Some inequalities based on a general quantum difference operator

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R E S E A R C H

Open Access

Some inequalities based on a general

quantum difference operator

Alaa E Hamza

1

_{and Enas M Shehata}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Faculty of Science, Menouﬁa University, Shibin El-Kom, Egypt Full list of author information is available at the end of the article

Abstract

In this paper, some integral inequalities based on the general quantum diﬀerence operatorDβ are deduced. Here,Dβis deﬁned byDβf(t) = (f(

β

(t)) –f(t))/(

β

(t) –t), where

β

is a strictly increasing continuous function, deﬁned on an intervalI⊆R, that has one ﬁxed points0∈I. The

β

-Hölder and

β

-Minkowski inequalities are proved. Also, the

β

-Gronwall,

β

-Bernoulli, and some related inequalities are shown. Finally, the

β

-Lyapunov inequality is established. MSC: 39A10; 39A13; 39A70; 47B39

Keywords: quantum diﬀerence operator; quantum calculus; Hölder inequality; Minkowski inequality; Gronwall inequality; Bernoulli inequality; Lyapunov inequality

1 Introduction

Quantum difference operators have an interest role due to their applications; see,e.g., [–] and references cited therein. A calculus based on a quantum difference operator is usu-ally known as calculus without limits. It substitutes the classical derivative by a difference operator, which allows one to deal with sets of nondifferentiable functions. In [] Hahn introduced his difference operator, as a tool for constructing families of orthogonal poly-nomials, which is defined by

Dq,ωf(t) =

f(qt+ω) –f(t)

t(q– ) +ω , t=ω, (.)

whereq∈(, ),ω>  are ﬁxed andω=_–ω_q. The derivative att=ωis deﬁned to bef(ω)

whenever it exists. In [], its inverse operator was constructed and a rigorous analysis of the calculus associated toDq,ωwas given. See also [–]. The Hahn quantum diﬀerence

oper-ator unifies two well-known difference operoper-ators. The first one is the Jacksonq-difference operator, which is defined by

Dqf(t) =

f(qt) –f(t)

t(q– ) , t= , (.) whereqis a fixed number,q∈(, ). The functionf is defined on aq-geometric setA⊆R (orC) such that whenevert∈A,qt∈A. Att= ,Dqf(t) =f(). The second difference

operator is the diﬀerence operatorDω, which is deﬁned by

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Dωf(t) =

f(t+ω) –f(t)

ω , t∈R, (.)

where the ﬁxed numberω= .

We refer the reader to the interesting books of Annaby and Mansour [], Kac and Che-ung [], and Bangerezako [].

In [] we introduced a general quantum diﬀerence operatorDβdeﬁned by

Dβf(t) =

f(β(t)) –f(t)

β(t) –t , (.)

for every t withβ(t)=t, where f is an arbitrary function defined, in general, on aβ -geometric setI⊆R, for whichβ(t)∈I,t∈I, andβ:I→Iis a strictly increasing con-tinuous function. For a fixed pointsofβ, theβ-derivativeDβf(t) att=sis defined to

bef(s), wheneverfis diﬀerentiable att=sin the usual sense. The functionβhas many

forms due to its properties. It may be linear or nonlinear. Accordingly, it has no fixed points or has at least one. Every choice of the functionβgives a new difference operator. Thus, we can obtain a wide class of difference operators with the corresponding quantum calculi. We have a space of forms of the functionβwhich may be classified into classes according to the number of fixed points beside the directions of the sequences{βk(t)}k∈N

towards and outwards these points. Theβ-difference operator yields the Hahn difference operator whenβ(t) =qt+ω,ω> , and the Jacksonq-difference operator whenβ(t) =qt,

q∈(, ). In these casesβis linear and has one fixed point. Also, the forward difference operatorDωwith the linear form ofβ(t) =t+ω,ω> , has no fixed points. Consequently,

the corresponding Hahn calculus,q-calculus, andω-calculus, respectively, are particular cases of theβ-calculus. In [] we considered the class of our functionβ which has one ﬁxed points∈Iand satisﬁes the following condition:

(t–s)

β(t) –t≤ for allt∈I.

This paper is devoted to deducing some basic integral inequalities based on the β -difference operator, whenβhas one fixed points∈Iand satisfies the same condition. The

paper is organized as follows. In Section , we exhibit the results that we need from [], concerning the calculus based on theβ-difference operator. Section  contains our main results. We prove theβ-Hölder,β-Minkowski,β-Gronwall, andβ-Bernoulli inequalities and some related ones. Finally, we show theβ-Lyapunov inequality. These inequalities are very important in establishing the theory ofβ-difference equations associated with the quantum difference operatorDβ.

