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NUMBERS WITH TWO PARAMETERS

ZHEN-GANG XIAO, ZHI-HUA ZHANG, AND FENG QI

Abstract. In this article, a type of mean values of several positive numbers with two parameters are defined by using the generalized Vandermonde deter-minant, some basic properties of them are given, and several inequalities for some mean values including the generalized symmetric means are established.

1. Introduction and notations

LetR= (−∞,∞), R+ = (0,∞), a = (a0, a1, . . . , an) ∈Rn++1, ϕbe a function

defined inR+, and

V(a;ϕ) =

1 a0 a2

0 · · · a

n−1

0 ϕ(a0)

1 a1 a21 · · · a

n−1

1 ϕ(a1) . . . .

1 an a2n · · · ann−1 ϕ(an)

. (1.1)

Takingϕ(x) =xn+r(lnx)k, then

V(a;ϕ) =

1 a0 a20 · · · a

n−1 0 a

n+r

0 (lna0)k

1 a1 a2

1 · · · a

n−1 1 a

n+r

1 (lna1)k . . . .

1 an a2n · · · ann−1 ann+r(lnan)k

,V(a;r, k). (1.2)

If lettingr= 0 andk= 0 in (1.2), then

V(a; 0,0) = n X

i=0

(−1)n+ian

iVi(a) = Y

0≤i<j≤n

(aj−ai) (1.3)

is the determinant of Vandermonde matrix of ordern+ 1, where

Vi(a) =

1 a0 a2

0 · · · a

n−1 0

1 a1 a21 · · · a

n−1 1 . . . .

1 ai−1 a2i−1 · · · ani−1−1

1 ai+1 a2i+1 · · · a

n−1

i+1 . . . .

1 an a2n · · · ann−1

(1.4)

2000Mathematics Subject Classification. Primary 26D15.

Key words and phrases. mean values, Vandermonde determinant, inequality, monotonicity, basic property, generalized symmetric mean.

The third author was supported in part by the Science Foundation of Project for Fostering Innovation Talents at Universities of Henan Province, China.

This paper was typeset usingAMS-LATEX.

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is the cofactor of the elementani in V(a; 0,0). So, we callV(a;ϕ) the generalized Vandermonde determinant.

Let a0 and a1 be two positive numbers. It is well known that the logarithmic mean L(a0, a1) and the identric or exponential mean I(a0, a1) of a0 and a1 are

respectively defined by

L(a0, a1) = 

a1−a0

lna1−lna0, a06=a1, a0, a0=a1

(1.5)

and

I(a0, a1) =

 

  1

e

aa1

1 aa0

0

a1−a1 0

, a06=a1,

a0, a0=a1.

(1.6)

LetG(a0, a1) =

a0a1 and A(a0, a1) = a0+2a1 denote the geometric mean and the

arithmetic mean ofa0and a1, respectively. Fora06=a1, the following inequalities

are proved in [4]:

G(a0, a1)< L(a0, a1)< I(a0, a1)< A(a0, a1). (1.7)

In [28, 29], Zh.-H. Zhang and Zh.-G. Xiao studied the identric mean and two logarithmic means ofn+ 1 positive numbersa0, a1, . . . , an which are defined as

I(a) = exp "

V(a; 0,1)

V(a; 0,0) − n X

k=1

1

k

#

, (1.8)

L(a) = V(a; 0,0)

nV(a;−1,1), (1.9)

l(a) =n!Vln(a; 1,0)

Vln(a; 0, n) , (1.10)

where

Vln(a;r, k),

1 lna0 (lna0)2 · · · (lna0)n−1 ar0(lna0)k

1 lna1 (lna1)2 · · · (lna1)n−1 ar1(lna1)k . . . .

1 lnan (lnan)2 · · · (lnan)n−1 arn(lnan)k

. (1.11)

They proved fora withai 6=aj for alli6=j the following inequalities

G(a)< L(a)< I(a)< A(a) (1.12)

and

G(a)< l(a)< I(a)< A(a), (1.13)

whereG(a) =Qn i=0a

1/(n+1)

i andA(a) =

1

n+1

Pn

i=0ai are the geometric mean and the arithmetic mean ofn+ 1 positive numbersa0, a1, . . . , an∈R+, respectively.

