In this paper, using the arithmetic-geometric mean inequality, we obtain some new mean value inequalities

ON SOME NEW MEAN VALUE INEQUALITIES

LIANG-CHENG WANG AND CAI-LIANG LI SCHOOL OFMATHEMATICSSCIENCE

CHONGQINGINSTITUTE OFTECHNOLOGY

NO. 4OFXINGSHENGLU

YANGJIAPING400050

CHONGQING, THEPEOPLE’SREPUBLIC OFCHINA. wangliangcheng@163.com

CHENGDUELECTROMECHANICALCOLLEGE

CHENGDU610031, SICHUANPROVINCE

THEPEOPLE’SREPUBLIC OFCHINA. dzlcl@163.com

Received 04 March, 2007; accepted 27 April, 2007 Communicated by P.S. Bullen

ABSTRACT. In this paper, using the arithmetic-geometric mean inequality, we obtain some new mean value inequalities. Finally, some applications are given, they are extension of Hölder’s inequalities.

Key words and phrases: Mean value inequality, Hölder’s inequality, Continuous positive function, Extension.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION ANDMAIN RESULTS

Leta > 0, b > 0and t ∈ (0,1). It is well-known that the following arithmetic-geometric mean inequality holds

(1.1) a^tb^1−t≤ta+ (1−t)b.

The arithmetic-geometric mean inequality is a classical inequality with many applications.

Also, there exist extensive works devoted to generalizing or improving the arithmetic-geometric mean inequality. In this respect, we refer the reader to [1] – [7] and the references cited therein for updated results.

In this paper, by (1.1), we obtain some new mean value inequalities. Finally, some applica- tions are given.

In this paper, we agree

q

X

i=q+1

b_i = 0, (b_i ∈R, q∈N).

The first author is partially supported by the Key Research Foundation of the Chongqing Institute of Technology under Grant 2004DZ94.

073-07

Theorem 1.1. Letx_i >0 (i= 1,2, . . . , n; n ≥2)andt∈(0,1).

(1) For the following B(k) 4 1

n²

"

k

k

X

i=1

x_i+

n

X

i=1

x^t_i

! _n X

i=k+1

x^1−t_i

!

+

n

X

i=k+1

x^t_i

! _k X

i=1

x^1−t_i

!#

, (k = 1,2, . . . , n) and

C(j) 4 1 n²

"

(n−j + 1)

n

X

i=j

x_i+

n

X

i=1

x^t_i

! _j−1 X

i=1

x^1−t_i

!

+

j−1

X

i=1

x^t_i

! _n X

i=j

x^1−t_i

!#

, (j = 1,2, . . . , n), we have

(1.2) 1 n

n

X

i=1

x^t_i

! 1 n

n

X

i=1

x^1−t_i

!

=B(1) ≤B(2)≤ · · · ≤B(k)≤B(k+ 1) ≤ · · · ≤B(n) = 1 n

n

X

i=1

xi

and (1.3) 1

n

n

X

i=1

x^t_i

! 1 n

n

X

i=1

x^1−t_i

!

=C(n)≤C(n−1)≤ · · · ≤C(j)≤C(j−1)≤ · · · ≤C(1) = 1 n

n

X

i=1

x_i.

(2) For1≤j < k < l ≤n(n ≥3), we have (1.4) (k−j+ 1)

k

X

i=j

x_i+ (l−k+ 1)

l

X

i=k

x_i+

l

X

i=j

x^t_i

! _l X

i=j

x^1−t_i

!

≤(l−j+ 1)

l

X

i=j

x_i+

k

X

i=j

x^t_i

! _k X

i=j

x^1−t_i

! +

l

X

i=k

x^t_i

! _l X

i=k

x^1−t_i

! .

Corollary 1.2. Letxi >0 (i= 1,2, . . . , n, n≥2)andp, q be any two positive numbers.

(1) For D(k) 4 1

n²

"

k

k

X

i=1

x^p+q_i +

n

X

i=1

x^p_i

! _n X

i=k+1

x^q_i

!

+

n

X

i=k+1

x^p_i

! _k X

i=1

x^q_i

!#

, (k = 1,2, . . . , n)

and

E(j) 4 1 n²

"

(n−j+ 1)

n

X

i=j

x^p+q_i +

n

X

i=1

x^p_i

! _j−1 X

i=1

x^q_i

!

+

j−1

X

i=1

x^p_i

! _n X

i=j

x^q_i

!#

, (j = 1,2, . . . , n), we have

(1.5) 1 n

n

X

i=1

x^p_i

! 1 n

n

X

i=1

x^q_i

!