Throughout this paperIis an interval ofRcontaining the ﬁxed pointsofβandXis a

Banach space with norm · .

2 Preliminaries

In this section we present some needed results from [] concerning the calculus associ-ated withDβ. Here, we consider the class of the functionβ when it has one ﬁxed point

s∈Iand satisﬁes the following condition:

(t–s)

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Lemma . The following statements are true:

(i) The sequence of functions{βk₍_t₎_}

k∈Nconverges uniformly to the constant function ˆ

β(t) :=son every compact intervalJ⊆Icontainings.

(ii) The series∞_k_=|βk(t) –βk+(t)|is uniformly convergent to|t–s|on every compact intervalJ⊆Icontainings.

Lemma . If f :I→Xis continuous at s, then the sequence{f(βk(t))}k∈N converges

uniformly to f(s)on every compact interval J⊆I containing s.

Theorem . If f :I→Xis continuous at s, then the series

∞

k=(βk(t) –βk+(t))×

f(βk(t))is uniformly convergent on every compact interval J⊆I containing s.

Definition . For a functionf :I→X, we define theβ-difference operator off as

Dβf(t) =

f(β(t))–f(t)

β(t)–t , t=s, f(s), t=s,

provided that the ordinary derivativefexists ats. In this case, we say thatDβf(t) is the

β-derivative off att. We say thatf isβ-diﬀerentiable onIiff(s) exists.

Theorem . Assume that f :I→Xand g:I→Rareβ-diﬀerentiable at t∈I.Then:

(i) The productfg:I→Xisβ-diﬀerentiable attand

Dβ(fg)(t) =

Dβf(t)

g(t) +fβ(t)Dβg(t)

=Dβf(t)

gβ(t)+f(t)Dβg(t). (ii) f/gisβ-diﬀerentiable attand

Dβ(f/g)(t) =

(Dβf(t))g(t) –f(t)Dβg(t)

g(t)g(β(t)) ,

provided thatg(t)g(β(t))= .

Lemma . Let f :I→Xbeβ-diﬀerentiable and Dβf(t) = for all t∈I,then f(t) =f(s)

for all t∈I.

Corollary . Suppose that f,g:I→Xareβ-diﬀerentiable on I.If Dβf(t) =Dβg(t)for all

t∈I,then

f(t) –g(t) =f(s) –g(s) for all t∈I.

Theorem . Assume f :I→Xis continuous at s.Then the function F deﬁned by

F(t) =

∞

k=

βk(t) –βk+(t)fβk(t), t∈I, (.)

is aβ-antiderivative of f with F(s) = .Conversely,aβ-antiderivative F of f vanishing at

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Deﬁnition . Letf:I→Xanda,b∈I. We deﬁne theβ-integral off fromatobby

b

a

f(t)dβt= b

s

f(t)dβt– a

s

f(t)dβt, (.)

where

x

s

f(t)dβt=

∞

k=

βk(x) –βk+(x)fβk(x), x∈I, (.)

provided that the series converges atx=aandx=b.f is calledβ-integrable onIif the series converges ata, bfor alla,b∈I. Clearly, if f is continuous ats∈I, thenf isβ

-integrable onI.

Ifβ(t) =qtandβ(t) =qt+ω,q∈(, ),ω> , then (.) and (.) reduce to the Jackson

q-integral and Hahn integral, respectively; see [, , , , ].

Theorem . Let f :I→Xbe continuous at s.Deﬁne the function

F(x) =

x

s

f(t)dβt, x∈I. (.)

Then F is continuous at s,DβF(x)exists for all x∈I and DβF(x) =f(x).

Theorem . If f :I→Xisβ-diﬀerentiable on I,then

b

a

(Dβf)(t)dβt=f(b) –f(a), for all a,b∈I.

Theorem . Assume f,g areβ-diﬀerentiable functions on I and Dβf,Dβg both

contin-uous at s.Then b

a

f(t)Dβg(t)dβt=f(b)g(b) –f(a)g(a) – b

a

Dβf(t)

gβ(t)dβt, a,b∈I.

Here at least one of the functions f and g is a real valued function.

Deﬁnition . Lets∈[a,b]⊆I. We deﬁne theβ-interval by

[a,b]β=

βk(a);k∈N ∪

βk(b);k∈N ∪ {s}.

For any pointc∈I, we denote by

[c]β=

βk(c);k∈N ∪ {s}.