Forr∈R\{−1,0}, the extended logarithmic meanSr(a0, a1) and the generalized logarithmic mean Jr(a0, a1) of two positive numbers a0 and a1 are defined (See [1, 2, 10, 13, 15, 24]) respectively by

Sr(a0, a1) = 

 

 

ar1+1−ar0+1

(r+ 1)(a1−a0) 1/r

, a06=a1,

a0, a0=a1

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and

Jr(a0, a1) =

r r+ 1·

ar1+1−ar0+1 ar

1−ar0

, a06=a1,

a0, a0=a1.

(1.15)

These two means and other mean values of two positive numbers are special cases of the extended mean values

E(p, q;a0, a1) =                               

q(ap

1−a

p

0) p(aq1−aq0)

1/(p−q)

forpq(p−q)(a0−a1)6= 0,

ap

1−a

p

0 p(lna1−lna0)

1/p

forp(a0−a1)6= 0 andq= 0,

exp

−1

p+

ap1lna1−ap0lna0 ap1−ap0

forp(a0−a1)6= 0 and p=q,

a0a1 fora0−a16= 0 and p=q= 0, a0 forp=qanda0=a1,

(1.16) =       

h(q;a0, a1)

h(p;a0, a1)

1/(q−p)

, (p−q)(a0−a1)6= 0,

exp

∂h(p;a

0, a1)/∂p h(p;a0, a1)

, a0−a16= 0, p=q,

(1.17)

where

h(t;a0, a1) = 

at

1−at0

t , t6= 0

lna1−lna0, t= 0 =

Z a1

a0

ut−1du. (1.18)

There have been a lot of literature about the extended mean valuesE(p, q;a0, a1). For more information, please refer to [5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25] and the references therein.

In [26, 27], Zh.-G. Xiao and Zh.-H. Zhang introduced

Sr(a) =                              n! n Q k=1

(k+r)

· V(a;r,0)

V(a; 0,0) 

  

1/r

, r6= 0,−1, . . . ,−n,

exp

V(a; 0,1)

V(a; 0,0)− n P k=1 1 k

, r= 0,

n!V(a;r,1)

(−1)r+1(−r1)!(n+r)!V(a; 0,0)

1/r

, r=−1, . . . ,−n

(1.19)

and

Jr(a) =                      r n+r·

V(a, r; 0)

V(a;r−1,0), r6= 0,−1, . . . ,−n,

V(a,0; 0)

nV(a,−1,1), r= 0,

r n+r·

V(a;r,1)

V(a;r−1,1), r=−1, . . . ,−(n−1), −nV(a; 0,1)

V(a;−n−1,0), r=−n

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withS0(a) =I(a) andS−1(a) =J0(a) =L(a).

The main purpose of this article is to define a type of mean values of several positive numbers with two parameters, from which many mean values mentioned above can be deduced, by utilizing the generalized Vandermonde determinant, to research their basic properties, and to establish several inequalities.

2. Lemmas

The following two lemmas are keys for us to research the basic properties of the mean valuesE(p, q;a) defined by Definition 3.1 in next section.

Lemma 2.1. Let n∈N,a= (a0, a1, . . . , an)∈Rn++1 and ϕbe an-times

differen-tiable function inR+. Then

V(a;ϕ) =V(a; 0,0) Z

D

ϕ(n)

n X

i=0 aixi

!

dx, (2.1)

where x0 = 1−Pn

i=1xi and dx= dx1dx2· · ·dxn denotes the differential of the

volume in

D= (

(x1, x2, . . . , xn)

n X

i=1

xi≤1 withxi≥0fori= 1,2, . . . , n )

. (2.2)

Proof. Let ¯a = (an, a1, . . . , an−1) ∈ Rn+ and ˜a = (a0, a1, . . . , an−1) ∈ Rn+. From

(1.3), one finds that

V(a; 0,0) = (−1)n−1 n Y

j=1

(aj−a0)V(¯a; 0,0)