=D(1)≤D(2) ≤ · · · ≤D(k)≤D(k+ 1)≤ · · · ≤D(n) = 1 n

n

X

i=1

x^p+q_i and

(1.6) 1 n

n

X

i=1

x^p_i

! 1 n

n

X

i=1

x^q_i

!

=E(n)≤E(n−1)≤ · · · ≤E(j)≤E(j−1)≤ · · · ≤E(1) = 1 n

n

X

i=1

x^p+q_i .

(2) For1≤j < k < l ≤n(n ≥3), we have (1.7) (k−j+ 1)

k

X

i=j

x^p+q_i + (l−k+ 1)

l

X

i=k

x^p+q_i +

l

X

i=j

x^p_i

! _l X

i=j

x^q_i

!

≤(l−j+ 1)

l

X

i=j

x^p+q_i +

k

X

i=j

x^p_i

! _k X

i=j

x^q_i

! +

l

X

i=k

x^p_i

! _l X

i=k

x^q_i

! .

2. PROOF OFTHEOREM ANDCOROLLARY

Proof of Theorem 1.1. (1) Two equalities are clear in (1.2). To complete the proof of (1.2), we only need to prove thatB(k)≤B(k+ 1)(1≤k ≤n−1). Indeed, from (1.1) we have

(2.1) x^t_k+1

k

X

i=1

x^1−t_i =

k

X

i=1

x^t_k+1x^1−t_i ≤

k

X

i=1

(tx_k+1+ (1−t)x_i), and

(2.2) x^1−t_k+1

k

X

i=1

x^t_i =

k

X

i=1

x^1−t_k+1x^t_i ≤

k

X

i=1

((1−t)x_k+1+tx_i). Using (2.1) and (2.2), after a simple manipulation we get

(2.3) x^t_k+1

k

X

i=1

x^1−t_i +x^1−t_k+1

k

X

i=1

x^t_i ≤kx_k+1+

k

X

i=1

x_i.

Fork = 1,2, . . . , n−1, by (2.3) we get

B(k) = 1 n²

"

k

k

X

i=1

x_i+

n

X

i=1

x^t_i

! _n X

i=k+1

x^1−t_i

! +

n

X

i=k+1

x^t_i

! _k X

i=1

x^1−t_i

!#

= 1 n²

"

k

k

X

i=1

x_i+x_k+1 +

n

X

i=1

x^t_i

! _n X

i=k+2

x^1−t_i

!

+

k

X

i=1

x^t_i+

n

X

i=k+2

x^t_i

!

x^1−t_k+1+

n

X

i=k+2

x^t_i

! _k X

i=1

x^1−t_i

!

+x^t_k+1

k

X

i=1

x^1−t_i

#

= 1 n²

"

k

k

X

i=1

x_i+x_k+1 +x^t_k+1

k

X

i=1

x^1−t_i +x^1−t_k+1

k

X

i=1

x^t_i

+

n

X

i=1

x^t_i

! _n X

i=k+2

x^1−t_i

! +

n

X

i=k+2

x^t_i

! _k+1 X

i=1

x^1−t_i

!#

≤ 1 n²

"

k

k

X

i=1

x_i+x_k+1 +kx_k+1+

k

X

i=1

x_i

+

n

X

i=1

x^t_i

! _n X

i=k+2

x^1−t_i

! +

n

X

i=k+2

x^t_i

! _k+1 X

i=1

x^1−t_i

!#

= 1 n²

"

(k+ 1)

k+1

X

i=1

x_i +

n

X

i=1

x^t_i

! _n X

i=k+2

x^1−t_i

! +

n

X

i=k+2

x^t_i

! _k+1 X

i=1

x^1−t_i

!#

=B(k+ 1).

By same arguments of proof for (1.2), we can also get inequalities in (1.3).

(2) For1≤j < k < l ≤n, from (1.1) we have

k−1

X

i=j

x^t_i

! _l X

i=k+1

x^1−t_i

!

=

k−1

X

i=j l

X

s=k+1

x^t_ix^1−t_s (2.4)

≤

k−1

X

i=j l

X

s=k+1

(tx_i+ (1−t)x_s)

= (l−k)

k−1

X

i=j

tx_i+ (k−j)

l

X

i=k+1

(1−t)x_i

and

(2.5)

l

X

i=k+1

x^t_i

! _k−1 X

i=j

x^1−t_i

!

