Generalized integral inequalities on time scales

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R E S E A R C H

Open Access

Generalized integral inequalities on time

scales

Tooba Fayyaz

1*

_{, Nazia Irshad}

1

_{, Asif R Khan}

1

_{, Ghaus ur Rahman}

2

_{and Gholam Roqia}

3

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

University of Karachi, University Road, Karachi, 75270, Pakistan Full list of author information is available at the end of the article

Abstract

The theory of dynamic equations on time scales which was formulated by Hilger is an area of mathematics which is currently receiving profuse attention. Despite the fact that the basic objective of times scales is to bring together the study of difference and differential equations, it also extends these classical cases to ‘in-between’. In the present article we present a version of Feng Qi integral inequalities on time scales which are in fact generalizations of results given in different articles.

MSC: 39A13; 26D15; 26E70

Keywords: time scale; Feng Qi inequality; non-negative increasing function;

and ∇integrals; (q,h)-derivative

1 Introduction

In the past few years, time scales calculus has fascinated scientists by virtue of its tremen-dous application in many branches of sciences,e.g.population dynamics, ﬁnance, physics,

etc.[, ]. During the last decade, with the evolution of integral and differential equations as in including difference equations, voluminous integral and differential inequalities have been disclosed [–]. These handy discoveries concern a significant part in the research of boundedness, comprehensive existence, stability of solutions of integral and differential equations, and likewise of difference equations. Subsequently, Hilger [] put forward the time scales’ theory as a theory that combines differential and difference calculus in a so-phisticated manner. Afterwards several researchers have talked about various facets of the dynamic equations on time scales including various inequalities in detail [, ]. However, the Feng Qi integral inequalities on time scales have been given limited attention.

In , Feng Qi explored some new types of integral inequalities and presented his results through an analytic approach and mathematical induction. His work motivated many researchers and a group of people at different times continued the work of Feng Qi. Mohamed Akkouchi, [] studied a general version of a problem posed by Feng Qi [] in the context of a measured space equipped with a positive finite measure. Hong Yong [] in his short note introduced some parameters and imposed sufficient conditions so that Qi type integral inequality holds.

Moreover, Kamel Brahimet al.[] introduced some Feng Qi typeq-integral inequali-ties, in quantum calculus.

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Motivated by the afore-mentioned discussion, the present article is concerned with Feng Qi integral inequalities on time scales. To the best of the authors’ knowledge, no contri-bution is available in the literature about Feng Qi type integral inequalities on time scales.

2 Preliminaries

For the convenience of the reader, we provide some significant and main characteristics of time scale calculus, basic definitions, and results. By a time scale we understand any non-empty and closed subset of real numbers with the ordering inherited from the set of real numbers. Although [] and [] are good sources to study the basics of time scale and quantum calculus, we just state some definitions, important results, and notations necessary for our article. Throughout this paper we will denote a time scale by the symbol

Tand we mean [a,b]∩Tby [a,b]T, where [a,b]⊂R. From [], we have extracted some

deﬁnitions and results which are as follows.

Deﬁnition . Foru∈Twe deﬁne the forward jump operatorσ:T→Tby

σ(u) :=inf{v∈T:v>u}

and the backward jump operatorρ:T→Tis deﬁned by

ρ(u) :=sup{v∈T:v<u}

while the graininess functionμ:T→[,∞) is deﬁned by

μ(u) :=σ(u) –u

andν:T→[,∞) is deﬁned as

ν(u) :=u–ρ(u).

Here we deﬁneTk_,_T

kas follows:

Tk₌

T\(ρ(supT),supT) if supT<∞,

T if supT=∞,

and

Tk=

T\[infT,σ(infT)) if infT> –∞,

T if infT= –∞.

Proposition . Letφ:T→Rbe a continuous function.

(i) Ifσ(u) >u,thenφis-diﬀerentiable atu∈Tkwith

φ(u) = φ(σ(u)) –φ(u)

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(ii) Ifρ(u) <t,thenφis∇-diﬀerentiable atu∈Tkwith

φ∇(u) =φ(u) –φ(ρ(u))

ν(u) .

Proposition . Assumeφ,f:T→Rare-diﬀerentiable at u∈Tk_._Then_: (i) The sumφ+f :T→Ris-diﬀerentiable atuwith

(φ+f)₍_u_{) =}_φ₍_u_{) +}_f₍_u_).

(ii) For any constantc,cφ:T→Ris-diﬀerentiable atuwith

(cφ)(u) =cφ(u).

