A sharpened and generalized version of Aczél Vasić Pečarić inequality and its application

R E S E A R C H

Open Access

A sharpened and generalized version of

Aczél-Vasi´c-Peˇcari´c inequality and its

application

Jingfeng Tian

*

*_{Correspondence:}

[email protected] College of Science and Technology, North China Electric Power University, Baoding, Hebei Province 071051, P.R. China

Abstract

In this paper, we present a sharpened and generalized version of Aczél-Vasi´c-Peˇcari´c inequality. As an application, an integral type of Aczél-Vasi´c-Peˇcari´c inequality is obtained.

MSC: Primary 26D15; secondary 26D10

Keywords: Aczél’s inequality; Aczél-Vasi´c-Peˇcari´c inequality; Hölder’s inequality

1 Introduction

In , Aczél [] established the following inequality, which is of wide application in the theory of functional equations in non-Euclidean geometry.

Theorem A If ai, bi (i= , , . . . ,n)are positive numbers such that a –

n

i=ai > or

b_–n_i=b_i > ,then

a_–

n

i=

a_i

b_–

n

i=

b_i

≤

ab–

n

i=

aibi 

. ()

Later, in , Popoviciu [] gave a generalization of the above inequality.

Theorem B Let p> ,q> ,

p+



q= ,and let ai,bi(i= , , . . . ,n)be positive numbers such

that ap_–n_i=ap_i > and bq_–n_i=bq_i > .Then

ap_–

n

i=

ap_i

 p

bq_–

n

i=

bq_i

 q

≤ab–

n

i=

aibi. ()

In , Vasić and Pečarić [] presented the following reversed version of inequality ().

Theorem C Let p<  (p= ), _p+_q = ,and let ai,bi(i= , , . . . ,n)be positive numbers

such that ap_–n_i=ap_i > and bq_–n_i=bq_i > .Then

ap_–

n

i=

ap_i

 p

bq_–

n

i=

bq_i

 q

≥ab–

n

i=

aibi. ()

Recently inequalities () and () were generalized and refined in many different ways; see, for example, [–] and []. In [], Wu established an interesting generalization of Aczél-Popoviciu inequality () as follows.

Theorem D Let p,q> ,ai,bi>  (i= , , . . . ,n),let k(≤k<n)be a positive integer such

thatk_i=ap_i –n_i=k+ap_i > andk_i=bq_i –n_i=k+bq_i > .Then

_k

i=

ap_i –

n

i=k+

ap_i

 pk

i=

bq_i –

n

i=k+

bq_i

 q

≤max{–p–q,}

_k

i=

ap_i

 pk

i=

bq_i

 q

–

_n

i=k+

ap_i

 pn

i=k+

bq_i

 q

–

max{–_p–_q,}

max{p,q, }

_k

i=

ap_i

 pk

i=

bq_i



qn

i=k+a

p i k

i=a

p i

–

n i=k+b

q i k

i=b

q i



, ()

and equality holds if and only if

k i=a

p i n

i=k+a

p i

=

k i=b

q i n

i=k+b

q i

= 

for_p+_q< ,or

k i=a

p i n

i=k+a

p i

=

k i=b

q i n

i=k+b

q i

for_p+_q= .

The main purpose of this work is to give a sharpened and generalized version of Aczél-Vasić-Pečarić inequality (). Moreover, a new Aczél-Aczél-Vasić-Pečarić type integral inequality is established.

2 A sharpened and generalized version of Aczél-Vasi´c-Peˇcari´c inequality

We begin this section with some lemmas, which will be used in the sequel.

Lemma .[] If x> –,α≥orα< ,then

( +x)α≥ +αx. ()

The inequality is reversed for <α< .The sign of equality holds if and only if x= or

α= .

Lemma .[] Let arj>  (r= , , . . . ,n,j= , , . . . ,m),letλ= ,λj<  (j= , , . . . ,m),

and letτ=max{m_j=λ

j, }.Then

n

r=

m

j=

arj≥n–τ m

j=

_n

r=

aλ_rjj



λ_j

The sign of equality holds if and only if the m sets(ar), (ar), . . . , (arm)are proportional for

m

j=λj ≤,or aj=aj=· · ·=anj,j= , , . . . ,m,for

m

j=λj > .