= (an−a0)

n−1

Y

j=1

(an−aj)V(˜a; 0,0),

(2.3)

V0(a) = n−1

Y

j=1

(an−aj)V0(˜a), (2.4)

Vi(a) = (−1)n n−1

Y

j=1

(aj−a0)Vi(¯a) + n−1

Y

j=1

(an−aj)Vi(˜a), 1< i < n, (2.5)

Vn(a) = n−1

Y

j=1

(aj−a0)V0(¯a). (2.6)

The expansion of the generalized Vandermonde determinant (1.1) equals

V(a;ϕ) = n X

i=0

(−1)n+iϕ(a

i)Vi(a). (2.7)

It is easy to see that

Z

D

ϕ(n)

n X

i=0 aixi

!

dx

= Z 1

0

Z 1−x1

0

· · ·

Z 1−Pn−i=11xi 0

ϕ(n) a0+

n X

i=1

(ai−a0)xi !

(5)

Forn= 1, formula (2.1) is true, since

(a1−a0) Z 1

0

ϕ0(a0+ (a1−a0)x1) dx1=ϕ(a1)−ϕ(a0) =V(a;ϕ).

Assume that (2.1) holds forn−1 withn≥2. Then

V(¯a;ϕ) = (−1)n−1ϕ(an)V0(¯a) +

n−1

X

i=1

(−1)n−1+iϕ(ai)Vi(¯a)

=V(¯a; 0,0) Z 1

0

Z 1−x1

0

· · ·

Z 1−Pn−i=12xi

0

ϕ(n−1) an+ n−1

X

i=1

(ai−an)xi !

dx1dx2· · ·dxn−1 (2.9)

and

V(˜a;ϕ) = n−1

X

i=0

(−1)n−1+iϕ(a i)Vi(˜a)

=V(˜a; 0,0) Z 1

0

Z 1−x1

0

· · ·

Z 1−Pn−i=12xi 0

ϕ(n−1) a0+ n−1

X

i=1

(ai−a0)xi !

dx1dx2· · ·dxn−1. (2.10)

Combination of (2.3) to (2.10) shows that

V(a; 0,0) Z

D

ϕ(n)

n X

i=0 aixi

!

dx

=V(a; 0,0) Z 1

0

Z 1−x1

0

· · ·

Z 1−Pn−i=11xi 0

ϕ(n) a0+ n X

i=1

(ai−a0)xi !

dx1dx2· · ·dxn

=V(a; 0,0)

an−a0

Z 1

0

Z 1−x1

0

· · ·

Z 1−Pn−i=12xi 0

"

ϕ(n−1) an+ n−1

X

i=1

(ai−an)xi !

−ϕ(n−1) a0+

n−1

X

i=1

(ai−a0)xi !#

dx1dx2· · ·dxn−1

= (−1)n−1

n−1

Y

j=1

(aj−a0)V(¯a; 0,0)

Z 1

0

Z 1−x1

0

· · ·

Z 1−Pn−i=12xi 0

ϕ(n−1) an+ n−1

X

i=1

(ai−an)xi !

dx1· · ·dxn−1

− n−1

Y

j=1

(an−aj)V(˜a; 0,0) Z 1

0

Z 1−x1

0

· · ·

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ϕ(n−1) a0+ n−1

X

i=1

(ai−a0)xi !

dx1· · ·dxn−1

= (−1)n−1

n−1

Y

j=1

(aj−a0) "

(−1)n−1ϕ(a

n)V0(¯a) + n−1

X

i=1

(−1)n−1+iϕ(a i)Vi(¯a)

#

− n−1

Y

j=1

(an−aj) n−1

X

i=1

(−1)n−1+iϕ(a i)Vi(˜a)

=ϕ(an) n−1

Y

j=1

(aj−a0)V0(¯a)−(−1)n−1ϕ(a0) n−1

Y

j=1

(an−aj)V0(˜a)

+ n−1

X

i=1

(−1)n−1+iϕ(ai) 

(−1)n−1 n−1

Y

j=1

(aj−a0)Vi(¯a)− n−1

Y

j=1

(an−aj)Vi(˜a) 

=ϕ(an)Vn(a) + n−1

X

i=1

(−1)n+iϕ(a

i)Vi(a) + (−1)nϕ(a0)V0(a)

=V(a;ϕ).