≤(l−k)

k−1

X

i=j

(1−t)x_i+ (k−j)

l

X

i=k+1

tx_i.

Using (2.4) and (2.5), after a simple manipulation we have

(2.6)

k−1

X

i=j

x^t_i

! _l X

i=k+1

x^1−t_i

! +

l

X

i=k+1

x^t_i

! _k−1 X

i=j

x^1−t_i

!

≤(l−k)

k−1

X

i=j

x_i+ (k−j)

l

X

i=k+1

x_i.

From (2.6) we obtain

(k−j + 1)

k

X

i=j

x_i+ (l−k+ 1)

l

X

i=k

x_i+

l

X

i=j

x^t_i

! _l X

i=j

x^1−t_i

!

= (k−j+ 1)

k

X

i=j

x_i+ (l−k+ 1)

l

X

i=k+1

x_i+ (l−k)x_k

+

k

X

i=j

x^t_i

! _k X

i=j

x^1−t_i

! +

l

X

i=k+1

x^t_i

! _l X

i=k+1

x^1−t_i

!

+x^t_k

l

X

i=k+1

x^1−t_i +x^1−t_k

l

X

i=k+1

x^t_i +xk

+

k−1

X

i=j

x^t_i

! _l X

i=k+1

x^1−t_i

! +

l

X

i=k+1

x^t_i

! _k−1 X

i=j

x^1−t_i

!

≤(k−j + 1)

k

X

i=j

x_i+ (l−k+ 1)

l

X

i=k+1

x_i+ (l−k)x_k

+

k

X

i=j

x^t_i

! _k X

i=j

x^1−t_i

! +

l

X

i=k

x^t_i

! _l X

i=k

x^1−t_i

!

+ (l−k)

k−1

X

i=j

x_i+ (k−j)

l

X

i=k+1

x_i

= (l−j+ 1)

l

X

i=j

x_i+

k

X

i=j

x^t_i

! _k X

i=j

x^1−t_i

! +

l

X

i=k

x^t_i

! _l X

i=k

x^1−t_i

! ,

which implies (1.4).

This completes the proof of Theorem 1.1.

Proof of Corollary 1.2. Replacet, 1−t andx_i in Theorem 1.1 by _p+q^p , _p+q^q andx^p+q_i , respec-

tively. We obtain Corollary 1.2.

3. APPLICATIONS

Proposition 3.1. Letxir >0 (i= 1,2, . . . , n,n ≥ 2;r = 1,2, . . . , m,m ≥2)andt ∈ (0,1).

For

F(k) 4 1 n²



k

k

X

i=1 m

X

r=1

xir

! +

n

X

i=1 m

X

r=1

xir

!t!



n

X

i=k+1 m

X

r=1

xir

!^1−t



+

n

X

i=k+1 m

X

r=1

x_ir

!t!



k

X

i=1 m

X

r=1

x_ir

!1−t





, (k = 1,2, . . . , n) and

G(h) 4 1 n²





n

X

i=1 n

X

j=1





h

X

r=1

xir

!^t _h X

r=1

xjr

!^1−t +

m

X

r=h+1

x^t_irx^1−t_jr







, (h= 1,2, . . . , m), we have

1 n²

n

X

i=1 n

X

j=1 m

X

r=1

x^t_irx^1−t_jr

! (3.1)

=G(1)≤G(2)≤ · · · ≤G(h)≤G(h+ 1) ≤ · · · ≤G(m)

= 1

n

n

X

i=1 m

X

r=1

xir

!t!

 1 n

n

X

i=1 m

X

r=1

xir

!^1−t



=F(1)≤F(2)≤ · · · ≤F(k)≤F(k+ 1) ≤ · · · ≤F(n)

= 1 n

n

X

i=1 m

X

r=1

xir.

Proof. Forxir >0,xjr >0(1≤i, j ≤n, r= 1,2, . . . , m) andt ∈(0,1). We write P(i, j;h) 4

h

X

r=1

x_ir

!^t _h X

r=1

x_jr

!^1−t +

m

X

r=h+1

x^t_irx^1−t_jr (h = 1,2, . . . , m).

The first named author of this paper showed in [8] that the following chain of Hölder’s inequal- ities holds

m

X

r=1

x^t_irx^1−t_jr =P(i, j; 1) (3.2)

≤P(i, j; 2)≤ · · · ≤P(i, j;h)≤P(i, j;h+ 1)≤ · · · ≤P(i, j;m)

=

m

X

r=1

x_ir

!t m

X

r=1

x_jr

!1−t

. From the properties of inequality and (3.2), we have

1 n²

n

X

i=1 n

X

j=1 m

X

r=1

x^t_irx^1−t_jr

!