Lemma . Let x> ,y> ,and let p< ,q< .Then

xy+  –xp



p _{ –y}qq ≥_min{–p–q,} _{ –minp}–_,q– _xp_–yq_{, ()}

and equality holds if and only if xp_=yq₌ .

Proof Case (I). Whenp<q< , it implies that _q< , _p– _q> . By applying Lemma . and Lemma ., we have

xy+  –xpp _{ –y}qq

= xpq _yqq _xpp–q _{+  –y}qq _{ –x}pq _{ –x}pp–q

≥min{–p–q,} _xp_{+  –y}q 

q _yq_{+  –x}pq _xp_{+  –x}pp–q

= min{–p–q,} _{ – x}p_–yq  q

≥min{–p–q,} _{ –q}– _xp_–yq_{. ()}

Case (II). When q<p< , it implies that _p < , _q – _p > . By using Lemma . and Lemma ., we obtain

xy+  –xp



p _{ –y}qq

= yqp _xpp _yqq–p_{+  –x}pp _{ –y}qp _{ –y}qq–p

≥min{–p–q,} _yq_{+  –x}p 

p _xp_{+  –y}qp _yq_{+  –y}qq–p

= min{–p–q,} _{ – x}p_–yq  p

≥min{–p–q,} _{ –p}– _xp_–yq_{. ()}

Case (III). Whenp=q,p< ,q< . From Lemma . and Lemma . we have

xy+  –xpp _{ –y}qq

= ypp _xpp_{+  –x}pp _{ –y}pp

≥min{–p,} _yp_{+  –x}p 

p _xp_{+  –y}pp

= min{–p,} _{ – x}p_–yp  p

= min{–p–q,} _{ – x}p_–yq  p

≥min{–p–q,} _{ –p}– _xp_–yq_{. ()}

Combining inequalities ()-() yields inequality (). The condition of equality in () fol-lows immediately from Lemma . and Lemma .. The proof of Lemma . is

Lemma . Let <x< ,y> ,and let p> ,q< .Then

xy+  –xp



p _{ –y}qq ≥_min{–p–q,} _{ –minp}–_,q– _xp_–yq_{, ()}

and equality holds if and only if xp_=yq₌ for



p+



q < ,or xp=yqfor



p+



q= .

Proof By using Lemma . and Lemma ., we have

xy+  –xp



p _{ –y}qq

= xp



q _yqq _xpp–q _{+  –y}qq _{ –x}pq _{ –x}pp–q

≥min{–p–q,} _xp_{+  –y}q 

q _yq_{+  –x}pq _xp_{+  –x}pp–q

= min{–p–q,} _{ – x}p_–yq  q

≥min{–p–q,} _{ –q}– _xp_–yq

= min{–p–q,} _{ –minp}–_,q– _xp_–yq_{. ()}

In addition, the condition of equality for inequality () can easily be obtained by Lemma . and Lemma .. The proof of Lemma . is completed.

Theorem . Let ai> ,bi>  (i= , , . . . ,n),let p= ,q< ,and let k (≤k<n)be a

positive integer such thatk_i=ap_i –n_i=k+ap_i > andk_i=bq_i –n_i=k+bq_i > .Then

_k

i=

ap_i –

n

i=k+

ap_i

 pk

i=

bq_i –

n

i=k+

bq_i

 q

≥min{–p–q,}

_k

i=

ap_i

 pk

i=

bq_i

 q

–

 n–k

min{–_p–_q,}n

i=k+

aibi

– min{–p–q,}_minp–_,q–

_k

i=

ap_i

 pk

i=

bq_i

 q

×

n i=k+a

p i k

i=a

p i

–

n i=k+b

q i k

i=b

q i



, ()

and equality holds if and only if (n– k)–k i=a

p i =a

p k+=a

p

k+ =· · ·=a

p

n and(n–

k)–k

i=b

q i =b

q k+=b

q

k+=· · ·=b

q

n for _p + _q < ,or

k

i=a p i

k

i=b q i

=a

p k+ bq_k+ =

ap_k+

bq_k+ =· · ·= apn bqn for



p+



q = .