Thus, by induction, the required formula (2.1) is proved.

Lemma 2.2. Let rbe an integer and a= (a0, a1, . . . , an)∈Rn++1. Then

V(a;r,0) = 

       

       

V(a; 0,0) P i0+i1+···+in=r

i0, i1, . . . , in≥0are integers

n Q

k=0 aik

k forr >0,

0 forr= 0,−1, . . . ,−(n−1),

(−1)nV(a; 0,0) P i0+i1+···+in=−r

i0,i1,...,in∈N n Q

k=0 a−ik

k forr≤ −n.

(2.11)

Proof. TakingVn(a;r,0) ,V(a;r,0). It is obvious that, if r=−1, . . . ,−n, then

Vn(a;r,0) =V(a;r,0) = 0.

In the case of n ∈ N and r ≥ 0, we will verify Lemma 2.2 by mathematical

induction. It is clear that identity (2.11) holds trivially forn= 1, since

V2(a;r,0) =ar1+1−ar0+1

= (a1−a0) ar1+a1r−1a0+· · ·+ar0

= (a1−a0) X i0+i1=r

i0,i1≥0 are integers

ai0

0a

i1

1.

Suppose identity (2.11) is true forn−1 and positive integerst. That is

Vn−1(a;t,0) =Vn−1(a; 0,0)

X

i0+i1+···+in−1=t

i0,i1,...,in−1≥0 are integers

n−1

Y

k=0 aik

(7)

By (1.2) and (2.12), one reveals

Vn(a;r,0) =

1 a0−an · · · an0−1−a

n−2

0 an an0+r−a

n−1 0 arn+1 1 a1−an · · · an1−1−a

n−2

1 an an1+r−a

n−1 1 arn+1

. . . .

1 an−1−an · · · ann−1−1−ann−1−2an ann+−1r−ann−1−1arn+1

1 0 · · · 0 0

= (−1)n+2

a0−an · · · an0−1−a

n−2

0 an an0+r−a

n−1 0 arn+1

a1−an · · · an1−1−a

n−2

1 an an1+r−a

n−1 1 a

r+1

n

. . . . an−1−an · · · ann−1−1−a

n−2

n−1an ann+−1r−a

n−1

n−1arn+1 = n−1 Y i=0

(an−ai)

1 a0 a20 · · · a

n−1 0

P

t+in=ra

n−1+t

0 a

in

n 1 a1 a21 · · · an1−1 P

t+in=ra

n−1+t

1 a

in

n

. . . .

1 an−1 a2n−1 · · · ann−1−1

P

t+in=ra

n−1+t n−1 ainn

= n−1 Y i=0

(an−ai) X

t+in=r ain

n

1 a0 a2

0 · · · a

n−1 0 a

n−1+t

0

1 a1 a21 · · · a

n−1 1 a

n−1+t

1 . . . .

1 an−1 a2n−1 · · · ann−1−1 ann−1+−1 t

= n−1 Y i=0

(an−ai) X

t+in=r

ain

nVn−1(a;t,0)

= n−1

Y

i=0

(an−ai)·Vn−1(a; 0,0)

X

t+in=r ain

n

X

i0+i1+···+in−1=t,

i0,i1,...,in−1≥0 are integers

n−1

Y

k=0 aik

k

=V(a; 0,0) X i0+i1+···+in=r,

i0,i1,...,in≥0 are integers

n Y

k=0 aik

k .