Proposition . Assumeφ,f :T→Rare∇-diﬀerentiable at u∈Tk.Then: (i) The sumφ+f :T→Ris∇-diﬀerentiable atuwith

(φ+f)∇(u) =φ∇(u) +f∇(u).

(ii) For any constantc,cf :T→Ris∇-diﬀerentiable atuwith

(cf)∇(u) =cf∇(u).

Proposition . Assume that f :T→Ris strictly increasing andT˜ :=f(T)is a time scale.

Letφ:f(R)→R.If f_and_φ˜ _{exist for t}_∈_Tk_then

(φ◦f)₌_φ˜ _◦_f_f_.

Proposition . Every continuous function has an antiderivative.In particular if u∈T, then the pre-diﬀerentiable functionϕdeﬁned by

ϕ(u) =

u

u

φ(θ)θ for u∈T

is an antiderivative ofφ.

Proposition . Let u∈T, v∈Randφ:Tk→Rbe a continuous map.Then there exists a unique pre-diﬀerentiable functionϕsatisfying

ϕ(u) =φ(u) for all u∈D, φ(u) =v,

where D is the region of diﬀerentiation.

Proposition .

(i) Assume thatφ:T→Ris-diﬀerentiable onTk_._Then_φ_is_∇_{-diﬀerentiable at}_u_and

φ∇(u) =φρ(u)

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(ii) Assume thatφ:T→Ris∇-diﬀerentiable onTk,thenφis-diﬀerentiable atuand

φ∇(u) =φρ(u)

foru∈Tk_{such that}_ρ(σ₍_u_{)) =}_u_.

In this article, we are interested in a discrete time scale and the most signiﬁcant dis-crete time scales are those originating from arithmetic and geometric sequences of real numbers, namelyTa

h:={a+kh,k∈Z},h> , andTaq:={aqk,k∈Z} ∪ {},q> . In [], the

two-parameter time scaleTa

(q,h)is introduced which generalizes the time scalesTahandTaq.

For a givena∈R+_{, the two-parameter time scale is deﬁned as}

Ta (q,h):=

aqk+ [k]qh,k∈Z

∪

h

 –q , q≥,h≥,q+h> ,

where [k]q=–q k

–q.

SurelyTa

(q,h)=Taq,Ta(q,h)=Tahprovidedh=  andq= , respectively (we put h

–q:= –∞). It

can be noted that the time scaleTa

(q,h)witha∈R–can be considered quite analogously. For

the sake of deﬁniteness, we restrict our consideration toa> . Now, the (q,h)-derivative ofφ:Ta

(q,h)→Ris deﬁned atvby

(q,h)φ(v) =

φ(qv+h) –φ(v) (q– )v+h

and

∇(q,h)φ(x) =

φ(v) –φ(q–₍_v_–_h₎₎ q–₍₍_q_{– )}_v_–_h₎ .

Letv∈Ta

(q,h)and letv>v,i.e.there existsn∈Z+such thatv=vqn+ [n]qh, then we deﬁne

the delta (q,h)-integral by

v

v

φ(u)(q,h)u=

(q– )v+h

n–

k=

qkfvqk+ [k]qh

. ()

If we consider the time scaleTa

(q,h)withvreplaced byv, then the integration over [_–h_q,t]

yields the deﬁnition of the nabla (q,h)-integral in the form

v

h/(–q)

φ(u)∇(q,h)u=

 –q–v+q–h

∞

k=

q–kφq–kv+ [–k]qh,

provided that the inﬁnite series converges. More generally, we can consider the nabla (q,h)-integral with the lower limith/( –q) replaced bya∈Ta

(q,h), h/( –q)≤a<t. If h/( –q) <a, then the infinite series occurring in () is reduced to the corresponding fi-nite series. These definitions extend standard integral introductions: in particular, ifh= , then () becomes the Jacksonq-integral [].

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sequel, and the third section is reserved for results related to Fenq Qi type-integral in-equalities deﬁned on a discrete time scale, which generalizes some results given in [] and []. In [] and [], there are given theq-analog and theh-analog, respectively, of some Feng Qi inequalities given in [, ], and []; therefore, the fourth section will explore the results related to Feng Qi type∇-integral inequalities deﬁned on a discrete time scale. The last section highlights the important results of present article.

3 Feng Qi type

-integral inequalities

This section is devoted to the results related to Fenq Qi type-integral inequalities de-ﬁned on discrete time scales, which generalize some results given in [] and [].

Lemma . Suppose p≥and f is a non-negative and monotonic function on[a,b]T.Then

pfp–(v)f(v)≤fp(v)≤pfp–σ(v)f(v).

