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volume 6, issue 4, article 101, 2005.

Received 18 May, 2005;

accepted 04 August, 2005.

Communicated by:P.S. Bullen

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Journal of Inequalities in Pure and Applied Mathematics

ON THE HOMOGENEOUS FUNCTIONS WITH TWO PARAMETERS AND ITS MONOTONICITY

ZHEN-HANG YANG

Zhejiang Electric Power Vocational Technical College Hangzhou, Zhejiang, China, 311600.

EMail:yzhkm@163.com

c

2000Victoria University ISSN (electronic): 1443-5756 155-05

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On the Homogeneous Functions with Two Parameters

and Its Monotonicity Zhen-Hang Yang

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J. Ineq. Pure and Appl. Math. 6(4) Art. 101, 2005

http://jipam.vu.edu.au

Abstract

Supposef(x, y)is a positive homogeneous function defined onU(jR₊×R+), callH_f(a, b;p, q) =

hf(a^p,b^p) f(a^q,b^q)

i_p−q¹

homogeneous function with two parameters.

Iff(x, y)is 2nd differentiable, then the monotonicity in parameterspandqof Hf(a, b;p, q)depend on the signs ofI1= (lnf)xy, for variableaandbdepend on the sign ofI2a = [(lnf)xln(y/x)]_yandI_2b = [(lnf)yln(x/y)]_xrespectively.

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 two- parameter, Monotonicity, Estimate for lower and upper bounds.

Thanks for Mr. Zhang Zhihua

1. Introduction

The so-called two-parameter mean or extended mean values between two unequal positive numbers aand bwere defined first by K.B. Stolarsky in [10] as

(1.1) E(a, b;p, q) =











q(a^p−b^p) p(a^q−b^q)

_p−q¹

p6=q, pq 6= 0 a^p−b^p

p(lna−lnb)

¹_p

p6= 0, q = 0 a^q−b^q

q(lna−lnb)

¹_q

p= 0, q 6= 0 exp

a^plna−b^plnb a^p−b^p − ¹_p

p=q 6= 0

√

ab p=q = 0

.

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:

(1.2) G(a, b;p, q) =











a^p+b^p a^q+b^q

_p−q¹

p6=q exp(^a^p^lna+b_ap+b^p^p^ln^b) p=q 6= 0

√

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 f(a^p,b^p)

f(a^q,b^q)

_p−q¹

, wheref(x, y)is a homogeneous function ofxandy.

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 f(a^p,b^p)

f(a^q,b^q)

_p−q¹

. As applications of the main results, we will deduce three inequality 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.

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2. Basic Concepts and Main Results

Definition 2.1. Assume that f : U(jR+ × R+) → R+ is a homogeneous function of variable x and y, and is continuous and exists first order partial derivative,(a, b)∈R+×R+witha6=b,(p, q)∈R×R. If(1,1)∈/ U, then we define

H_f(a, b;p, q) =

f(a^p, b^p) f(a^q, b^q)

_p−q¹

(p6=q, pq6= 0), (2.1)

H_f(a, b;p, p) = lim

q→pH_f(a, b;p, q) =G_f,p(a, b) (p=q6= 0), (2.2)

where

G_f,p(a, b) = G

1 p

f(a^p, b^p), G_f(x, y) (2.3)

= exp

xf_x(x, y) lnx+yf_y(x, y) lny f(x, y)

,

f_x(x, y)andf_y(x, y)denote partial derivative to 1st and 2nd variable off(x, y) respectively.

If(1,1)∈U, then define further H_f(a, b;p,0) =

f(a^p, b^p) f(1,1)

¹_p

(p6= 0, q= 0), (2.4)

H_f(a, b; 0, q) =

f(a^q, b^q) f(1,1)

¹_q

(p= 0, q 6= 0), (2.5)

H_f(a, b; 0,0) = lim

p→0H_f(a, b;p,0) = a^fx(1,1)^f(1,1)b

fy(1,1)

f(1,1) (p=q = 0).

(2.6)

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From Lemma3.1, H_f(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 bwith two parameterspandq, and call it a two-parameter homogeneous function for short. To avoid confusion, we also denote it byH_f(p, q)orH_f(a, b) orHf.

Iff(x, y)is a positive1-order homogeneous mean function defined onR+× R+, then call H_f(a, b;p, q) the two-parameter f-mean of positive numbers a andb.

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

G_f,0(a, b) = H_f(a, b; 0,0) =√ ab.

In fact, by (2.3), we have

G_f,0(a, b) = exp

f_x(1,1) lna+f_y(1,1) lnb f(1,1)

=H_f(a, b; 0,0).