By induction, it is showed that (2.11) holds fornand positive integersr. From (1.3), it is deduced easily that

V(a−1; 0,0) = n X

i=0

(−1)n+ia−n

i Vi(a−1)

= Y

0≤i<j≤n

(a−1j −a−1i )

= Y

0≤i≤n

a−in Y 0≤i<j≤n

(ai−aj)

= (−1)n(n+1)/2 Y

0≤i≤n

a−inV(a; 0,0),

(2.13)

wherea−1= (a−1 0 , a

−1

(8)

In the case ofr <−n, we have−(n+r)>0. From (1.2) and (2.13), one finds

V(a;r,0) = Y

0≤i≤n

ani−1

a−(0 n−1) a−(0 n−1) · · · 1 ar0+1 a−(1 n−1) a−(1 n−2) · · · 1 ar1+1 . . . . an−(−1n−1) a−(n−1n−2) · · · 1 arn+1−1

= (−1)n(n−1)/2 Y

0≤i≤n

ani−1

1 a−10 · · · a0−(n−1) a−[0 n−(n+r+1)]

1 a−11 · · · a1−(n−1) a−[1 n−(n+r+1)] . . . .

1 a−1n−1 · · · an−(−1n−1) a−[n−1n−(n+r+1)]

= (−1)n(n−1)/2 Y

0≤i≤n

ani−1V(a−1; 0,0) X

i0+i1+···+in=−(n+r+1)

i0,i1,...,in≥0 are integers

n Y

k=0 a−ik

k

= (−1)n Y

0≤i≤n

a−1i V(a; 0,0) X

i0+i1+···+in=−(n+r+1)

i0,i1,...,in≥0 are integers

n Y

k=0 a−ik

k

= (−1)nV(a; 0,0) X i0+i1+···+in=−r

i0,i1,...,in≥1 are integers

n Y

k=0 a−ik

k .

The proof of Lemma 2.2 is completed.

Lemma 2.3 ([19, 21, 23]). Letf be a continuous function andpa positive contin-uous weight onI. Then the weighted arithmetic mean of function f with weightp

defined by

F(x, y) = 

 

 

Ry

x p(t)f(t) dt Ry

x p(t) dt

, x6=y,

f(x), x=y

(2.14)

is increasing (decreasing)on I2 iff is increasing (decreasing)onI.

3. Definition and basic properties

Now we are in a position to give the definition of mean values of several positive numbers with two parameters.

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q∈Rare defined as

E(p, q;a) =                                                                          n Q k=1 k+q k+p·

V(a;p,0)

V(a;q,0)

1/(p−q)

for (p−q) n Q

k=1

[(k+p)(k+q)]6= 0,

  

(−1)q+1(−q1)!(q+n)!

n Q

k=1

(k+p)

· V(a;p,0)

V(a;q,1) 

  

1/(p−q)

forp6=q=−1,−2, . . . ,−n,

(−1)q−p(−q1)!(q+n)!

(−p−1)!(p+n)! ·

V(a;p,1)

V(a;q,1)

1/(p−q)

forp6=qandp, q=−1,−2, . . . ,−n,

exp

V(a;p,1)

V(a;p,0)− n P

k=1

1

k+p

forp=q6=−1,−2, . . . ,−n,

exp 

V(a;p,2) 2V(a;p,1) −

n P

k=1

k6=−p 1

k+p

forp=q=−1,−2, . . . ,−n.

(3.1)

The mean valuesE(p, q;a) have the same usual properties as many other means listed in [3].

Theorem 3.1. The following basic properties hold:

(1) E(p, q;a) =E(q, p;a),

(2) limp→∞E(p, q;a) = limq→∞E(p, q;a) = max{a0, a1, . . . , an}, (3) limp→−∞E(p, q;a) = limq→−∞E(p, q;a) = min{a0, a1, . . . , an}, (4) limp→qE(p, q;a) =E(q, q;a),

(5) min{a0, a1, . . . , an} ≤E(p, q;a)≤max{a0, a1, . . . , an}, (6) E(p, q;a) =a0 if and only ifa0=a1=· · ·=an, (7) Fort >0,E(p, q;ta) =tE(p, q;a),

(8) [E(p, q;a)]p−q = [E(p, t;a)]p−t[E(t, q;a)]t−q,

whereta= (ta0, ta1, . . . , tan).

Proof. These follow straightforwardly from Definition 3.1 and Lemma 2.1 by

stan-dard arguments.

The theorem below gives an alternative expression of the mean valuesE(p, q;a).