Proof The deﬁnition of the-derivative may be quoted from [] in the following way:

fp(v)=f

p_(σ₍_v_{)) –}_fp₍_v₎

μ(v) =

p

μ(v)

f(σ(v))

f(v)

up–du.

Sincef is a monotonic increasing function, we obtain

fp–(v)fσ(v)–f(v)≤

f(σ(v))

f(v)

up–du≤fp–σ(v)fσ(v)–f(v),

pfp–(v)f(σ(v)) –f(v)

μ(v) ≤

fp(v)≤pfp–σ(v)f(σ(v)) –f(v)

μ(v) ,

pfp–(v)f(v)≤fp(v)≤pfp–σ(v)f(v).

Remark . If we setσ(v) =qv+hin the preceding lemma deﬁned onTa

(q,h), then we get

the following result.

Supposep≥ andf is a non-negative and monotonic function on [a,b]∩Ta

(q,h). Then

pfp–(v)(q,h)f(v)≤(q,h)fp(v)≤pfp–(qv+h)(q,h)f(v).

Here, we observe that if we setq=  in this result, we get Lemma . of [] and if we set

h=  in this result, we get Lemma . of [].

Theorem . Letφbe a function deﬁned on[a,b]Tsuch thatφ(v)≥and

φp–(v)φ(v)≥σ(v)(p– )σ(v) –ap–φp–σ(v)

∀v∈[a,b]Tand p≥.Then

b

a

φp(v)v≥

b

a

φ(v)v

p–

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Proof Forv∈[a,b]T, we consider the diﬀerence

F(v) =

v

a

φp(u)u–

v

a

φ(u)u

p–

,

by setting

f(v) =

v

a

φ(u)u

and by using Lemma ., we have

F(v) =φp(v) –fp–(v)

≥φp(v) – (p– )fp–σ(v)f₍_v₎

=φp(v) – (p– )fp–σ(v)φ(v)

=φ(v)F(v),

whereF(v) =φp–(v) – (p– )fp–(σ(v))≥.

Now, again using Lemma . we get

F_(v) =φp–(v)– (p– )fp–σ(v)

≥(p– )φp–₍_v_)φ₍_v_{) – (}_p_{– )(}_p_{– )}_fp–_σ₍_v₎_f_σ₍_v₎

= (p– )φp–(v)φ(v) – (p– )(p– )fp–σ(v)φσ(v)σ(v).

Since the functionφincreases, we have

fσ(v)=

σ(v)

a

φ(u)u≤σ(v) –aφσ(v)

and therefore

F_(v)≥(p– )φp–(v)φ(v) –σ(v)(p– )σ(v) –ap–φt–σ(v)φσ(v)

≥(p– )φp–(v)φ(v) –σ(v)(p– )σ(v) –ap–φp–σ(v)≥.

So, we conclude that the functionF(v) increases on [a,b]T. Therefore,F(v)≥F(a)≥,

which shows the non-negativity ofF₍_v_{). Hence, we conclude that}_F _{is increasing and} F(v)≥, so we obtain the required result.

If we chooseσ(v) =qv+hdeﬁned onTa

(q,h)then we get the following corollary.

Corollary . Letφbe a function deﬁned on[a,b]∩T₍a_q_,_h₎such thatφ(v)≥and

φp–(v)(q,h)φ(v)≥q(p– )

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∀v∈[a,b]Tand p≥.Then

b

a

φp(v)(q,h)v≥

b

a

φ(v)(q,h)v

p–

.

Remark . Observation shows that Theorem  of [] is a special case of Corollary . if we simply setq= . Further we note that our result is also a reﬁnement of Proposition . of [], and if we set h=  in the above theorem then we get a more reﬁned result as compared to Proposition . of [].

Theorem . Ifφis a non-negative function deﬁned on[a,b]Tsuch thatφ(v)≥and

φt(v)φ(v)≥ασ

₍_v_)(σ₍_v_{) –}_a₎α–

(b–a)α– φ

α_σ₍_v₎

for v∈[a,b]Tandα≥,then

b

a

φα+(v)v≥ 

(b–a)α–

b

a

φ(v)v

α+

.

Proof We deﬁne the following diﬀerence, forv∈[a,b]T:

F(v) =

v

a

φα+(v)v– 

(b–a)α–

v

a

φ(v)v

α+

and assume

f(v) =

v

a

φ(u)u.