Finally, forA⊆R, we denote by

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LetDdenote the set of all real valued functions deﬁned on [c,d]βand continuous ats,

wherec,d∈Iandc<d.

Lemma . Let f ∈D. Then_cdf(t)h(β(t))dβt= for all functions h∈D with h(c) =

h(d) = if and only if f(t) = for all t∈[c,d]β.

Lemma . Let f :I→X,g:I→Rbeβ-integrable functions on I.If

f(t)≤g(t) for all t∈[a,b]β,a,b∈I,a≤b,

then for x,y∈[a,b]β,x<s<y,we have

_sy 

f(t)dβt

≤

y

s

g(t)dβt, (.)

x

s

f(t)dβt

≤

s

x

g(t)dβt, (.)

and

_xyf(t)dβt

≤

y

x

g(t)dβt. (.)

Consequently, if g(t)≥ for all t ∈[a,b]β, then the inequalities y

sg(t)dβt≥  and y

xg(t)dβt≥hold for all x,y∈[a,b]β,a,b∈I,a≤b.

Lemma . Let f :I→Xand g:I→Rbeβ-diﬀerentiable on I.If

Dβf(t)≤Dβg(t), t∈[a,b]β,a,b∈I,a≤b,

then

f(y) –f(x)≤g(y) –g(x), (.)

for every x,y∈[a,b]β,x<s<y.

Deﬁnition .(β-exponential functions) Assume thatp:I→Cis a continuous func-tion ats. We deﬁne theβ-exponential functionsep,β(t) andEp,β(t) by

ep,β(t) =



∞

k=[ –p(βk(t))(βk(t) –βk+(t))]

(.)

and

Ep,β(t) =

∞

k=

 +pβk(t)βk(t) –βk+(t), (.)

whenever both inﬁnite products are convergent to a non-zero number for every t∈I. Clearly, we have

ep,β(t) =



E–p,β(t)

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Theorem . Theβ-exponential functions ep,β(t)and Ep,β(t)are the unique solutions of

the ﬁrst orderβ-diﬀerence equations

Dβy(t) =p(t)y(t), y(s) = , (.)

and

Dβy(t) =p(t)y

β(t), y(s) = , (.)

respectively.

3 Main results

3.1

β

-Hölder and

β

-Minkowski inequalities

Inspired in the work by Agarwalet al.[], we present theβ-Hölder,β-Cauchy-Schwarz, andβ-Minkowski inequalities.

Deﬁnition . Letp≥ anda,b∈I⊆R,a≤b. We denote byLp_β[a,b]β the space of all

functionsf: [a,b]β→Rsuch that

sup

y

x

f(t)pdβt:x,y∈[a,b]β,s∈[x,y]

<∞.

Theorem . (β-Hölder inequality) Let f ∈ Lp_β[a,b]β and g ∈ Lqβ[a,b]β. Then |fg| ∈

L

β[a,b]βand y

x

f(t)g(t)dβt≤

y

x

f(t)pdβt

/p y

x

g(t)qdβt /q

, (.)

where x,y∈[a,b]β,s∈[x,y],and p> ,q=p/(p– ).The equality holds if|f(t)|p/|g(t)|qis

constant.

Proof Forα,γ ∈[,∞), we have

α/pγ/q≤α p+

γ

q. (.)

Letα(t) =y|f(t)|p

x|f(t)|pdβt andγ(t) =

|g(t)|q

y

x|g(t)|qdβt, with

y

x

f(t)pdβt

y

x

g(t)qdβt

= .

Substituting in (.) and applying Lemma ., we get

y

x

|f(t)| (_xy|f(t)|p_d

βt)/p

|g(t)| (_xy|g(t)|q_d

βt)/q

dβt≤



p

y

x

|f(t)|p

y

x|f(t)|pdβt dβt

+

q

y

x

|g(t)|q

y

x|g(t)|qdβt

dβt

=

p+



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Then

y

x

f(t)g(t)dβt≤

y

x

f(t)pdβt

/p y

x

g(t)qdβt /q

.

It is obvious that if|f(t)|p_/_|_g₍_t₎_|q_{is constant, the equality holds.}

Actually, if we putp=q=  in theβ-Hölder inequality we get theβ-Cauchy-Schwarz inequality.

Corollary .(β-Cauchy-Schwarz inequality) Let f,g∈L

β[a,b]β. Then|fg| ∈Lβ[a,b]β

and

y

x

f(t)g(t)dβt≤

y

x

f(t)dβt

y

x

g(t)dβt

,

where x,y∈[a,b]β,s∈[x,y].