Theorem 3.2. Let n∈N,a= (a0, a1, . . . , an)∈Rn++1, andp, q∈R. Then

E(p, q;a) =            " R Dϕ

(n) 1 p,

Pn i=0aixi

dx

R Dϕ

(n) 1 q,

Pn i=0aixi

dx

#1/(p−q)

, p6=q,

exp ( R

(n) 2 p,

Pn i=0aixi

dx

R Dϕ

(n) 1 p,

Pn i=0aixi

dx

)

, p=q,

(3.2)

whereD is the simplex defined by (2.2),dx= dx1dx2· · ·dxn is the differential of

the volume inD,x0= 1−P

n i=1xi,ϕ

(n)

1 (p, t) =t

p andϕ(n)

2 (p, t) =t

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Proof. This follows from substituting

ϕ1(p, t) = 

  

  

tn+p

n Q

k=1

1

k+p forp6=−1, . . . ,−n, tn+plnt

(−1)p+1(−p1)!(n+p)! forp=−1, . . . ,−n

(3.3)

and

ϕ2(p, t) = 

      

      

tn+p

lnt−

n P

k=1

1

k+p

n Q

k=1

1

k+p forp6=−1, . . . ,−n,

tn+plnt

(−1)p+1(−p−1)!(n+p)! 

 lnt

2 − n P

k=1

k6=−p 1

k+p

forp=−1, . . . ,−n

(3.4)

into Lemma 2.1.

Corollary 3.1. Let n∈Nanda= (a0, a1, . . . , an)∈Rn++1. Then Sr(a) =E(r,0;a) =

(

n!R Dϕ

(n) 1 r,

Pn i=0aixi

dx1/r

, r6= 0,

exp

n!R Dϕ

(n) 2 0,

Pn i=0aixi

dx , r= 0, (3.5)

Jr(a) =E(r, r−1;a) = R

(n) 1 r,

Pn i=0aixi

dx

R Dϕ

(n) 1 r−1,

Pn i=0aixi

dx

, (3.6)

E(p, q;a) = [S

p(a)]p [Sq(a)]q

1/(p−q)

, (3.7)

where D is defined by (2.2), dx = dx1dx2· · ·dxn denotes the differential of the

volume inD, andx0= 1−Pn i=1xi.

Some mean values are special cases of the mean values E(p, q;a) as Corollary 3.1 reveals. From Definition 3.1 and Lemma 2.2, we further obtain the following

Theorem 3.3. Let n∈Nanda= (a0, a1, . . . , an)∈Rn++1. Then

(1) E(1,0;a) =S1(a) =J1(a) =A(a),

(2) E(0,−(n+ 1);a) =S−(n+1)(a) =G(a),

(3) E(−(n+ 1),−(n+ 2);a) =J−(n+1)(a) =H(a),

(4) E(0,0;a) =S0(a) =I(a),

(5) E(0,−1;a) =S−1(a) =J0(a) =L(a),

where A(a), G(a), H(a), I(a), and L(a) denote respectively the arithmetic, geo-metric, harmonic, exponential, and logarithmic means of n+ 1 positive numbers

a0, a1, . . . , an.

Proof. The proof is straightforward.

Some special cases ofE(p, q;a) have relationships with the generalized symmetric

means k P

n

(a) of orderkintroduced in [7] by D. M. DeTemple and J. M. Robertson.

Theorem 3.4. Let r be a positive integer and a = (a0, a1, . . . , an) ∈ Rn++1 for n∈N. Then

E(r,0;a) =Sr(a) = r X

n+1

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E(r, r−1;a) =Jr(a) = r P

n+1

(a)

r−1

P

n+1

(a)

, (3.9)

where

k X

n+1

(a) = n+k

n

−1

X

i0+i1+···+in=k

i0,i1,...,in≥0

ai0

0a

i1

1 · · ·a

in

n (3.10)

denotes thek-th generalized symmetric mean ofn+1positive numbersa0, a1, . . . , an.

4. Monotonicity and applications

Theorem 4.1. The mean valuesE(p, q;a) are increasing strictly with bothp∈R

andq∈R.

