Some generalizations of Aczél, Bellman's inequalities and related power sums

Abstract

In this paper, we establish some functional generalizations and refinements of Aczél's inequality and of Bellman's inequality. We also establish several mean value theorems for the related power sums.

Mathematics Subject Classification (2010): 26D15; 26D20; 26D99.[PUBLICATION ABSTRACT]

Zhou Journal of Inequalities and Applications 2012, 2012:130 http://www.journalofinequalitiesandapplications.com/content/2012/1/130

RESEARCH Open Access

Some generalizations of Aczl, Bellmans inequalities and related power sums

Xiaoyan Zhou

Abstract

In this paper, we establish some functional generalizations and refinements of Aczls inequality and of Bellmans inequality. We also establish several mean value theorems for the related power sums.

Mathematics Subject Classification (2010): 26D15; 26D20; 26D99.

Keywords: generalization, Aczls inequality, Bellmans inequality, mean value theorem, power sums

1 Introduction

Let n be a positive integer, and let ai, bi (i = 1, 2, ..., n) be real numbers such that a21

Correspondence: mailto:[email protected]

Web End [email protected] The School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

ni=2 a2i 0 or b21

ni=2 b2i 0. Then Aczls inequality [1] can be stated as fol-

lows

a21 n

i=2a2i

b21

n

i=2b2i

a1b1

n

i=2aibi

2, (1)

with equality if and only if the sequences ai and bi are proportional.

The Aczl inequality (1) plays an important role in the theory of functional equations in non-Euclidean geometry. During the past years, many authors have given considerable attention to this inequality, its generalizations and applications [2-11].

As an example, the Hlder-like generalization of the Aczl inequality (1), derived by Popoviciu [12], takes

ap1 n

i=2api

1 p

bq1 n

i=2bqi

1q a1b1 n

i=2aibi, (2)

where n is a positive integer, and p, q, ai, bi (i = 1, 2, ..., n) are positive numbers such that p1 + q-1 = 1, ap1

ni=2 api > 0 and bq1

ni=2 bqi > 0.

One application of Aczls inequality is the following Bellmans inequality [13]

ap1 n

i=2api

1p+

bp1 n

i=2bpi

1p

(a1 + b1)p n

i=2(ai + bi)p

1p. (3)

Here n is a positive integer, and p 1, ai, bi (i = 1, 2, ..., n) are positive numbers such that ap1

ni=2 api > 0 and bp1

ni=2 bpi > 0.

2012 Zhou; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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In this paper, inspired by the functional generalizations of the Cauchy-Schwarz inequality [14] and of the Hlder inequality [15,16], we will establish some functional generalizations of the Aczl inequality and of the Bellman inequality. Refinements of these inequalities will also be presented.

As we seen, the following theorem is very useful to give results related to power sums and Aczls inequality.

Theorem 1.1 (see e.g. [9,17]). Let n be a positive integer, and xi (i = 1, 2, ..., n) be positive numbers such that x1

ni=2 xi > 0.If f : + is a function such that f(x)/x

is increasing on +, then

f

x1 n

i=2xi

f (x1)

n

i=2f (xi). (4)

The inequality is reversed if f(x)/x is decreasing on +. The inequalities are strict if f (x)/x is strictly increasing or decreasing on +.

Several mean value theorems for the related power sums of (4) have been established in [9,18-20]. In this paper, we will also generalize two of them in the last section.

2 Aczl and Bellmans inequalities

In order to establish the functional generalization of Aczls inequality, we need the following lemma.

Lemma 2.1 (power means inequality, see [21]). Let n be a positive integer, p >0 and let ai >0 (i = 1, 2, ..., n). Then

n

i=1api n1min{p,1}

n

i=1ai

p. (5)

Theorem 2.1. Let n, m be positive integers, and let pj 1, xij (i = 1, 2, ..., n; j = 1, 2, ..., m) be positive numbers such that x1j

ni=2 xij > 0 for j = 1, 2, ..., m. If fj : + + is a function such that fj(x)/x is increasing on +. Then we have

m

j=1fj

x1j n

i=2xij

m

j=1

1 pj

fj(x1j)

pj

n

i=2

fj(xij)

pj

(6)

C

m

j=1fj(x1j)

n

i=2

m

j=1 fj(xij),

mj=1 p1j.