Proof Case (I). Whenp> ,q< . From the hypotheses of Theorem ., we find that

 <

k i=a

p i –

n i=k+a

p i k

i=a

p i

 p

< ,

k i=b

q i –

n

i=k+b

q i k

i=b

q i

 q

Thus, by using Lemma . with a substitution

x=

k i=a

p i –

n i=k+a

p i k

i=a

p i

 p

, y=

k i=b

q i–

n i=k+b

q i k

i=b

q i

 q

in (), we have

k i=a

p i –

n i=k+a

p i k

i=a

p i



pk

i=b

q i –

n i=k+b

q i k

i=b

q i  q + n i=k+a

p i k

i=a

p i



pn

i=k+b

q i k

i=b

q i

 q

≥min{–p–q,}

×

 –minp–,q–

k i=a

p i –

n i=k+a

p i k

i=a

p i

–

k i=b

q i –

n i=k+b

q i k

i=b

q i  , () which implies _k i=

ap_i –

n

i=k+

ap_i

 pk

i=

bq_i –

n

i=k+

bq_i

 q

+

_n

i=k+

ap_i

 pn

i=k+

bq_i

 q

≥min{–p–q,}

_k

i=

ap_i

 pk

i=

bq_i

 q

×

 –minp–_,q–

n i=k+a

p i k

i=a

p i

–

n i=k+b

q i k

i=b

q i



. ()

Hence, we obtain

_k

i=

ap_i –

n

i=k+

ap_i

 pk

i=

bq_i –

n

i=k+

bq_i

 q

≥min{–p–q,}

_k

i=

ap_i

 pk

i=

bq_i

 q

–

_n

i=k+

ap_i

 pn

i=k+

bq_i

 q

– min{–p–q,}_minp–_,q–

×

_k

i=

ap_i

 pk

i=

bq_i



qn

i=k+a

p i k

i=a

p i

–

n i=k+b

q i k

i=b

q i



, ()

where the equality holds if and only if

n

i=k+a p i

k

i=a p i

=

n

i=k+b q i

k

i=b q i

=_ for _p+_q< , or

n

i=k+a p i

k

i=a p i

=

n

i=k+b q i

k

i=b q i

for_p+_q= .

On the other hand, by using Lemma ., we obtain

_n

i=k+

ap_i

 pn

i=k+

bq_i

 q

≤

 n–k

min{–_p–_q,}n

i=k+

aibi

, ()

where the equality holds if and only ifak+=ak+=· · ·=anandbk+=bk+=· · ·=bnfor



p+



q < , or ap_k+ bq_k+ =

ap_k+ bq_k+ =· · ·=

apn bqn for



p+



Combining the above two inequalities gives the desired result.

Case (II). Whenp< ,q< . By the same method as in the above case (I) and using Lemma . and Lemma ., we get that inequality () is also valid. The proof of

Theo-rem . is completed.

Remark . If we setk= ,_p+_q= , and

n

i=a p i ap_ =

n

i=b q i

bq_ in Theorem ., then inequality

() reduces to inequality ().

If we setk= , then from Theorem . we obtain the following sharpened and general-ized version of Aczél-Vasić-Pečarić inequality ().

Corollary . Let p= ,q< ,and let ai> ,bi> ,ap –

n

i=a

p i > , b

q

 –

n

i=b

q i > 

(i= , , . . . ,n).Then

ap_–

n

i=

ap_i

 p

bq_–

n

i=

bq_i

 q

≥min{–p–q,}_a

b–

 n– 

min{–_p–_q,}n

i=

aibi

– min{–p–q,}_minp–_,q–_a

b

_n

i=

ap_i ap_ –

bq_i bq_



, ()

and equality holds if and only if(n– )–p_a

=a=· · ·=anand(n– )–  p_b

=b=· · ·=bn

for_p+_q< ,ora

p  bq_ =

ap bq_ =· · ·=

apn bqn for



p+



q= .

