NEWDYNAMICINEQUALITIES FORDECREASING FUNCTIONS AND THEOREMS OF HIGHER INTEGRABILITY∗

S.H.Saker,D.O’Regan,M.M.Osman,R.P.Agarwal

(1.Dept.of Math.,Faculty of Science,Mansoura University,Mansoura-Egypt;2.School of Math.,Statistics and Applied Math.,National University of Ireland,Galway,Ireland;3.Dept.of Math.,Texas A&M University-Kingsvilie,Texas,78363,USA)

1 Introduction

The classical integral Hölder inequality states that if f and g are positive measurable functions defined on

For generalizations and extensions in the literature we refer the reader to the papers[6–9,12–15].In this paper we establish some new dynamic inequalities on time scales.For related dynamic inequalities on time scales,we refer the reader to the books[1,2]and the papers[16,17].

The paper is organized as follows.In Section 2 we give a brief overview on time scales.In Section 3,we prove some new dynamic inequalities via convexity functions which contain inequalities given by Hardy,Littlewood and Pólya.We also establish some new dynamic inequalities on time scales for decreasing functions.These inequalities contain in particular generalizations of integral inequalities due to Hardy,D’Apuzzo and Sbordone and Popoli.We apply also these inequalities to prove a higher integrability theorem for monotone decreasing function on time scales.

2 Preliminaries on Time Scales

For a good introduction on time scales we refer the reader to the book[4].The forward jump operator and the backward jump operator are defined byσ(t):=inf{s∈T:s>t},andρ(t):=sup{s∈T:sinf T,right-dense ifσ(t)=t,left-scattered ifρ(t)t.A function f:T → R is said to be rightdense continuous(rd-continuous)provided f is continuous at right-dense points and at left-dense points in T,whose left hand limits exist and are finite.The set of all such rd-continuous functions is denoted by Crd(T).The graininess functionµfor a time scale T is defined by µ(t):= σ(t)−t,and for any function f:T → R,f(σ(t))is defined by fσ(t).We now recall the product and quotient rules for the derivative of the product f g and the quotient f/g(where ggσ0,here gσ=g ◦ σ)of two differentiable functions f and g

One chain rule is

which is a simple consequence of Keller’s chain rule[4,Theorem 1.90].Another chain rule is as follows:Let f:R→R be continuously differentiable and suppose g:T→R is delta differentiable,then f◦g:T→R is delta differentiable and

In this paper we will consider the(delta)integral which we can be defined as follows.If F∆(t)=f(t),then the Cauchy(delta)integral of f is defined byf(s)∆s:=F(t)−F(t0).It can be shown(see[4])that if f∈Crd(T),then the Cauchy integralAn infinite integral is defined asIntegration on discrete time scales is defined by

The integration by parts formula on time scales is given by

Hölder’s inequality[4,Theorem 6.13]states that two rd-continuous functions f,g:T→R satisfy

where p>1,1/p+1/q=1 and a,b∈T.

Throughout this paper,we will assume that the functions in the statements of the theorems are positive and the integrals considered are assumed to exist and are finite.

Next we present a weighted Hardy type inequality[16]on time scales which is useful in the proof of our main results.Without loss of generality,we assume that sup T=∞,and define the time scale interval[a,b]Tby[a,b]T:=[a,b]∩T.

Theorem 2.1LetTbe a time scale witha,b∈T(a

3 Main Results

In our next two results we will assume thatϕis a differentiable convex function.

Theorem 3.1LetTbe a time scale with0,a∈T(a>0).Iffis decreasing then

ProofLet x∈(0,a)T.Since f is a decreasing function,

Let F(x)=so F is increasing.Note thatis a convex function,then

From the chain rule(2.3)note

Since F is increasing,

Now from(3.2)and(3.3)we obtain

so integration from 0 to a yields

From this inequality we have(3.1).The proof is complete.

A special case of Theorem 3.1 is given as follows whenϕ(u)=up,p>1.

Corollary 3.1LetTbe a time scale with0,a∈T(a>0).Iffis decreasing then

Remark 3.1Applying Hölder’s inequality on the right hand side of(3.6)we get the inequality

Remark 3.2We consider the interval[a,b]Tand immediately we have

Note if f is decreasing then xγ−1f(with γ ≤ 1)is decreasing.Use xγ−1f in(3.6)to obtain the following result.