Proof. Applying Theorem 1.1 on each fj yields

m

j=1fj

x1j n

i=2xij

where C = n1min{r,1}, r =

m

j=1

fj(x1j) n

i=2 fj(xij)

.

Reusing Theorem 1.1 on xpj and replacing xij by fj (xij) in Theorem 1.1, we obtain

m

j=1

fj(x1j) n

i=2 fj(xij)

m

j=1

fj(x1j)

pj

n

i=2

fj(xij)

pj

1 pj ,

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which completes the first inequality of (6). To proof the second inequality of (6), let us denote

fj =

fj(x1j)

pj

n

i=2

fj(xij)

pj

1 pj .

By using the power means inequality (5) and the well known Hlder inequality, we have

m

j=1

fj +

n

i=2

j=1fj(xij) n1min{r,1}

m

j=1

r

m

f

1 r

j +

n

i=2

m

j=1

1 r

fj(xij)

r

1 rpj

(7)

n1min{r,1}

m

j=1

fpjj +n

i=2

fj(xij)

pj

= n1min{r,1}

m

j=1 fj(x1j).

Rearranging the terms of (7) immediately leads to the second inequality of (6). This completes the proof.

Remark 2.1. From the proof we have that the second inequality of (6) still holds if

fj(x1j)

pj

ni=2 (fj(xij))pj > 0for pj > 0, j = 1, 2, ..., m.

From Theorem 2.1, by taking fj(x) = x, we getCorollary 2.1. Under the assumptions of Theorem 2.1, and letting fj(x) = x, we have

m

j=1

x1j n

i=2xij

m

j=1

xpj1j n

i=2xpjij

1 pj

C

m

j=1x1j

n

i=2

m

j=1xij. (8)

The first inequality of (8) gives a lower bound of Aczls inequality. And the second is a generalized Aczl inequality obtained in [22].

The following theorem is the functional generalization of Bellmans inequality. Theorem 2.2. Let n, m be positive integers, and let p 1, xij (i = 1, 2, ..., n; j = 1, 2,

..., m) be positive numbers such that x1j

ni=2 xij > 0for j = 1, 2, ..., m. If fj : + +

is a function such that fj (x)/x is increasing on +. Then we have

m

j=1fj

x1j n

i=2xij

m

j=1

fj(x1j)

p

n

i=2

fj(xij)

p

1 p

(9)

m

j=1 fj(x1j)

p

n

i=2

p

1 p

,

m

j=1 fj(xij)

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Proof. The proof of the first inequality of (9) is similar to the proof of Theorem 2.1 and we omit it. The second inequality is an identity when p = 1. Hence we only need to prove the second inequality of (9) for p >1 below.

From the assumptions and Theorem 1.1, we have

fj(x1j)

n

i=2fj(xij) > fj

x1j n

i=2xij

> 0, j = 1, 2, . . . , m.

Applying the above inequality, the power means inequality (5) and the Minkowski inequality (see [21]), we obtain

m

j=1 fj(x1j)

p

m

j=1

n

i=2

fj(xij)

p

1 p

p

n

i=2

m

j=1 fj(xij)

p

.

we now deduce from Theorem 2.1 that

fl(x1l)

1 p

p

n

i=2

1 p

p

m

j=1 fj(x1j)

n

i=2

p

m

j=1 fj(xij)

fl(xil)

p

1

m

j=1 fj(xij)

m

j=1 fj(x1j)

fl(x1l)

p1

n

i=2 fl(xil)

p1

,

for l = 1, 2, ..., m. This leads to

m

l=1

1 p

1 p

fl(x1l)

p

n

i=2

fl(xil)

p

p

n

i=2

p

1

m

j=1 fj(x1j)

m

j=1 fj(xij)

m

l=1 fl(x1l)

m

j=1 fj(x1j)

p1

n

i=2

m

j=1 fj(xij)

p1

m

l=1 fl(xil)

m

j=1 fj(x1j)

m

j=1 fj(xij)

=

p

n

i=2

p

,

which yields immediately the desired inequality. This completes the proof. Taking fj(x) = x in Theorem 2.2, we obtain

Corollary 2.2. Under the assumptions of Theorem 2.2, and letting fj (x) = x, we have

m

j=1

m

j=1

p

1 p

x1j n

i=2xij

xp1j n

i=2xpij

1 p

m

j=1x1j

p

n

i=2

m

j=1xij

. (10)

The first inequality of (10) gives a lower bound of Bellmans inequality. And the second is a generalized Bellman inequality obtained in [3,8].

