Multidimensional Hardy-type inequalities on time scales with variable exponents

J ournal of

M athematical I nequalities

With Compliments of the Author

JMI-13-49 725–736 Zagreb, Croatia Volume 13, Number 3, September 2019

O. O. Fabelurin, J. A. Oguntuase and L.-E. Persson

Multidimensional Hardy-type inequalities on time

scales with variable exponents

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Journal of Mathematical

Inequalities

Volume 13, Number 3 (2019), 725–736 doi:10.7153/jmi-2019-13-49

MULTIDIMENSIONAL HARDY–TYPE INEQUALITIES ON TIME SCALES WITH VARIABLE EXPONENTS

O. O. FABELURIN, J. A. OGUNTUASE ANDL.-E. PERSSON

(Communicated by J. Peˇcari´c)

Abstract. A new Jensen inequality for multivariate superquadratic functions is derived and proved.

The derived Jensen inequality is then employed to obtain the general Hardy-type integral inequal- ity for superquadratic and subquadratic functions of several variables.

1. Introduction

Hardy’s discrete inequality reads: if p>1 and {a_k}^∞_k=1 is a sequence of nonneg- ative real numbers, then

∑

1 n

∑

a_{k p}

p

p−1 _{p ∞}

n=1

∑

a_n^p. (1.1)

Furthermore, G. H. Hardy [9] announced (without proof) that ifp>1 and the function f is nonnegative and integrable over the interval(0,x), then

_∞ 0

1 x

_x 0 f(t)dt

_p dx

p p−1

_{p ∞}

0 f^p(x)dx. (1.2) Inequality (1.2) was finally proved by Hardy [10] in 1925. Thus, inequality (1.2) is usually referred to in the literature as the classical Hardy integral inequality while in- equality (1.1) is its discrete analogue. The constant

p−1p

_p

on the right hand sides of both inequalities (1.1) and (1.2) is the best possible.

Note that (1.1) follows from (1.2), which was pointed out by Hardy [9] but there he also informed that a proof of (1.1) was given to him already in a private letter from E. Landau in 1921. More information concerning the interesting prehistory of Hardy’s inequality can be found in [15].

In the lastfive decades, the Hardy inequality (1.2) has been extensively studied and generalized. A lot of information as regarding applications, alternative proofs, variants, generalizations and refinements abound in the literature (see e.g. the books [11,16,17]

and the references cited therein).

Mathematics subject classification(2010): 26D10, 26D20, 26E70.

Keywords and phrases: Multidimensional inequalities, Jensen’s inequality, Hardy-type inequalities, time scales, superquadratic functions.

c , Zagreb

Paper JMI-13-49 725

In his PhD thesis, S. Hilger [12] (see also [6,13,14]) initiated the calculus of time scales in order to create a theory that will unify discrete and continuous analysis. This new concept has inspired researchers to study Hardy inequalities on time scales. The first known work in this direction is probably due to P.Reh´ak [19] who in 2005 derivedˇ Hardy integral inequality on time scales. Indeed, he showed that

_∞ a

1 σ(x)−a

_σ(x) a f(t)Δt

p

Δx<

p p−1

_{p ∞}

a f^p(x)Δx, wherea>0,p>1 and f is a nonnegative function.

For notations here and in the sequel see Section 2.

In 2001, R. P. Agarwal et al. [1] obtained the following Jensen’s inequality on time scales

Φ 1

b−a _b

a f(x)Δx

1

b−a _b

a Φ(f(x))Δx.

Moreover, T. Donchev et al. [8] employed the above result to derive the following Hardy-type inequality involving multivariate convex functions on time scales:

THEOREM1.1. Let (Ω1,M,μΔ) and (Ω2,L,λΔ) be two time scale measure spaces and U⊂Rⁿ be a closed convex set. Let K:Ω1→R be defined by K(x):=

Ω2k(x,y)Δy<∞, x∈Ω1,where k(x,y)0is a kernel. Moreover, letζ^:Ω1→R and the weight w=w(y)be defined by

w(y):=

Ω1

k(x,y)ζ(x) K(x)

Δx, y∈Ω2. Then for each convex functionΦ,

Ω1ζ(x)Φ 1

K(x)

Ω2

k(x,y)f(y)Δy

Δx

Ω2

w(y)Φ(f(y))Δy (1.3) holds for allλΔ-integrable functionsf:Ω2→Rⁿ such thatf(Ω2)⊂U⊂Rⁿ.

In a recent paper, Oguntuase and Persson [18] presented a number of Hardy-type in- equalities on time scales using superquadraticity technique which is based on the ap- plication of Jensen dynamic inequality. For some recent developments on Hardy-type inequalities on time scales and related results we refer interested reader to the book [3].