Sincef(x, y)is a positive1-order homogeneous function, from (3.1) of Lemma 3.2, we obtain

(2.7) 1·f_x(1,1)

f(1,1) +1·f_y(1,1) f(1,1) = 1.

Iff(x, y) = f(y, x), thenf_x(x, y) =f_y(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

f_x(1,1)

f(1,1) = f_y(1,1) f(1,1) = 1

2, therebyG_f,0 =√

ab.

Thus it can be seen that despite the form off(x, y)we always haveH_f(a, b; 0,0)

=G_f,0(a, b) =√

ab, so long asf(x, y)is a positive1-order homogeneous sym- metric function defined onR+×R+.

Example 2.1. In Definition2.1, letf(x, y) =L(x, y) = _ln^x−y_x−ln_y (x, y >0, x6=

y), we get (1.1), i.e.

(2.9) H_L(a, b;p, q) =











q(a^p−b^p) p(a^q−b^q)

_p−q¹

p6=q, pq6= 0 L¹^p(a^p, b^p) p6= 0, q = 0 L¹^q(a^q, b^q) p= 0, q 6= 0 G_L,p(a, b) p=q6= 0

G(a, b) p=q= 0

,

where

G_L,p(a, b) = E_p(a, b) = E¹^p(a^p, b^p) =E_p, E(a, b) =e⁻¹

a^a b^b

_a−b¹

, G(a, b) =

√ ab.

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Remark 2. That

E(a, b) = e⁻¹ a^a

b^b _a−b¹

(a, b >0witha6=b)

is called the exponential mean of unequal positive numbersaandb, and is also called the identical mean and denoted byI(a, b). To avoid confusion, we adopt our terms and notations in what follows.

Example 2.2. In Definition2.1, letf(x, y) =A(x, y) = ^x+y₂ (x, y >0, x6=y), we get (1.2), i.e.

(2.10) HA(a, b;p, q) =











a^p+b^p a^q+b^q

_p−q¹

p6=q GA,p(a, b) p=q6= 0 G(a, b) p=q= 0

,

where G_A,p(a, b) = Z_p(a, b) = Z¹^p(a^p, b^p) = Z_p. Z(a, b) = aâ+bâ bâ+b^b is called the power-exponential mean between positive numbersaandb.

Example 2.3. In Definition2.1, letf(x, y) =E(x, y) = e⁻¹

x^x y^y

_x−y¹

(x, y >

0, x6=y), then

(2.11) H_E(a, b;p, q) =











E(a^p,b^p) E(a^q,b^q)

_p−q¹

p6=q G_E,p(a, b) p=q6= 0 G(a, b) p=q= 0

,

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where G_E,p(a, b) = Y_p(a, b) = Y ¹^p(a^p, b^p) = Y_p. Y(a, b) = Ee¹⁻^G

2

L2 is called the exponential-geometric mean between positive numbersaandb, whereE = E(a, b), L=L(a, b), G=G(a, b).

Example 2.4. In Definition2.1, letf(x, y) = D(x, y) =|x−y|(x, y >0, x6=

y), then

(2.12) HD(a, b;p, q) =





 ^a

p−b^p a^q−b^q

1

p−q p6=q, pq 6= 0 G_D,p(a, b) p=q6= 0

,

whereG_D,p(a, b) =G_D,p=e¹^pE¹^p(a^p, b^p) = e^p¹E_p.

In order to avoid confusion, we renameH_L(a, b;p, q) (orE(a, b;p, q)) and H_A(a, b;p, q)(orG(a, b;p, q)) as the two-parameter logarithmic mean and two- parameter arithmetic mean respectively. In the same way, we callH_E(a, b;p, q) in Example2.3the two-parameter exponential mean.

In Example 2.4, since D(x, y) = |x − y| is not a certain mean between positive numbersxandy, but one absolute value function of difference of two positive numbers, we callH_D(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 H_f(a, b;p, q), we have the following main results.

Theorem 2.1. Letf(x, y)be a positiven−order homogenous function defined onU(jR+×R⁺), and be second order differentiable. IfI1 = (lnf)xy >(<)0,

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thenH_f(p, q)is strictly increasing (decreasing) in both pandq on(−∞,0)∪ (0,+∞).

Corollary 2.2.

1. H_L(p, q),H_A(p, q),H_E(p, q) are strictly increasing both p and q on (−∞,+∞),

2. H_D(p, q)is strictly decreasing bothpandqon(−∞,0)∪(0,+∞).

Theorem 2.3. Letf(x, y)be a positive1-order homogeneous function defined onU(jR+×R⁺), and be second order differentiable.