Following the similar methods from [8,10], we will establish some refinements of inequalities (6) and (9). Since the proofs are trivial by breaking the corresponding sums in the following form

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x1j

n

i=2xij = x1j

l

i=2xij

n

i=l+1 xij,

and reusing the corresponding theorems, we present these refinements below without proofs.

Theorem 2.3. Under the assumptions of Theorem 2.1, for 2 l <n, we have

m

j=1

fj

x1j n

i=2xij

m

j=1

fj

x1j l

i=2xij

pj

n

i=l+1

fj(xij)

pj

1 pj

1 pj

m

j=1

fj(x1j)

pj

n

i=2

fj(xij)

pj

(11)

pj

1 pj

C1

m

j=1

f

j(x1j)

pj

l

i=2

fj(xij)

n

i=l+1

m

j=1 fj(xij)

C1C2

m

j=1fj(x1j)

n

i=2

m

j=1 fj(xij),

where C1 = (n l + 1)1min{r,1}, C2 = l1min{r,1}. In particular r =

mj=1 p1j 1, we

have C1 = C2 = 1, hence

m

j=1

p

1 pj

fj(x1j)

p

n

i=2

fj(xij)

p

1 pj

m

j=1

f

j(x1j)

p

l

i=2

fj(xij)

n

i=l+1

m

j=1 fj(xij)

m

j=1fj(x1j)

n

i=2

m

j=1 fj(xij),

leading to a refinement of (6).

Remark 2.2. The third and fourth inequality of (11) still holds if

fj(x1j)

pj

n i=2

fj(xij)

pj > 0for pj > 0, j = 1, 2, ..., m.

Taking fj (x) = x in Theorem 2.3, we get

Corollary 2.3. Under the assumptions of Theorem 2.1 and letting fj (x) = x, for 2 l <n, we have

m

j=1(x1j

n

i=2xij)

m

j=1

x1j l

i=2xij

pjn

i=l+1 xpjij

1 pj

m

j=1

xpj1j n

i=2xpjij

1 pj

(12)

C1

m

j=1

xpj1j l

i=2xpjij

1 pj

n

i=l+1

m

j=1xij

C1C2

m

j=1x1j

n

i=2

m

j=1xij.

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In particular r =

mj=1 p1j 1, we have

m

j=1

xpj1j n

i=2xpjij

1 pj

m

j=1

xpj1j l

i=2xpjij

1 pj

n

i=l+1

m

j=1xij

m

j=1x1j

n

i=2

m

j=1xij, (13)

leading (12) to a refinement of (8).

Remark 2.3. The third and fourth inequality of (12) still holds for pj > 0 if

xpj1j

ni=2 xpjij > 0, j = 1, 2, ..., m. The inequality (13) is also obtained in [8].

Theorem 2.4. Under the assumptions of Theorem 2.2, for 2 l <n, we have

m

j=1fj

1 p

x1j n

i=2xij

m

j=1

fj

x1j l

i=2xij

p

n

i=l+1 (fj(xij))p

fj(x1j)

1 p

m

j=1

p

n

i=2 (fj(xij))p

(14)

1 p

m

j=1

fj(x1j)

p

l

i=2

fj(xij)

n

i=l+1

1 p

p

m

j=1 fj(xij)

p

p

m

j=1 fj(xij)

p

n

i=2

p

1 p

,

m

j=1 fj(x1j)

Taking fj(x) = x in Theorem 2.4, we have the following.

Corollary 2.4. Under the assumptions of Theorem 2.2, and letting fj(x) = x, we have

m

j=1(x1j

n

i=2xij)

m

j=1

x1j l

i=2xij

pn

i=l+1xpij

1 p

m

j=1

xp1j n

i=2xpij

1 p

(15)

m

j=1

xp1j l

i=2xpij

1 p

p

n

i=l+1

m

j=1xij

p

1 p

1 p

.

m

j=1x1j

p

p

n

i=2

m

j=1xij

The third and fourth inequality (15) is also obtained in [8].