In particular, if we set _p+_q≤, then from Corollary . we get the sharpened version of Aczél-Vasić-Pečarić inequality () as follows.

Corollary . Let p> ,q< , 

p +



q ≤,and let ai> ,bi> ,a p

 –

n i=a

p i > ,b

q

 –

n

i=b

q

i >  (i= , , . . . ,n).Then

ap_–

n

i=

ap_i

 p

bq_–

n

i=

bq_i

 q

≥ab–

_n

i=

aibi

–ab q

_n

i=

ap_i ap_ –

bq_i bq_



, ()

and equality holds if and only ifa

p  bq_ =

ap_ bq_ =· · ·=

apn bqn and



p+



q = .

3 Application

As application of the above results, we establish here an integral type of Aczél-Vasić-Pečarić inequality.

Theorem . Let p> ,q< , _p+_q = ,let A> ,B> ,and let f(x),g(x)be positive Rie-mann integrable functions on[a,b]such that Ap_–b

a fp(x) dx> and Bq– b

Then

Ap–

b

a

fp(x) dx

 p

Bq–

b

a

gq(x) dx

 q

≥AB–

b

a

f(x)g(x) dx–AB q

b

a

fp_(x)

Ap –

gq_(x)

Bq

dx



. ()

Proof For any positive integern, we choose an equidistant partition of [a,b] as

a<a+b–a

n <· · ·<a+ b–a

n k<· · ·<a+ b–a

n (n– ) <b,

xi=a+

b–a

n i, i= , , . . . ,n, xk= b–a

n , k= , , . . . ,n.

Since

Ap–

b

a

fp(x) dx> , Bq–

b

a

gq(x) dx> ,

we have

Ap– lim

n→∞

n

k=

fp

a+k(b–a) n

b–a n > ,

and

Bq– lim

n→∞ n

k=

gq

a+k(b–a) n

b–a n > .

Hence, there exists a positive integerNsuch that

Ap–

n

k=

fp

a+k(b–a) n

b–a n > ,

and

Bq–

n

k=

gq

a+k(b–a) n

b–a

n >  for alln>N.

By using Corollary ., we obtain that for anyn>N, the following inequality holds:

Ap–

n

k=

fp

a+k(b–a) n

b–a n

 p

Bq–

n

k=

gq

a+k(b–a) n

b–a n

 q

≥AB–

n

k=

fp

a+k(b–a) n

gq

a+k(b–a) n

b–a n

 p+q

–AB q

_n

k=

 Apf

p

a+k(b–a) n

b–a n –

 Bqg

q

a+k(b–a) n

b–a n



Since

 p+

 q= ,

we have

Ap–

n

k=

fp

a+k(b–a) n

b–a n

 p

Bq–

n

k=

gq

a+k(b–a) n

b–a n

 q

≥AB–

n

k=

fp

a+k(b–a) n

gq

a+k(b–a) n

b–a n

–AB q

_n

k=

 Apf

p

a+k(b–a) n

–  Bqg

q

a+k(b–a) n

b–a n



. ()

In view of the hypotheses thatf(x),g(x) are positive Riemann integrable functions on [a,b], we conclude thatf(x)g(x),fp_{(x) andg}q_{(x) are also integrable on [a,b]. Passing the}

limit asn→ ∞in both sides of inequality (), we obtain inequality (). The proof of

Theorem . is completed.

Competing interests

The author declares that he has have no competing interests.

Acknowledgements

The author would like to express his sincere thanks to the anonymous referees for making great efforts to improve this paper. This work was supported by the NNSF of China (Grant No. 61073121), and the Fundamental Research Funds for the Central Universities (No. 13ZD19).

Received: 27 June 2013 Accepted: 3 October 2013 Published:08 Nov 2013

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Cite this article as:Tian:A sharpened and generalized version of Aczél-Vasi´c-Peˇcari´c inequality and its application.