As in the classical Minkowski inequality, we can deduce the following result.

Theorem . (β-Minkowski inequality) Let ≤p<∞and a,b∈I, a≤b. Let f,g ∈ Lp_β[a,b]β.Then|f +g| ∈Lpβ[a,b]βand

y

x

(f +g)(t)pdβt /p

≤

y

x

f(t)pdβt /p

+

y

x

g(t)pdβt /p

, (.)

where x,y∈[a,b]β,s∈[x,y].The equality holds if f(t)/g(t)is constant. 3.2

β

-Gronwall and

β

-Bernoulli inequalities

Hamza and Ahmed, in [], deduced the Hahn, Gronwall, and Bernoulli inequalities. In the following we present the correspondingβ-version.

Lemma . Let y,f,p are real valued functions deﬁned on I and continuous at s.If

Dβy(t)≤p(t)y(t) +f(t) for all t, (.)

then

y(t)≤y(s)ep,β(t) +ep,β(t) t

s

f(τ)E–p,β

β(τ)dβτ. (.)

Proof We have

Dβ

y(t)E–p,β(t)

=Dβy(t)E–p,β

β(t)+y(t)DβE–p,β(t)

=Dβy(t)E–p,β

β(t)–p(t)y(t)E–p,β

β(t) =E–p,β

β(t)Dβy(t) –p(t)y(t)

≤E–p,β

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Integrating both sides fromstotwe get

y(t)E–p,β(t) –y(s)E–p,β(s)≤ t

s

f(τ)E–p,β

β(τ)dβτ.

In view ofE–p,β(s) =  andep,β(t) = 

E–p,_β(t), we conclude

s

f(τ)E–p,β

β(τ)dβτ.

Theorem .(β-Gronwall inequality) Let p≥and y,f, p be real valued continuous functions at sdeﬁned on I.If

y(t)≤f(t) +

t

s

y(τ)p(τ)dβτ, (.)

then

y(t)≤f(t) +ep,β(t) t

s

p(τ)f(τ)E–p,β

β(τ)dβτ. (.)

Proof Deﬁne

z(t) =

t

s

y(τ)p(τ)dβτ. (.)

Thenz(s) =  andDβz(t) =y(t)p(t). Therefore, inequality (.) yields

y(t)≤f(t) +z(t) (.) and

Dβz(t)≤

f(t) +z(t)p(t). (.) By Lemma ., we obtain

z(t)≤z(s)ep,β(t) +ep,β(t) t

s

p(τ)f(τ)E–p,β

β(τ)dβτ. (.)

Inequality (.) implies

y(t)≤f(t) +ep,β(t) t

s

p(τ)f(τ)E–p,β

β(τ)dβτ.

As a direct consequence, we obtain the following results.

Corollary . Let p,y,f are continuous functions at sand p(t)≥.Then

y(t)≤

t

s

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implies

y(t)≤.

Proof This is due to Theorem . withf(t)≡.

Corollary . Let p(t)≥andα∈R.Then

y(t)≤α+

t

s

y(τ)p(τ)dβτ, for all t>s,

implies

y(t)≤αep,β(t).

Proof By theβ-Gronwall inequality if we putf(t) =α, then

y(t)≤α+

t

s

y(τ)p(τ)dβτ for allt.

Consequently,

y(t)≤α+ep,β(t) t

s

αp(τ)E–p,β

β(τ)dβτ

=α

 –ep,β(t) t

s

DβE–p,β(τ)dβτ

=α( –ep,β(t)

E–p,β(t) –E–p,β(s)

=α–αep,β(t)E–p,β(t) +αep,β(t).

Therefore,y(t)≤αep,β(t).

Theorem .(β-Bernoulli inequality) Forα∈(,∞),the following inequality is true:

ep,β(t)≥ +α(t–s), t>s. (.)

Proof Supposey(t) =α(t–s),t>s. ThenDβy(t) =α. We haveαy(t) +α=α(t–s) +α≥

α=Dβy(t), which implies thatDβy(t)≤αy(t) +α. By Lemma . we get

s

αE–p,β

β(τ)dβτ

=ep,β(t) t

s

–DβE–p,β(τ)dβτ

= –ep,β(t)

E–p,β(t) – 

= – +ep,β(t).

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3.3

β

-Lyapunov Inequality

Lyapunov inequality has many applications in the theory of differential and difference equations. These applications include bounds for eigenvalues, stability criteria for peri-odic differential equations, and estimates for intervals of disconjugacy; see [, ]. In this section we introduce the Lyapunov inequality based onDβ.