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3 Mean value theorem

As we seen, Theorem 1.1 is very useful to give results related to Aczls inequality. It is also useful to give results related to power sums [9,17]. In this section, we first present a generalized version of Theorem 1.1, then use it to establish our first generalized mean value theorem related to power sums. We conclude this section with a mean value theorem which generalize the recent result obtained by Peari and Rehman [18].

Lemma 3.1. Let n be a positive integer, and xi (i = 1, 2, ..., n) be positive numbers such thatx1

ni=2 xi > 0. If f : + is a function such that f(x)/xp is increasing on +. Then the inequality (4) holds for p 1, f 0 on + or p 1, f 0 on +. If f(x)/xp is decreasing on +, the inequality (4) is reversed for p 1, f 0 on + or p 1, f 0 on +.

Remark 3.1. If p 1, f 0 or f(x)/x is strictly increasing or decreasing on +, then strictly inequalities hold.

Proof. This Lemma is an easy corollary of Theorem 1.1, so we omit the proof. Theorem 3.1. Let p 2. Let (x1, x2, ..., xn) In, where I = [a, b] (0, ) and x1

ni=2 xi I. If f : + is a function such that f C1(I) and map f(a) Map, where m, M are defined by (17) below. Then there exists I such that

f (x1)

n

i=2f (xi)f

x1 n

i=2xi

= f () (p 1)f ()p

xp1 n

i=2xpi x1

n

i=2xi

p . (16)

Proof. Let

F(x) = xf (x) (p 1)f (x) xp .

Since I is compact and f C1(I), there exist x, x I such thatM := F(x) = max

xI

F(x), m := F(x) = min

xI

F(x). (17)

We define two auxiliary functions as follows

1(x) = Mxp f (x), 2(x) = f (x) mxp. It is easily deduced that

1(x) xp1

0,

0,

hence the two functions 1(x)xp1 and 2(x)xp1 are all increasing on I. From the above

inequalities, we also have

1(x) =

xp1 1(x) xp1

2(x) xp1

= p 1x 1(x) + xp1 1(x) xp1

p 1x 1(x),

and

2(x)

x 2(x).

By the famous Grownwall inequality and j1(a) 0 and j2(a) 0 from the assumption, we find

p 1

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1(x) 0, 2(x) 0.

Now applying Lemma 3.1 on j1(x) and j2(x) respectively and rearranging the terms, we have

f (x1)

n

i=2f (xi) f

x1 n

i=2xi

M

xp1 n

i=2xpi x1

n

i=2xi

p , (18)

f (x1)

n

i=2f (xi) f

x1 n

i=2xi

m

xp1 n

i=2xpi x1

n

i=2xi

p . (19)

Applying Lemma 1.1 on the function xp we obtain

xp1

n

i=2xpi x1

n

i=2xi

p> 0.

Combining (18) and (19) leads to

m

f (x1)

ni=2 f (xi) f (x1

ni=2 xi)

xp1

ni=2 xpi

M. (20)

For our definition, F(x) is continuous on I and m F(x) M. Hence, there exists I such that

f (x1)

ni=2 f (xi) f (x1

p

x1

ni=2 xi

= f () (p 1)f ()

p , (21)

which immediately leads to (16). This completes the proof.

We present the Cauchy type mean value theorem of Theorem 3.1 below without proof for the proof is quite standard and coincides with the proof of Theorem 3.14 in[9].

Theorem 3.2. Let p 2. Let (x1, x2, ..., xn) In, where I = [a, b] (0, ) and

x1

ni=2 xi)

xp1

ni=2 xpi (x1

ni=2 xi)p

ni=2 xi I. If f, g : + are functions such that f, g C1(I) and mf ap f(a) Mfap, mgap g(a) Mgap, where mf, Mf and mg, Mg are defined by (17) with corresponding function f and g. Then there exists I such that

f (x1) n

i=2f (xi) f

x1 n

i=2xi

[g () (p 1)g()]

= g(x1) n

i=2g(xi) g

x1 n

i=2xi

[f () (p 1)f ()].

(22)

Remark 3.2. If p = 2, the conditions mf f(a)ap Mf, mg g(a)ap Mgcould be

removed, then Theorem 3.1 and Theorem 3.2 reduce to Theorem 3.13 and Theorem3.14 of [9], respectively.