1. If I_2a = [(lnf)_xln(y/x)]_y > (<)0, then H_f(a, b) is strictly increasing (decreasing) ina.

2. If I_2b = [(lnf)_yln(x/y)]_x > (<)0, then H_f(a, b) is strictly increasing (decreasing) inb.

Corollary 2.4. H_L(a, b),H_D(a, b)is strictly increasing in bothaandb.

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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 overΩrespectively.

If for a certainpwith(x^p, y^p)∈Ω,andf^p(x, y)exists, thenf(x^p, y^p), f^p(x, y) are bothnp-order homogeneous functions overΩ.

Lemma 3.2. Letf(x, y)be an-order homogeneous function overΩ, andf_x, f_y both exist, then fx, fy are both(n−1)-order homogeneous functions overΩ, furthermore we have

(3.1) xf_x+yf_y =nf.

In particular, whenn = 1andf(x, y)is second order differentiable overΩ, then

xf_x+yf_y =f, (3.2)

xf_xx+yf_xy = 0, (3.3)

xf_xy +yf_yy = 0.

(3.4)

Lemma 3.3. Let f(x, y) be a positive n−order homogenous function defined onU(jR+×R+), and be second order differentiable. Set

T(t) = lnf(a^t, b^t), wherex=a^t, y =b^t, a, b >0,

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then

T⁰⁰(t) =−xyI₁(lnb−lna)², whereI₁ = ∂²lnf(x, y)

∂x∂y = (lnf)_xy. Proof. Since f(x, y) is a positive n-order homogeneous function, from (3.1), we can obtainx(lnf)x+y(lnf)y =norx(lnf)x =n−y(lnf)y, y(lnf)y = n−x(lnf)_x, so

T⁰(t) = a^tf_x(a^t, b^t) lna+b^tf_y(a^t, b^t) lnb f(a^t, b^t)

(3.5)

= xfx(x, y) lna+yfy(x, y) lnb f(x, y)

(3.6)

=x(lnf)xlna+y(lnf)ylnb.

(3.7) Hence

T⁰⁰(t) = ∂T⁰(t)

∂x dx

dt + ∂T⁰(t)

∂y dy dt

= [y(lnf)_y(lnb−lna) +nlna]_xa^tlna

+ [x(lnf)_x(lna−lnb) +nlnb]_yb^tlnb

=y(lnf)yx(lnb−lna)xlna+x(lnf)xy(lna−lnb)ylnb

=−xy(lnf)xy(lnb−lna)²

=−xyI₁(lnb−lna)².

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Lemma 3.4. Let f(x, y) be a positive 1-order homogeneous function defined onU(jR+×R⁺), and be second order differentiable. Set

S(t) = txf_x(x, y)

f(x, y) , wherex=a^t, y =b^ta, b >0, then

S⁰(t) =xyI_2a, whereI_2a= [(lnf)_xln(y/x)]_y. Proof.

S⁰(t) = xf_x(x, y) f(x, y) +td

dt

xf_x(x, y) f(x, y)

=x(lnf)_x+t

∂(x(lnf)_x)

∂x

dx

dt + ∂(x(lnf)_x)

∂y

dy dt

=x(lnf)_x+t

∂(x(lnf)_x)

∂x a^tlna+∂(x(lnf)_x)

∂y b^tlnb

=x(lnf)x+t[x(x(lnf)x)xlna+y(x(lnf)x)ylnb].

By Lemma 3.1, that x(lnf)_x = ^xf_f^x_(x,y)^(x,y) is a 0-order homogeneous function, from (3.1) of Lemma3.2, we obtain

x[x(lnf)_x]_x+y[x(lnf)_x]_y = 0 or x[x(lnf)_x]_x =−y[x(lnf)_x]_y,

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hence

S⁰(t) =x(lnf)_x+ty[x(lnf)_x]_y(lnb−lna)

=x(lnf)_x+txy(lnf)_xy(lnb−lna)

=x(lnf)_x+xy(lnf)_xy(lnb^t−lna^t)

=x(lnf)_x+xy(lnf)_xy(lny−lnx)

=xy

y⁻¹(lnf)_x+ (lnf)_xyln(y/x)

=xy[(lnf)_xln(y/x)]_y =xyI_2a.

Based on the above lemmas, then next we will go on proving the main results in this paper.

Proof of Theorem2.1. SinceH_f(p, q)is symmetric with respect topandq, we only need to prove the monotonicity forpoflnH_f.

1) Whenp6=q,

lnH_f = 1

p−qlnf(a^p, b^p)

f(a^q, b^q) = T(p)−T(q) p−q ,

∂lnH_f

∂p = (p−q)T⁰(p)−T(p) +T(q)

(p−q)² .