Proof. Letg(x) =Pn

i=0aixi. Ifp=q, then

E(p, p;a) = exp ( R

(n) 2 p, g(x)

dx

R Dϕ

(n) 1 p, g(x)

dx

)

and

Z

D

ϕ(1n) p, g(x) dx

2 d

dp[lnE(p, p;a)]

= Z

D

ϕ2(n) p, g(x)ϕ2(n) 0, g(x)dx

Z

D

ϕ(1n) p, g(x)dx

− Z

D

ϕ(1n) p, g(x)

ϕ(2n) 0, g(x) dx

2

= Z

D

[g(x)]p[lng(x)]2dx

Z

D

[g(x)]pdx− Z

D

[g(x)]pln[g(x)] dx

2

>0

by using the Cauchy-Schwartz-Buniakowski integral inequality. This implies that the mean valuesE(p, p;a) are strictly increasing withp∈R.

Ifp6=q, then

E(p, q;a) = " R

(n) 1 p, g(x)

dx

R Dϕ

(n) 1 q, g(x)

dx

#1/(p−q)

and

lnE(p, q;a) = 1

p−q

Z p

q R

(n) 2 θ, g(x)

dx

R Dϕ

(n) 1 θ, g(x)

dx

= 1

p−q

Z p

q

lnE(θ, θ;a) dθ.

From Lemma 2.3 and the fact that E(p, p;a) are strictly increasing with p ∈ R,

it is not difficult to see that the mean valuesE(p, q;a) are increasing strictly with

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As applications of the monotonicity of the mean valuesE(p, q;a), the following inequalities about some mean values are obtained readily.

Corollary 4.1. If p1< p2 andq1< q2, then

Ep1,q1(a)< Ep2,q2(a). (4.1)

If q0, then

Sq(a)≶E(p, q;a). (4.2)

If qp−1, then

Jp(a)≶E(p, q;a). (4.3)

If pq, then

Jp(a)≶Jq(a) (4.4)

and

Sp(a)≶Sq(a). (4.5)

If p1, then

Sp(a)≶Jp(a). (4.6)

Corollary 4.2. LetG(a),L(a),I(a)andA(a)denote the geometric, logarithmic, exponential and arithmetic means of n+ 1 positive numbers respectively. Then

G(a)< L(a)< I(a)< A(a). (4.7)

Proof. From (4.5) in Corollary 4.1, it follows that

S−(n+1)(a)< S−1(a)< S0(a)< S1(a).

Thus, inequalities in (4.7) is obtained immediately from Theorem 3.3.

Corollary 4.3. Let r∈N,n∈N, anda= (a0, a1, . . . , an)∈Rn++1. Then

"[r]

X

n+1

(a) #2

[r+1]

X

n+1

(a)

[r−1]

X

n+1

(a) (4.8)

and

"[r]

X

n+1

(a) #1r

≤ "[r+1]

X

n+1

(a) #r+11

. (4.9)

Equalities in (4.8) and (4.9)are valid if and only if a0=a1=· · ·=an.

Proof. By using Corollary 4.1, we find

Jr(a)≤Jr+1(a), Sr(a)≤Sr+1(a).

Further considering Theorem 3.4 proves the required results.

Corollary 4.4. If r1, then

Sr(a)≶Mr(a), (4.10)

where

Mr(a) = 

 

 

1

n+ 1 n P

i=0 ar

i 1/r

, r6= 0,

G(a), r= 0.

(13)

Proof. Let g(x) =Pn

i=0aixi. In the case of r(r−1) >0, using the arithmetic-geometric mean inequality, we obtain

ϕ(1n) r, g(x) =

"

a0+ n X

i=1

(ai−a0)xi #r

= "

1− n X

i=1 xi

!

a0+

n X

i=1 xiai

#r

< 1− n X

i=1 xi

!

ar0+ n X

i=1 xiari

=ar0+ n X

i=1

(ari −ar0)xi,

which is equivalent to

Srr(a) =n! Z

D

ϕ(1n) r, g(x) dx

< n! Z

D

ar0+

n X

i=1

(ari −ar0)xi

dx

= 1

n+ 1 n X

i=0 ari

=Mrr(a).

Therefore, it is deduced that Sr(a) < Mr(a) for r > 1 and Sr(a) > Mr(a) for

r <0.