Motivated by the above results, our main aim in this paper is to first establish a Jensen inequality for multivariate superquadratic functions and then employ it to derive some new general Hardy-type inequalities for multivariate superquadratic functions in- volving more general kernels on arbitrary time scales.

The paper is organized as follows: In Section 2, we recall some basic notions, def- initions and results on multivariate superquadratic functions on time scales. In Section 3 we state and prove our main results.

HARDY-TYPE INEQUALITIES ON TIME SCALES 727 2. Preliminaries, definitions and some basic results

First, we recall that a time scale (or measure chain) T is an arbitrary nonempty closed subset of the real lineRwith the topology of the subspaceR.Examples of time scales are the real numbersRand the discrete time scaleZ. Since a time scaleTmay or may not be connected, we need the concept of jump operators. Fort∈T, we define the forward jump operatorσ^:T→Tby

σ(t) =inf{s∈T:s>t} and the backward jump operator by

ρ(t) =inf{s∈T:s<t}.

Ifσ(t)>t, we say thatt is right-scattered and ifρ(t)<twe say thatt is left-scattered.

The points that are both right-scattered and left-scattered are called isolated. Ifσ(t) =t, thent is said to be right-dense, and if ρ(t) =t then t is said to be left-dense. The points that are simultaneously right-dense and left-dense are called dense. The mapping μ^:T→[0,∞)defined by

μ(t) =σ(t)−t

is called the graininess function. If T has a left-scattered maximum M, then define T^k=T\{M}; otherwise T^k=T. Let f :T→R be a function. Then we define the function f^σ:T=Rby f^σ(t) = f(σ(t))for allt∈T. Also, for a function f:T→R, the delta derivative is defined by

f^Δ(t):= lim

s→t,σ(s)=t

f^σ(s)−f(t) σ(s)−t .

A function f:T→Ris called rd-continuous provided it is continuous at all right-dense points inT and its left-sided limits exists (finite) at all left-dense points in T. We refer interested readers to the books [2], [6] and [7] for more details concerning the calculus of time scales. Note that we have

σ(t) =t,μ(t) =0, f^Δ=f, ^b

a f(t)Δt= ^b

a f(t)dt,whenT=R, σ(t) =t+1, μ(t) =1, f^Δ=Δf,

_b

a f(t)Δt=^b−1_t=a

∑

The following Fubini’s theorem on time scale in [5] will be needed in the proof of our results in Section 3:

LEMMA2.1. Let (Ω,M,μΔ) and (Λ,L,λΔ), be two finite dimensional time scale measures spaces. If f:Ω×Λ→ℜ is a μΔ×λΔ-integrable function and define the function φ(y) = _Ωf(x,y)Δx for a.e. y ∈Λ and ϕ(x) = _λ f(x,y)Δy for a.e.

x ∈Ω, thenφ ^isλΔ−integrable onΛ,ϕ ^isμΔ-integrable onΩ and

ΩΔx

Λf(x,y)Δy=

ΛΔy

Ωf(x,y)Δx. (2.1)

Moreover, M. Anwar et al. [4] result on the Jensen inequality for convex functions in several variables on time scales will also be needed.

THEOREM2.2. Let (Ω1,Σ1,μΔ) and (Ω2,Σ2,λΔ) be two time scale measure spaces. Suppose U⊂Rⁿis a closed convex set andΦ∈C(U,R)is convex. Moreover, let k:Ω1×Ω2→Rbe nonnegative such that k(x,.)isλΔ−integrable. Then

Φ

Ω2k(x,y)f(y)Δy

Ω2k(x,y)Δy

^Ω2k(x,y)Φ(f(y))Δy

Ω2k(x,y)Δy (2.2)

holds for all functionsf:Ω2→U, where f_j(y) areμΔ2-integrable for all j∈ {1,2,...,n}, and _Ω2k(x,y)f(y)Δ(y)denotes the n-tuple

Ω2

k(x,y)f₁(y)Δ(y),

Ω2

k(x,y)f₂(y)Δ(y),...,

Ω2

k(x,y)f_n(y)Δ(y)

. In the sequel, we make the following definitions, assumptions and notations.

(A1.) Ω1=Ω2= [a,l) = [a1,l₁)T×[a2,l2)T...×[an,ln)T, where 0ai<li∞.

(A2.) a<b if componentwiseai<bi, i=1,2,...,n.

(A3.) k:[a,l)×[a,l)→R+is such that k(x,y) =

1 if ay<σ(x)l,

0 otherwise, (2.3)

that is

k(x1,..,xn,y₁,...,yn) =

1 if aiyi<σ(xi)li,i=1,...,n

0 otherwise, (2.4)

(A4.) Φ(u) =u^p,p>1.