= 1 n²

n

X

i=1 n

X

j=1

P(i, j; 1) (3.3)

≤ 1 n²

n

X

i=1 n

X

j=1

P(i, j; 2)

≤ · · · ≤ 1 n²

n

X

i=1 n

X

j=1

P(i, j;h)

≤ 1 n²

n

X

i=1 n

X

j=1

P(i, j;h+ 1)≤ · · ·

≤ 1 n²

n

X

i=1 n

X

j=1

P(i, j;m)

= 1 n²

n

X

i=1 n

X

j=1 m

X

r=1

xir

!t m

X

r=1

xjr

!^1−t

= 1

n

n

X

i=1 m

X

r=1

x_ir

!t!

 1 n

n

X

i=1 m

X

r=1

x_ir

!1−t

.

It is easy to see that

(3.4) G(h) = 1

n²

n

X

i=1 n

X

j=1

P(i, j;h), h= 1,2, . . . , m.

(3.3) and (3.4) imply inequalities between the first equality and the second equality in (3.1).

Replacingx_i in (1.2) byPm

r=1x_ir, we obtain inequalities between the third equality and the fourth equality in (3.1).

This completes the proof of Proposition 3.1.

Proposition 3.2. Letf_i : [a, b]7→(0,+∞) (a < b)be continuous functions(i= 1,2, . . . , n, n≥ 2)andt ∈(0,1). For

H(k) = 1 n²

"

k

k

X

i=1

Z b a

f_i(x)dx

+

n

X

i=1

Z b a

f_i(x)dx

^t! _n X

i=k+1

Z b a

f_i(x)dx ^1−t!

+

n

X

i=k+1

Z b a

fi(x)dx

^t! _k X

i=1

Z b a

fi(x)dx

^1−t!#

, (k = 1,2, . . . , n) and anyy∈[a, b], we have

1 n²

n

X

i=1 n

X

j=1

Z b a

(f_i(x))^t(f_j(x))^1−tdx (3.5)

≤ 1 n²

" _n X

i=1 n

X

j=1

Z y a

f_i(x)dx

tZ y a

f_j(x)dx 1−t

+ Z b

y

(f_i(x))^t(f_j(x))^1−tdx

!#

≤ 1 n

n

X

i=1

Z b a

f_i(x)dx ^t!

1 n

n

X

i=1

Z b a

f_i(x)dx ^1−t!

=H(1)≤H(2)≤ · · · ≤H(k)≤H(k+ 1)≤ · · · ≤H(n)

= 1 n

n

X

i=1

Z b a

f_i(x)dx.

Proof. For1 ≤ i, j ≤ n, t ∈ (0,1), y ∈ [a, b]and continuous functionsf_i : [a, b] 7→ (0,+∞) (i = 1,2, . . . , n;n ≥ 2), in [8], Wang also obtained the following refinement for the integral form of Hölder’s inequalities:

Z b a

(f_i(x))^t(f_j(x))^1−tdx (3.6)

≤ Z y

a

f_i(x)dx

tZ y a

f_j(x)dx 1−t

+ Z b

y

(f_i(x))^t(f_j(x))^1−tdx

≤ Z b

a

f_i(x)dx

^tZ b a

f_j(x)dx ^1−t

. Using the properties of inequality and (3.6), we have

1 n²

n

X

i=1 n

X

j=1

Z b a

(f_i(x))^t(f_j(x))^1−tdx

≤ 1 n²

n

X

i=1 n

X

j=1

Z y a

f_i(x)dx

tZ y a

f_j(x)dx 1−t

+ Z b

y

(f_i(x))^t(f_j(x))^1−tdx

!

≤ 1 n²

n

X

i=1 n

X

j=1

Z b a

f_i(x)dx

^tZ b a

f_j(x)dx ^1−t

= 1

n

n

X

i=1

Z b a

f_i(x)dx ^t!

1 n

n

X

i=1

Z b a

f_i(x)dx ^1−t!

, which is two inequalities of left hand in (3.2).

Replacing x_i in (1.2) by Rb

af_i(x)dx, we obtain inequalities between the two equalities in (3.2).

This completes the proof of Proposition 3.2.

Remark 3.3. (3.1) and (3.2) are extensions of Hölder’s inequalities.

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