Using Lemma . shows

F(v) =φα+(v) –  (b–a)α–

fα+(v)

≥φα+(v) – α+  (b–a)α–f

α_σ₍_v₎_f₍_v₎

=φα+(v) – α+  (b–a)α–f

α_σ₍_v₎_φ(_v₎

=φ(v)F(v),

whereF(v) =φα+(v) –₍_b_–α+_a₎α–fα(σ(v)). Now, again by using Lemma . we obtain

F  (v) =

φα+(v)– α+  (b–a)α–

fα_σ(_v₎

≥(α+ )φα(v)φ(v) – α(α+ ) (b–a)α–f

α–_σ₍_v₎_f_σ(_v₎

= (α+ )

φα(v)φ₍_v_{) –} α

(b–a)α–f

α–_σ₍_v₎_φ_σ₍_v₎_σ₍_v₎

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Sinceφincreases on [a,b]T, forv∈[a,b]T, we have

fσ(v)=

σ(v)

a

φ(u)u≤σ(v) –aφσ(v).

And therefore,

F_(v)≥(α+ )

φα(v)φ(v) –ασ(v)

σ(v) –a b–a

α–

φα–σ(v)φσ(v)

≥(α+ )

φα(v)φ(v) –ασ(v)

σ(v) –a b–a

α–

φασ(v)≥.

So,F(v) is an increasing function. Therefore, F(v)≥F(a)≥, which shows the

non-negativity ofF₍_v_{). So, we conclude that}_F_{is increasing and in addition to}_F₍_v₎_≥_{, hence}

we get the required result.

We end this section with a corollary and some remarks. By settingσ(v) =qv+hdeﬁned onTa

(q,h), then we get the following corollary.

Corollary . Ifφis a non-negative function deﬁned on[a,b]∩Ta

(q,h) such thatφ(v)≥ and

φα(v)(q,h)φ(v)≥

αq(q_v_{+ []}_qh_–_a₎α–

(b–a)α– φ

α_q_v_{+ []} qh

b

a

φα+(v)(q,h)v≥

 (b–a)α–

b

a

φ(v)(q,h)v

α+

.

Remark . It is worth noting that Theorem  of [] becomes a special case of Corol-lary . if we simply setq= . Furthermore, it can be noted that this result is a reﬁnement of Proposition . of [] and if we seth=  in the above corollary then we get a more reﬁned result than Proposition . of [].

4 Feng Qi type∇-integral inequalities

In [] and [] there are given theq-analog and theh-analog, respectively, of some Feng Qi ∇-integral inequalities, which are given in [, ], and []. Hence, this part of the article is concerned with the results related to Fenq Qi type∇-integral inequalities deﬁned on a discrete time scale.

Lemma . Suppose p≥is a real number and f be a non-negative and monotonic func-tion on[a,b]T,then

pfp–ρ(v)f∇(v)≤fp(v)∇≤pfp–(v)f∇(v).

Proof The deﬁnition of the∇-derivative may be quoted from [] in the following way:

fp(v)∇=f

p₍_v_{) –}_fp_(ρ(_v₎₎

ν(v) =

p

ν(v)

f(v)

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Sincef is an increasing function, we get

fp–ρ(v)f(v) –fρ(v)≤

f(v)

f(ρ(v))

tp–dt≤fp–(v)f(v) –fρ(v),

pfp–ρ(v)f(v) –f(ρ(v))

ν(v) ≤

fp(v)∇≤pfp–(v)f(v) –f(ρ(v))

ν(v) ,

pfp–ρ(v)f∇(v)≤fp(v)∇≤pfp–(v)f∇(v).

Remark . If in the preceding lemma we setρ(v) =q–₍_v_–_h_{) deﬁned on}_Ta

(q,h), then we

obtain the following result.

Supposep≥ is a real number andf is a non-negative and monotonic function on [a,b]∩Ta

(q,h), then

pfp–q–(v–h)∇(q,h)f(v)≤ ∇(q,h)fp(v)≤pfp–(v)∇(q,h)f(v).

Theorem . Suppose thatφis a function deﬁned on[a,b]Tsatisfying the conditionφ(v)≥ and

φ∇(v)≥(p– )(v–a)p–

for v∈[a,b]Tand p≥.Then

b

a

φp(v)∇v≥

b

a

φ(v)∇v

p–

.

Proof Forv∈[a,b]T, we consider the diﬀerence

F(v) =

v

a

φp(u)∇u–

v

a

φ(u)∇u

p–

and we deﬁne

f(v) =

v

a

φ(u)∇u.

Now by using Lemma ., we have

F∇(v) =φp(v) –fp–(v)∇

≥φp(v) – (p– )gp–(v)f∇(v)

=φp(v) – (p– )fp–₍_v_)φ(_v₎

=φ(v)F(v),

whereF(v) =φp–(v) – (p– )fp–(v)≥.