REMARK2.3. Under the assumptions (A1- A4), form=1, Theorem2.2yields the inequality

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi)−ai) _σ(x1)

a1

... ^σ(x1)

a1

f(y1,...,yn)Δy1...Δyn

⎞

⎟⎟

⎠

p

1

∏n

i=1(σ(x_i)−a_i) _σ(x1)

a1

... ^σ(x1)

a1

f^p(y1,...,yn))Δy1...Δyn. (2.5)

We will also need the following Lemmas for the proof of our main results in the paper.

HARDY-TYPE INEQUALITIES ON TIME SCALES 729 LEMMA2.4. Letβ >0and a,b,l∈Tbe such that0a<bl.

(i) Ifβ >1, then _l

b(s−a)^β−1Δs 1 β

(t−a)^β−(b−a)^β

^l

b(σ(s)−a)^β−1Δs. (2.6) (ii) If β<1,then

_l

b(s−a)^β−1Δs 1 β

(l−a)^β−(b−a)^β

_l

b(σ(s)−a)^β−1Δs. (2.7) Proof. For case (i), letβ >1. Then by applying Keller’s chain [6], wefind that

(t−a)^β_Δ

=β ¹

0 [h(σ(t)−a) + (1−h)(t−a)]^β−1dh

β ¹

0 [h(t−a) + (1−h)(t−a)]^β−1dh

=β(t−a)^β−1. Integrating, we obtain

_l

b(t−a)^β−1Δt 1 β

l−a)^β−(b−a)^β

. (2.8)

On the other hand, (t−a)^β_Δ

=β ¹

0 [h(σ(t)−a) + (1−h)(t−a)]^β−1dh

β ¹

0 [h(σ(t)−a) + (1−h)(σ(t)−a)]^β−1dh

=β(σ(t)−a)^β−1, yielding

1 β

(l−a)^β−(b−a)^β

^l

b(σ(t)−a)^β−1Δt. (2.9) Finally, combining inequalities (2.8) and (2.9) yields the desired result.

(ii). For the caseβ<1, the proof is similar to the proof of (i), except that the inequali- ties signs are reversed.

LEMMA2.5. Let n∈N. If0xiyi, for1in.Then

∏

(yi−xi)

∏

i=1

y_i−

∏

i=1

x_i. (2.10)

Proof. The proof is performed by induction and just noting that (y2−x₂)(y1−x₁) =y₂y₁−x₂x₁−x₂(y1−x₁)−x1(y2−x₂)

y₂y₁−x₂x₁.

3. Multidimensional Hardy-type inequalities for convex functions on time scales

Ourfirst main result reads:

THEOREM3.1. Let 0a<b<∞. Let the functions p,β ^:[a,b)T→R be de- fined, respectively, by

p(x) =

p_o, 0xb,

p₁, x>b, and β(x) =

βo, 0xb,

β1, x>b. (3.1) Moreover, assume that po,p1∈R\{0} are such that po1,p11 or po1,p1<0 or po<0,p11or po<0,p1<0.If f:[a,l]→Ris non-negativeΔ-integrable and

f∈C_rd([a,l],R)for which _b1

a1 ...

_bn an

f^p(x)(y₁,...,y_n) n

∏

1 β(x)

(y_i−a_i)^−β(x)

×

1−

∏

i=1

y_i−a_i l_i−a_i

_β(x)

Δy1...Δyn<∞,

(3.2)

then _l1

a1

... ^ln

an

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi)−a_i) _σ(x1)

a1

... ^σ(x1)

a1

f(y1,...,y_n)Δy1...Δyn

⎞

⎟⎟

⎠

p(x)

×

∏

i=1(σ(x_i)−a_i)^−β(x)Δx₁...Δx_n

_b1 a1

...

_bn an

f^p(x)(y₁,...,y_n) n

∏

1 β(x)

(y_i−a_i)^−β(x)

×

1−

∏

i=1

y_i−a_i l_i−a_i

_β(x)

Δy₁...Δy_n+I_o, (3.3)

where Io=0 if lb (so thatβ(x) =βoand p(x) =po) and I_o= ^b1

a1

... ^bn

an

f^p1(y₁,...,y_n)

∏

i=1

1 β1

(y_i−a_i)^−β1−(l_i−a_i)^−β1

Δy₁...Δy_n

− ^bo

a1

... ^bn

an

f^po(y1,...,yn)

∏

i=1

1 βo

(yi−ai)^−βo−(li−ai)^−βo

Δy1...Δyn. (3.4) If0<p(x)1, then (3.3) holds in the reverse direction.

HARDY-TYPE INEQUALITIES ON TIME SCALES 731 Proof. Letbl. By applying Jensen’s inequality (see Remark2.3), Lemma2.1 and Lemma2.4, wefind that

_l1 a₁ ...

_ln an

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi)−a_i) _σ(x1)

a₁ ...

