . Obviously ,if
∂ ln H
I2a > 0, then ∂a f > 0, so Hf (a, b) is strictly increasing in a; If I2a < 0, then
∂ ln Hf
< 0, so Hf (a, b) is strictly decreasing in a.
∂a
2) It can be proved in the same way.
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Proof of Corollary 2.4.
1) For f (x, y) = L(x, y),
I2a = [(ln f )x ln(y/x)]y =
x/y − 1 − ln(x/y)
.
(x − y)2
By the well-known inequality ln x < x − 1 (x > 0, x 6= 1), we have I2a > 0.
2) For f (x, y) = D(x, y),
I2a = [(ln f )x ln(y/x)]y =
x/y − 1 − ln(x/y)
> 0.
(x − y)2
Since HL (a, b), HD (a, b) are both symmetric with respect to a and b, applying mechanically Theorem 2.3, we immediately obtain Corollary 2.4.
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4.
Some Applications
As direct applications of theorems and lemmas in this paper, we will present
several examples as follows.
Example 4.1 (a G-A inequality chain). By 1) of Corollary 2.2, for f (x, y) =
A(x, y), L(x, y) and E(x, y), Hf (p, q) are strictly increasing in both p and q.
So there are
1
(4.1)
Hf (a, b; 0, 0) < Hf (a, b; 1, 0) < Hf a, b; 1,
2
< Hf (a, b; 1, 1) < Hf (a, b; 1, 2).
From it we can obtain the following inequalities respectively, that are
√ !2
√
√
a+b
a+ b
(4.2)
< E(a, b) <
;
ab < L(a, b) <
2
2
2
√
a+b
a+b
a2 + b 2
√
(4.3)
< √
;
ab <
< Z(a, b) <
2
a+b
a+ b
2

√
E(a, b)
E(a2 , b2 )
(4.4)
ab < E(a, b) <  √ √  < Y (a, b) <
.
E(a, b)
E
a, b
Notice
(4.5)
E(a2 ,b2 )
E(a,b)
= Z(a, b), then (4.4) can be written into that
√ √ √
G2
2
ab < E(a, b) < Z
a, b < E exp 1 − 2 < Z(a, b).
L
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The inequality (4.2) was proved by [13], which shows that can be inserted L,
and E between G and A, so we call (4.2) the G-A inequality chain. (4.3)
and (4.4) are the same in form completely, so we call (4.1) the G-A inequality
chain for homogeneous functions.
A+G
2
Remark 3. That
fact,
E(a2 ,b2 )
E(a,b)
= Z(a, b) is a new identical equation for mean. In
E(a, b)Z(a, b) = e−1
bb
aa
1
b−a
b
b
a
b b+a a b+a
b
a
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
a
= e−1 b b+a + b−a a b+a − b−a
−1
=e b
2b2
b2 −a2
a
2
= e−1
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−2a2
b2 −a2
(b2 )b
(a2 )a2
!
1
b2 −a2
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= E(a2 , b2 ).
Contents
It shows that Z(a, b) is not only one “geometric mean”, but also one ratio of
one exponential mean to another. Thus inequalities involving Z(a, b) may be
translated into inequalities involving exponential mean.
Example 4.2 (An estimation for upper bound of Stolarsky mean). From 2)
of Corollary 2.2, we can prove expediently an estimation for the upper bound of
the Stolarsky mean presented by [12]:
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Sp (a, b) < p
1
1−p
(a + b) with p > 2, where Sp (a, b) =
p
p
b −a
p(b − a)
1
p−1
.
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In fact, from 2) of Corollary 2.2, when p, q ∈ (−∞, 0) ∪ (0, +∞), HD (p, q)
is strictly decreasing in both p and q, so when p > 2, we have HD (a, b; 1, p) <
HD (a, b; 1, 2).
Notice
p
1
1
a − bp p−1
(4.6)
HD (a, b; p, 1) =
= p p−1 Sp (a, b) (p > 0),
a−b
1
1
thus when p > 2, we obtain p p−1 Sp (a, b) < 2 2−1 S2 (a, b) = a+b, i.e. Sp (a, b) <
1
p 1−p (a + b).
Example 4.3 (Reversed inequalities and estimations for exponential mean).
By 1) of Corollary 2.2, HL (p, q) is strictly increasing in both p and q, so when
p1 ∈ (0, 1), p2 ∈ (1, +∞), we have
HL (a, b; p1 , 1) < HL (a, b; 1, 1) < HL (a, b; p2 , 1),
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i.e.
Sp1 (a, b) < E(a, b) < Sp2 (a, b).