Now, by virtue of Lemma . we get

F_∇(v) =φp–(v)∇– (p– )fp–(v)∇

≥(p– )φp–(v)φ∇(v) – (p– )(p– )fp–(v)f∇(v)

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f(v) =

v

a

φ(u)∇u≤(v–a)φ(v)

and hence we get

F_∇(v)≥(p– )φp–(v)φ∇(v) – (p– )(v–a)p–φp–(v)f(v)

≥(p– )φp–(v)φ∇(v) – (p– )(v–a)p–≥.

So,F(v) is an increasing function. Therefore, F(v)≥F(a)≥, which shows the

non-negativity ofF∇(v). So, we conclude thatFis increasing andF(v)≥, hence the assertion

follows.

If we chooseρ(v) =q–(v–h) deﬁned onTa₍_q_,_h₎, then we get the following corollary.

Corollary . Suppose thatφbe a function deﬁned on[a,b]∩Ta₍_q_,_h₎satisﬁes the condition

φ(v)≥and

φ∇(q,h)₍_v₎≥₍_p_{– )(}_v_–_a₎p–

for v∈[a,b]Tand p≥.Then

b

a

φp(v)∇(q,h)v≥

b

a

φ(v)∇(q,h)v

p–

.

Theorem . Suppose thatφis a function deﬁned on[a,b]Tsatisfying the conditionφ(v)≥ and

φ∇(v)≥t

v–a b–a

α–

b

a

φα+(v)∇v≥ 

(b–a)α–

b

a

φ(v)∇v

α+

.

Proof Assume the following diﬀerence, forv∈[a,b]T:

F(v) =

v

a

φα+(v)∇v–  (b–a)α–

v

a

φ(v)∇v

α+

.

Also we deﬁne the functionf,

f(v) =

v

a

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Now by using Lemma ., we have

F∇(v) =φα+(v) –  (b–a)α–

fα+(v)∇

≥φα+(v) – α+  (b–a)α–f

α

(v)f∇(v)

=φα+(v) – α+  (b–a)α–f

α

(v)φ(v)

=φ(v)F(v),

whereF(v) =φα+(v) –₍_b_–α+_a₎α–fα(v)≥. Again using Lemma . we get

F_∇(v) =φα+(v)∇– α+  (b–a)α–

fα₍_v₎∇

≥(α+ )φα(v)φ∇(v) – α(α+ ) (b–a)α–f

α–₍_v₎_g∇₍_v₎

= (α+ )φα(v)φ∇(v) – α(α+ ) (b–a)α–f

α–₍_v_)φ(_v_).

f(v) =

v

a

φ(u)∇u≤(v–a)φ(v),

and hence we obtain

F_∇(v)≥(α+ )

φα(v)φ∇(v) –α(v–a)

α–

(b–a)α– φ

α–₍_v_)φ₍_v₎

≥(α+ )φt(v)

φ∇(v) –α

v–a b–a

α–

≥.

Consequently,F(v) is an increasing function. Therefore,F(v)≥F(a)≥, which shows

the non-negativity ofF∇(v). It is also inferred thatFis increasing and sinceF(v)≥ we

get the required result.

If we chooseρ(v) =q–₍_v_–_h_{) deﬁned on}_Ta

(q,h)then we get the following corollary.

Corollary . Letφbe a function deﬁned on[a,b]∩T₍a_q_,_h₎such thatφ(v)≥and

φ∇(q,h)₍_v₎≥_α

v–a b–a

α–

for v∈[a,b]Tandα≥.Then

b

a

φα+(v)∇(q,h)v≥

 (b–a)α–

b

a

φ(v)∇(q,h)v

α+

(12)

5 Conclusion

The article is aimed to explore Feng Qi integral inequalities on time scales while consid-ering- and∇-integrals. The present article generalizes some results of various articles, Lemma . of [, ] is a special case of Lemma .. Moreover, observations show that Theorem  of [] is a special case of Corollary ., in addition to results of the present ar-ticle there is a reﬁnement of Proposition . in []. It can also be noted that Corollary . gives us Theorem  of [] as a particular case and reﬁnes Proposition . of [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ARK and GR made the main contribution in conceiving the presented research. TF, NI, and ARK worked jointly on the second, on the third, and on the fourth sections, while ARK drafted these sections. GUR has drafted and worked solely on the abstract, the Introduction section and the last section and proof-read the whole article. It should be noted that the authors’ names are in alphabetical/dictionary order. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, University of Karachi, University Road, Karachi, 75270, Pakistan.}2_{Department of}

Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan.3_{Comsats Institute of Information} Technology, Abbot Abad, Pakistan.

Received: 4 March 2016 Accepted: 7 September 2016 References

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