Letf :I→[,∞) be a continuous function ats∈I. Consider the Sturm-Liouvilleβ

-diﬀerence equation

Dβx(t) +f(t)x

β(t)= , t∈I. (.)

Deﬁne the functionFby

F(y) =

b

a

Dβy(t) 

–f(t)yβ(t)dβt. (.)

Lemma . Let x be a nontrivial solution of the Sturm-Liouvilleβ-diﬀerence equation

(.).Then for every y belonging to the domain of F,the following equality holds:

F(y) –F(x) –F(y–x) = (y–x)(b)Dβx(b) – (y–x)(a)Dβx(a). (.)

Proof Simple calculations show that

F(y) –F(x) –F(y–x) =

b

a

Dβy(t) 

–f(t)yβ(t)–Dβx(t) 

+f(t)xβ(t) –Dβ(y–x)(t)



+f(t)(y–x)β(t) dβt

= 

b

a

–Dβx(t) 

+f(t)xβ(t)+Dβy(t)Dβx(t)

–f(t)yβ(t)xβ(t) dβt

= 

b

a

Dβy(t)Dβx(t) +y

β(t)Dβx(t) –

Dβx(t) 

+Dβx(t)x

β(t) dβt

by using (.) = 

b

a

Dβ

y(t)Dβx(t)

–Dβ

x(t)Dβx(t) dβt

= 

b

a Dβ

y(t) –x(t)Dβx(t) dβt

= y(b) –x(b)Dβx(b) – 

y(a) –x(a)Dβx(a).

Lemma . Let y be in the domain of F,then for any c,d∈[a,b],a,b∈I such that a≤ c<d≤b,we have

d

c

Dβy(t) 

dβt≥

(y(d) –y(c))

(11)

Proof Letx(t) =y(d_d)–_–y_c(c)t+dy(c_d)–_–cy_c(d). ThenDβx(t) = y(d_d)–_–y_c(c) andDβx(t) = . This implies

thatx(t) is a solution of (.) withf(t) =  for allt∈Iand

F(y) =

d

c

Dβy(t) 

dβt

for anyyin domainF. From Lemma ., we getF(y) –F(x) –F(y–x) = , and consequently

F(y) =F(x) +F(y–x)≥F(x). Therefore,

d

c

Dβy(t) 

dβt≥ d

c

Dβx(t) 

dβt

=

d

c

y(d) –y(c)

d–c



dβt

=(y(d) –y(c))



d–c .

Theorem .(β-Lyapunov inequality) Let f :I→(,∞)be a continuous function,s∈I.

Let x be a nontrivial solution of(.)with x(a) =x(b) = ,where a,b∈I with a<b.Then

b

a

f(t)dβt≥



b–a. (.)

Proof From Lemma . withy= , we have

F(x) =

b

a

Dβx(t) 

–f(t)xβ(t)dβt= . (.)

Let M=max{x₍_t_);_t_∈ _[_a_,_b_]_} _and _c_∈_[_a_,_b_{] such that} _x₍_c_{) =}_M_{. Then} _M₌_x₍_c₎_≥

x(β(t)) > , and

M

b

a

f(t)dβt≥ b

a

f(t)xβ(t)dβt

=

b

a

Dβx(t) 

dβt

=

c

a

Dβx(t) 

dβt+ b

c

Dβx(t) 

dβt

≥(x(c) –x(a))

c–a +

(x(b) –x(c))

b–c

=M



c–a+



b–c

=M

(b+a– c) (c–a)(b–c)(b–a)+



b–a

≥M  b–a.

(12)

4 Conclusion and future direction

This paper was devoted to a presentation of some basic integral inequalities based on our general quantum diﬀerence operator,Dβ, which is deﬁned byDβf(t) =f(β_β(t₍_t))–_)–_tf(t),t=β(t),

where β is a strictly increasing continuous function deﬁned on an intervalI⊆Rthat has one ﬁxed point s∈I. These inequalities are theβ-Hölder,β-Cauchy-Schwarz,β

-Minkowski, β-Gronwall,β-Bernoulli,β-Lyapunov inequalities, and some related ones. We are looking forward to study in detail the theory of linearβ-difference equations based onDβ. This theory unifies the theory ofq-difference equations and Hahn difference

equa-tions. Also, it includes other types of quantum diﬀerence equaequa-tions.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript. Author details

1_{Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.}2_{Department of Mathematics, Faculty of}

Science, Menouﬁa University, Shibin El-Kom, Egypt.

Received: 25 June 2014 Accepted: 14 January 2015

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