(4.7)
On the other hand, By 2) of Corollary 2.2, when p, q ∈ (−∞, 0) ∪ (0, +∞),
HD (p, q) is strictly monotone decreasing in both p and q. So when p1 ∈ (0, 1),
p2 ∈ (1, +∞), we have
(4.8)
On the Homogeneous
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HD (a, b; p1 , 1) > HD (a, b; 1, 1) > HD (a, b; p2 , 1).
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From (4.6), (4.8) can be written into
1
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p2p2 −1 Sp2 (a, b) < eE(a, b) < p1p1 −1 Sp1 (a, b)
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or
(4.9)
1 p21−1
1 1
p2 Sp2 (a, b) < E(a, b) < p1p1 −1 Sp1 (a, b).
e
e
Combining (4.7) with (4.9), we have
(4.10)
1 1
Sp1 (a, b) < E(a, b) < p1p1 −1 Sp1 (a, b), where p1 ∈ (0, 1),
e
(4.11)
1 p21−1
Sp2 (a, b) < E(a, b) < Sp2 (a, b), where p2 ∈ (1, +∞).
p
e 2
1
In particular, when p1 = , p2 = 2, by (4.10), (4.11), we get
2
!
√ !2
√ 2
√
√
4
a+ b
a+ b
(4.12)
< E(a, b) <
,
2
2
e
2 a+b
a+b
(4.13)
< E(a, b) <
.
e
2
2
The inequalities (4.12) and (4.13) may be denoted simply by
On the Homogeneous
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Zhen-Hang Yang
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A+G
4A+G
<E<
,
2
e 2
2
A < E < A.
e
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The inequalities (4.14) and (4.15) make certain a bound of error that exponential mean E are estimated by A or A+G
.
2
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(4.14)
(4.15)
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References
[1] C. GINI, Diuna formula comprensiva delle media, Metron, 13 (1938), 3–
22.
[2] BAINI GUO, SHIQIN ZHANG AND FENG QI, An elementary proof of
monotonicity for extend mean values with two parameters, Mathematics
in Practice and Theory, 29(2) (1999), 169–173.
[3] P.A. HÄSTÖ, A Montonicity property of ratios of symmetric homogeneous means. J. Inequal. Pure and Appl. Math., 3(5) (2005), Art. 48. [ONLINE http://jipam.vu.edu.au/article.php?sid=223].
[4] E.B. LEACH AND M.C. SHOLANDER, Extended mean values, Amer.
Math. Monthly, 85 (1978), 84–90.
[5] E.B. LEACH AND M.C. SHOLANDER, Extended mean values, J. Math.
Anal. Appl. , 92 (1983), 207–223.
[6] FENG QI, Logarithmic convexities of the extended mean values, J. London Math. Soc., to appear.
[7] FENG QI, Generalized weighted mean values with two parameters, Proceedings of the Royal Society of London, Series, Mathematical, Physical
and Engineering Sciences, 454(1978) (1998): 2723–2732.
On the Homogeneous
Functions with Two Parameters
and Its Monotonicity
Zhen-Hang Yang
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[8] FENG QI, On a two-parameter family of nonhomogeneous mean values,
Tamkang Journal of Mathematics, 29(2) (1998), 155–163.
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[9] FENG QI, Generalized abstracted mean values, J. Inequal. Pure and Appl.
Math., 1(1) (2000), Art. 4. [ONLINE http://jipam.vu.edu.au/
article.php?sid=97].
[10] K.B. STOLARSKY, Generalizations of the logarithmic mean, Math. Mag.,
48 (1975), 87–92.
[11] MINGBAO SUN, Inequality for two-parameter mean of convex Function,
Mathematics in Practice and Theory, 27(3) (1997), 193–197.
[12] MINQI SHI AND HUANNAN SHI, An upper bound for extend logarithmic mean, J. Math., 5 (1997), 37–38.
[13] ZHENHANG YANG, Exponential mean and logarithmic mean, Mathematics in Practice and Theory, 4 (1987), 76–78.
[14] ZHENHANG YANG, Inequalities for power mean of convex function,
Mathematics in Practice and Theory, 20(1) (1990), 93–96
[15] ZHENHANG YANG, Inequalities for general mean of convex function,
Mathematics in Practice and Theory, 33(8) (2003), 136–141.
[16] ZHENHANG YANG, Simple discriminance for convexity of homogeneous functions and applications, Study in College Mathematics, 7(4)
(2004), 14–19.
[17] Zs. PALES, Inequalities for differences of powers, J. Math. Anal. Appl. ,
131 (1988), 271–281.
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