We conclude this section with a generalization of the mean value theorem obtained

in [18], which is a special case of the following theorem with k = 1. As given in [18], this theorem is also a generalization of Theorem 3.1 with p = 2.

Theorem 3.3. Let (x1, x2, ..., xn) In, where I is a compact interval, pi, qi (i = 1, 2, ..., n) be non-negative numbers such that

ni=1 pixi Iand

ni=1 pixi xj, j = 1, 2, ..., n. If f

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Ck(I), then there exists I such that

n

j=1qjf

n

i=1pixi

n

j=1qjf (xj)

(23)

n

i=1pixi xj

j=1qj f ()(xj) !

+ f (k)()k!n

j=1qj

n

i=1pixi xj

k .

n

=

k1

=1

Proof. Since I is compact and f Ck(I), there exist x, x I such thatM := f (k)(x) = max

xI

f (k)(x), m := f (k)(x) = min

xI

f (k)(x). (24)

We define 2n auxiliary functions as follows

j(x) = M

k! (x xj)k +

k1

=0f ()(xj)! (x xj) f (x),

and

=0f ()(xj)! (x xj) mk!(x xj)k,

for j = 1, 2, ..., n. Then we have

j(x) = M

(k 1)!

(x xj)k1 +

k1

j(x) = f (x)

k1

=1f ()(xj)( 1)!(x xj)1 f (x). (25)

Expanding f (x) at xj by the Taylor theorem, (25) can be rewritten as

j(x) =

M f (k)() (k 1)!

(x xj)k1,

where h I. Obviously, j(x) 0 for x xj, j = 1, 2, ..., n, which means jj(x) is

increasing on x xj. Similarly, we can deduce that j(x) is increasing on x xj, j = 1, 2, ..., n. Thus, from the assumption we obtain

j

n

i=1pixi

j(xj), j

n

i=1pixi

j(xj), j = 1, 2, . . . , n.

Rearranging the terms yield

f

n

i=1pixi

f (xj)

k1

=0f ()(xj) !

Mk! n

i=1pixi xj

k ,

n

i=1pixi xj

and

f

n

i=1pixi

f (xj)

k1

=0f ()(xj) !

mk! n

i=1pixi xj

k ,

n

i=1pixi xj

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for j = 1, 2, ..., n. Hence,

n

j=1qj

f n

i=1pixi

f (xj)

k1

=0f ()(xj) !

Mk!n

j=1qj

n

i=1pixi xj

k, (26)

n

i=1pixi xj

and

n

j=1qj

f n

i=1pixi

f (xj)

k1

=0f ()(xj) !

mk!n

j=1qj

n

i=1pixi xj

k. (27)

n

i=1pixi xj

For f Ck(I) and

m k!

n

j=1qj

n

i=1pixi xj

kf (k)(x)k!n

j=1qj

n

i=1pixi xj

k

Mk!n

j=1qj

n

i=1pixi xj

k ,

combining (26) and (27) immediately leads to (23). This completes the proof. Similarly, we present the Cauchy type mean value theorem of Theorem 3.3 below without proof. This theorem reduce to the Cauchy type mean value theorem of [18] with k = 1.

Theorem 3.4. Let (x1, x2, ..., xn) In, where I is a compact interval, pi, qi (i = 1, 2, ..., n) be non-negative numbers such that

ni=1 pixi I and

ni=1 pixi xj, j = 1, 2, ..., n. If f,

g Ck(I), then there exists I such that

n

j=1qjf

n

i=1pixi

n

j=1qjf (xj)

k1

=1

n

j=1qj f ()(xj) !

g(k)()

n

i=1pixi xj

n

i=1pixi

n

j=1qjg(xj)

n

i=1pixi xj

=

n

j=1qjg

k1

=1

n

j=1qj g()(xj) !

f (k)().

(28)

Competing interestsThe authors declare that they have no competing interests.

Received: 9 January 2012 Accepted: 8 June 2012 Published: 8 June 2012

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Zhou Journal of Inequalities and Applications 2012, 2012:130 http://www.journalofinequalitiesandapplications.com/content/2012/1/130

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doi:10.1186/1029-242X-2012-130Cite this article as: Zhou: Some generalizations of Aczl, Bellmans inequalities and related power sums. Journal of Inequalities and Applications 2012 2012:130.

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