Setg(p) = (p−q)T⁰(p)−T(p) +T(q), theng(q) = 0,g⁰(p) = (p−q)T⁰⁰(p), and then existξ =q+θ(p−q)withθ ∈ (0,1)by Mean-value Theorem, such that

∂lnH_f

∂p = g(p)−g(q)

(p−q)² = g⁰(ξ)

p−q = (ξ−q)T⁰⁰(ξ)

p−q = (1−θ)T⁰⁰(ξ).

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By Lemma3.3,T⁰⁰(ξ) =−xyI₁(lnb−lna)²,x=a^ξ, y =b^ξ. Obviously, when I1 <(>)0, we get ^∂^ln_∂p^H^f >(<)0.

2) Whenp=q, from (2.2) and (3.6), lnH_f = lnG

1 p

f(a^p, b^p) = xf_x(x, y) lnx+yf_y(x, y) lny

f(x, y) =T⁰(p),

∂lnH_f

∂p =T⁰⁰(p) = −xyI1(lnb−lna)². whenI1 <(>)0, we get ^∂^ln_∂p^H^f >(<)0.

Combining 1) with 2), the proof is completed.

Proof of Corollary2.2. It follows from Theorem 2.1 that the monotonicity of H_f(p, q)depends on the sign ofI₁ = (lnf)_xy.

1) Forf(x, y) = L(x, y),

I₁ = (lnf)_xy = 1

(x−y)² − 1

xy(lnx−lny)²

= 1

xy(x−y)² (√

xy)²−L²(x, y) . By the well-known inequalityL(x, y)>√

xy([13]), we haveI₁ <0.

2) Forf(x, y) = A(x, y),

I₁ = (lnf)_xy =− 1

(x+y)² <0.

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3) Forf(x, y) = E(x, y), I1 = (lnf)xy = 1

(x−y)³ [2(x−y)−(x+y)(lnx−lny)]

= 2(lnx−lny) (x−y)³

L(x, y)−x+y 2

.

By the well-known inequalityL(x, y)< ^x+y₂ ([13]), we haveI₁ <0.

4) Forf(x, y) = D(x, y),

I₁ = (lnf)_xy = 1

(x−y)² >0.

Applying mechanically Theorem2.1, we immediately obtain Corollary2.2.

Proof of Theorem2.3.

1) Since

∂lnH_f

∂a = 1

p−q

pa^p−1f_x(a^p, b^p)

f(a^p, b^p) − qa^q−1f_x(a^q, b^q) f(a^q, b^q)

= S(p)−S(q) a(p−q) , by the Mean-value Theorem, there existsξ =q+θ(p−q)withθ ∈(0,1), such

that ∂lnH_f

∂a = S(p)−S(q)

a(p−q) =a⁻¹S⁰(ξ).

From Lemma 3.4, S⁰(ξ) = xyI2a, where x = a^ξ, y = b^ξ. Obviously ,if I_2a >0, then ^∂^ln_∂a^H^f >0, soH_f(a, b)is strictly increasing ina; IfI_2a <0, then

∂lnH_f

∂a <0, soHf(a, b)is strictly decreasing ina.

2) It can be proved in the same way.

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Proof of Corollary2.4.

1) Forf(x, y) = L(x, y),

I_2a= [(lnf)_xln(y/x)]_y = x/y−1−ln(x/y) (x−y)² .

By the well-known inequalitylnx < x−1 (x >0, x6= 1),we haveI_2a >0.

2) Forf(x, y) = D(x, y),

I_2a= [(lnf)_xln(y/x)]_y = x/y−1−ln(x/y) (x−y)² >0.

SinceHL(a, b),HD(a, b)are both symmetric with respect toaandb, applying mechanically Theorem2.3, we immediately obtain Corollary2.4.

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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 Corollary2.2, forf(x, y) = A(x, y), L(x, y)and E(x, y), H_f(p, q) are strictly increasing in both pandq.

So there are

H_f(a, b; 0,0)<H_f(a, b; 1,0)<H_f

a, b; 1,1 2

(4.1)

<H_f(a, b; 1,1)<H_f(a, b; 1,2).

From it we can obtain the following inequalities respectively, that are

√

ab < L(a, b)<

√a+√ b 2

!2

< E(a, b)< a+b 2 ; (4.2)

√

ab < a+b 2 <

a+b

√a+√ b

2

< Z(a, b)< a²+b² a+b ; (4.3)

√

ab < E(a, b)<





E(a, b) E

√ a,√

b





2

< Y(a, b)< E(a², b²) E(a, b) . (4.4)

Notice ^E(a_E(a,b)²^,b²⁾ =Z(a, b), then (4.4) can be written into that

(4.5) √

ab < E(a, b)< Z²√ a,√

b

< Eexp

1−G² L²

< Z(a, b).