By the same argument as above, if 0< r <1, thenSr(a)> Mr(a) and

S0(a)< S−(n+1)(a) =G(a) =M0(a).

Thus, ifr <1, thenSr(a)> Mr(a). The proof is complete.

References

[1] H. Alzer, Uer eine einparametrige familie von mittewerten¨ (On a one-parameter family of means), Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber.1987(1988), 1–9. (German) [2] H. Alzer, Uer eine einparametrige familie von Mitlewerten¨ (On a one-parameter family of

mean values), II, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. 1988 (1989), 23–29. (German)

[3] P. S. Bullen,Handbook of Means and Their Inequalities, Mathematics and its Applications (Dordrecht)560, Kluwer Academic Publishers, Dordrecht, 2003.

[4] B. C. Carlson,Algorithms involving arithmetic and geometric means, Amer. Math. Mothly, 78(1971), 496–505.

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[10] C. E. M. Pearce, J. Peˇcari´c and V. ˇSimi´c,On weighted generalized logarithmic means, Houston J. Math.24(1998), 459–465.

[11] J. Peˇcari´c, F. Qi, V. ˇSimi´c and S.-L. Xu,Refinements and extensions of an inequality, III, J. Math. Anal. Appl.227(1998), no. 2, 439–448.

[12] F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl. Math.1 (2000), no. 1, Art. 4. Available online athttp://jipam.vu.edu.au/article.php?sid=97. RGMIA Res. Rep. Coll.2 (1999), no. 5, Art. 4, 633–642. Available online athttp://rgmia.vu.edu.au/v2n5. html.

[13] F. Qi,Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.454(1998), no. 1978, 2723–2732.

[14] F. Qi,On a two-parameter family of nonhomogeneous mean values, Tamkang J. Math.29 (1998), no. 2, 155–163.

[15] F. Qi,The extended mean values: definition, properties, monotonicities, comparison, convex-ities, generalizations, and applications, Cubo Mat. Educ.5(2003), no. 3, 63–90. RGMIA Res. Rep. Coll.5(2002), no. 1, Art. 5. Available online athttp://rgmia.vu.edu.au/v5n1.html. [16] F. Qi and Q.-M. Luo,A simple proof of monotonicity for extended mean values, J. Math.

Anal. Appl.224(1998), 356–359.

[17] F. Qi and Q.-M. Luo,Refinements and extensions of an inequality, Math. Inform. Quart.9 (1999), no. 1, 23–25.

[18] F. Qi, J.-Q. Mei, D.-F. Xia, and S.-L. Xu,New proofs of weighted power mean inequalities and monotonicity for generalized weighted mean values, Math. Inequal. Appl.3(2000), no. 3, 377–383.

[19] F. Qi, J. S´andor, S. S. Dragomir, and A. Sofo,Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math.9(2005), no. 3, 411–420. RGMIA Res. Rep. Coll.5(2002), no. 1, Art. 3, 19–27. Available online athttp://rgmia.vu.edu.au/v5n1.html.

[20] F. Qi and S.-L. Xu, Refinements and extensions of an inequality, II, J. Math. Anal. Appl. 211(1997), no. 2, 616–620.

[21] F. Qi, S.-L. Xu, and L. Debnath,A new proof of monotonicity for extended mean values, Internat. J. Math. Math. Sci.22(1999), no. 2, 415–420.

[22] F. Qi and Sh.-Q. Zhang,Note on monotonicity of generalized weighted mean values, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.455(1999), no. 1989, 3259–3260.

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[26] Zh.-G. Xiao and Zh.-H. Zhang, The Henleman mean of n positive numbers, J. Yueyang Normal Uinv.14(2001), no. 2, 1–5. (Chinese).

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(Zh.-G. Xiao) Hunan Institute of Science and Technology, Yueyang City, Hunan Province, 414000, China

(Zh.-H. Zhang)Zixing Educational Research Section, Chenzhou City, Hunan Province, 423400, China

(F. Qi)Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

E-mail address: qifeng@hpu.edu.cn, fengqi618@member.ams.org, qifeng618@hotmail.com, qifeng618@msn.com

References

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