Corollary 3.2LetTbe a time scale with0,a∈T(a>0),p>1andγ≤1.Iffis decreasing then

Remark 3.3Consider the interval[a,b]Tand assume that there exist positive numbers r and s with r

In our next results we will assume that there exists an m>0 such that

We apply now the weighted Hardy inequality(2.6)to prove a dynamic inequality.

Theorem 3.2SupposeTis a time scale anda,b∈T(b>a),and assume(3.11)holds.Let

Ifα≤1andβ>1,then

where

In particular

ProofApplying inequality(2.6)with p=q= β,u(x)=(x − a)α−β−1and υ(x)=(x − a)α−1,we obtain

here A=Cpand so

Let t∈(a,σ(b))T.From the chain rule(2.2)we see that(noteα≤1 andβ>1)

and as a result

Let t∈(a,b)T.Apply the chain rule(2.2)and(3.11),then we have(noteα<β)

and as a result

From(3.15),(3.16)and(3.17)we have

Putting(3.18)into(3.14),we obtain(3.13).The proof is complete.

Letα=q/p andβ=q with p≥q>1 in(3.13)and we obtain the following Hardy-type inequality.

Corollary 3.3SupposeTis a time scale anda,b∈T(b>a),p≥q>1,and assume(3.11)holds.Then

wheremandGare defined in(3.11)and(3.12)respectively.

Remark 3.4As a special case of Theorem 3.2 when T=R(note m=1),we have the inequality in[15,Theorem 2.1](note in our caseα≤1 andβ>1)

where

In particular when a=0 andα=1 we have

Remark 3.5As a special case of Theorem 3.2 when T=N,a,b∈N,noting σ(x)=x+1 andwe have the discrete inequality

Now we prove some lemmas that will be needed below.

Lemma 3.1Suppose0<λ≤1and assume(3.11)holds.Let

ProofFrom the definition of H and(3.11)we have

Let u(x)=.Integrating the right hand side of(3.22),using formula(2.4)with u∆(x)=(x − a)λ−1and υσ(x)=h(t)∆t,we obtain(note

Let t∈(a,b)T.Apply the chain rule(2.2)and we see that(note 0<λ≤1)

and as a result

The proof is complete.

Lemma 3.2Letλ<0and suppose(3.11)holds.In addition assume

Then

whereHis defined in Lemma3.1.

ProofProceed as in Lemma 3.1 and we obtain(note(3.23))

Let t∈(a,b)T.Apply the chain rule(2.2),and we see that(noteλ<0)

and as a result

The proof is complete.

Lemma 3.3Letλ>1and assume(3.11)holds.Then

whereHis defined in Lemma3.1.

ProofFrom the definition of H(x)and(3.11)we have

Let u(x)=Integrating the right hand side of(3.26),using formula(2.4)with u∆(x)=(σ(x)−a)λ−1and υσ(x)=∆t,we obtain(note

Let t∈(a,b)T.Apply the chain rule(2.2),and we see that(noteλ>1)

and as a result

The proof is complete.

In our next result we assume the nonnegative function f satisfies the reverse Hölder inequality,that is for x ∈ (a,b)Tthere exist some constants p,q with p0 such that

Theorem 3.3LetTbe a time scale witha,b∈T(b>a),α<1and assume(3.11)holds.Suppose there exist constantsp,qwithp0such that(3.27)holds.Also assume

and

hold.Then

ProofLet

where we let r=p and r=q.From(3.27)we have

Apply Lemma 3.2 on the left-hand side of(3.29)withλ=α−1<0,h=fq(x),then we have

Apply Theorem 3.2 on the right-hand side of(3.29)withβ=q/p(note p

Substituting this into(3.29),we obtain

Thus

The proof is complete.

Remark 3.6Consider the special case whenα=(and assume r>q)and p=1.Then we have

see[7,Lemma 3.2]when T=R.

Now we prove a higher integrability theorem for a monotone decreasing function on time scales.

Theorem 3.4SupposeTis a time scale anda,b∈T(b>a),and assume(3.11)holds.Letfbe a decreasing function and suppose there exist constantsp,qwithp0such that(3.27)holds.Also assume there exists a constants>qsuch that

and

holds.Then

ProofFrom Theorem 3.3(withα=)we obtain

Apply inequality(3.10)with r=q(note r=q

Thus

The proof is complete.

Remark 3.7Since f is decreasing,we have f(σ(x))≤ f(x),so inequality(3.31)becomes

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