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The inequality (4.2) was proved by [13], which shows that can be insertedL,

A+G

2 andE betweenGandA, 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.

Remark 3. That ^E(a_E(a,b)²^,b²⁾ = Z(a, b) is a new identical equation for mean. In fact,

E(a, b)Z(a, b) =e⁻¹ b^b

a^a b−a¹

b^b+a^b a^b+a^a

=e⁻¹b^b+a^b ⁺^b−a^b a^b+a^a ⁻^b−a^a

=e⁻¹b ^2b

2 b2−a2a

−2a2 b2−a2

=e⁻¹ (b²)^b² (a²)^a²

! ¹

b2−a2

=E(a², b²).

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 Corollary2.2, we can prove expediently an estimation for the upper bound of the Stolarsky mean presented by [12]:

S_p(a, b)< p^1−p¹ (a+b)withp > 2, whereS_p(a, b) =

b^p−a^p p(b−a)

_p−1¹ .

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In fact, from 2) of Corollary2.2, whenp, q ∈(−∞,0)∪(0,+∞),H_D(p, q) is strictly decreasing in both pandq, so whenp >2, we haveHD(a, b; 1, p)<

H_D(a, b; 1,2).

Notice

(4.6) H_D(a, b;p,1) =

a^p−b^p a−b

_p−1¹

=p^p−1¹ S_p(a, b) (p >0),

thus whenp >2, we obtainp^p−1¹ S_p(a, b)<2²⁻¹¹ S₂(a, b) =a+b, i.e. S_p(a, b)<

p^1−p¹ (a+b).

Example 4.3 (Reversed inequalities and estimations for exponential mean).

By 1) of Corollary2.2, H_L(p, q)is strictly increasing in bothpandq, so when p₁ ∈(0,1),p₂ ∈(1,+∞), we have

H_L(a, b;p₁,1)<H_L(a, b; 1,1)<H_L(a, b;p₂,1), i.e.

(4.7) S_p₁(a, b)< E(a, b)< S_p₂(a, b).

On the other hand, By 2) of Corollary2.2, whenp, q ∈(−∞,0)∪(0,+∞), H_D(p, q)is strictly monotone decreasing in bothpandq. So whenp₁ ∈ (0,1), p₂ ∈(1,+∞), we have

(4.8) H_D(a, b;p₁,1)>H_D(a, b; 1,1)>H_D(a, b;p₂,1).

From (4.6), (4.8) can be written into p

1 p2−1

2 Sp2(a, b)< eE(a, b)< p

1 p1−1

1 Sp1(a, b)

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or

(4.9) 1

ep

1 p2−1

2 S_p₂(a, b)< E(a, b)< 1 ep

1 p1−1

1 S_p₁(a, b).

Combining (4.7) with (4.9), we have S_p₁(a, b)< E(a, b)< 1

ep

1 p1−1

1 S_p₁(a, b), wherep₁ ∈(0,1), (4.10)

1 ep

1 p2−1

2 S_p₂(a, b)< E(a, b)< S_p₂(a, b), wherep₂ ∈(1,+∞).

(4.11)

In particular, whenp1 = 1

2, p2 = 2, by (4.10), (4.11), we get

√a+√ b 2

!2

< E(a, b)< 4 e

√a+√ b 2

!2

, (4.12)

2 e

a+b 2

< E(a, b)< a+b 2 . (4.13)

The inequalities (4.12) and (4.13) may be denoted simply by A+G

2 < E < 4 e

A+G 2 , (4.14)

2

eA < E < A.

(4.15)

The inequalities (4.14) and (4.15) make certain a bound of error that expo- nential meanE are estimated byAor ^A+G₂ .

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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 homoge- neous means. J. Inequal. Pure and Appl. Math., 3(5) (2005), Art. 48. [ON- LINEhttp://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 ANDM.C. SHOLANDER, Extended mean values, J. Math.

Anal. Appl. , 92 (1983), 207–223.

[6] FENG QI, Logarithmic convexities of the extended mean values, J. Lon- don Math. Soc., to appear.

[7] FENG QI, Generalized weighted mean values with two parameters, Pro- ceedings of the Royal Society of London, Series, Mathematical, Physical and Engineering Sciences, 454(1978) (1998): 2723–2732.

[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 logarith- mic mean, J. Math., 5 (1997), 37–38.

[13] ZHENHANG YANG, Exponential mean and logarithmic mean, Mathe- matics 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 homoge- neous 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.