_σ(x1)

a₁ f(y₁,...,y_n)Δy₁...Δy_n

⎞

⎟⎟

⎠

p(x)

×

∏

i=1(σ(xi)−ai)^−β(x)Δx1...Δxn

^l1

a1

... ^ln

an

⎡

⎢⎢

⎣ 1

∏n

i=1(σ(xi)−a_i) _σ(x1)

a1

... ^σ(x1)

a1

f^po(y1,...,y_n)Δy1...Δyn

⎤

⎥⎥

⎦

×

∏

i=1(σ(x_i)−a_i)^−βoΔx₁...Δx_n

_l1 a₁ ...

_ln an

f^po(y₁,...,y_n)

l1

y₁ ...

_ln yn

∏

i=1(σ(x_i)−a_i)^−(βo+1)Δx₁...Δx_n

×Δy1...Δyn

^b1

a1

... ^bn

an

f^p(x)(y1,...,y_n) n

∏

1 β(x)

(yi−a_i)^−β(x)

×

1−

∏

i=1

yi−ai

l_i−a_{i β(x)}

Δy1...Δyn.

Hence, (3.3) is proved for this case.

Next, letbl. By applying Jensen’s inequality (see Remark2.3) and Lemma2.1, wefind that

_l1 a1

... ^ln

an

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi)−ai) _σ(x1)

a1

... ^σ(x1)

a1

f(y1,...,yn)Δy1...Δyn

⎞

⎟⎟

⎠

p(x)

×

∏

i=1(σ(xi)−a_i)^−β(xi)Δx1...Δxn

^b1

a1

... ^bn

an

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi)−a_i) _σ(x1)

a1

... ^σ(x1)

a1

f(y1,...,y_n)Δy1...Δyn

⎞

⎟⎟

⎠

po

×

∏

i=1(σ(x_i)−a_i)^−βoΔx₁...Δx_n

+ ^l1

b1

... ^ln

bn

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi)−ai) _b1

a1

... ^bn

an

f(y1,...,y_n)Δy1...Δyn

⎞

⎟⎟

⎠

p1

×

∏

i=1(σ(x_i)−a_i)^−β1Δx₁...Δx_n + ^l1

b1

... ^ln

bn

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(x_i)−a_i) _σ(x1)

b1

... ^σ(xn)

bn

f(y₁,...,y_n)Δy₁...Δy_n

⎞

⎟⎟

⎠

p1

×

∏

i=1(σ(xi)−ai)^−β1Δx1...Δxn

^b1

a1

... ^bn

an

f^po(y1,...,yn)

b1

y1

... ^bn

yn

∏

i=1(σ(xi)−ai)^−βoΔx1...Δxn

Δy1...Δyn

+ ^b1

a1

... ^bn

an

f^p1(y₁,...,y_n)

l1

b1

... ^ln

bn

∏

i=1(σ(x_i)−a_i)^−βoΔx₁...Δx_n

Δy₁...Δy_n

+ _l1

b₁ ...

_ln bn

f^p1(y₁,...,y_n)

l1

y₁ ...

_ln yn

∏

i=1(σ(x_i)−a_i)^−βoΔx₁...Δx_n

Δy₁...Δy_n

:=I. (3.5)

By Lemma2.4and Lemma2.5, wefind that

I ^b1

a1

... ^bn

an

f^po(y1,...,yn)

∏

i=1

1 βo

(yi−ai)^−βo−(bi−ai)^−βo

Δy1...Δyn

+ ^b1

a1

... ^bn

an

f^p1(y1,...,yn)

∏

i=1

1 β1

(bi−ai)^−β1−(li−ai)^−β1

Δy1...Δyn

+ ^l1

b1

... ^ln

bn

f^p1(y1,...,yn)

∏

i=1

1 β1

(yi−ai)^−β1−(li−ai)^−β1

Δy1...Δyn

^b1

a1

... ^bn

an

f^po(y1,...,y_n) n

∏

1 βo

(yi−a_i)^−βo 1−

∏

i=1

y_i−a_i l_i−a_i

_βo

·Δy₁...Δy_n + ^l1

b1

... ^ln

bn

f^p1(y1,...,yn) n

∏

1 β1

(yi−ai)^−β1 1−

∏

i=1

yi−ai

l_i−ai

_β1

·Δy1...Δyn

+ _b1

a1

...

_bn an

f^p1(y₁,...,y_n)

∏

i=1

1 β1

(y_i−a_i)^−β1−(l_i−a_i)^−β1

Δy₁...Δy_n

HARDY-TYPE INEQUALITIES ON TIME SCALES 733

− _bo

a1 ...

_bn an

f^po(y₁,...,y_n)

∏

i=1

1 βo

(y_i−a_i)^−β1−(l_i−a_i)^−βo

Δy₁...Δy_n

= _b1

a1 ...

_bn an

f^p(x)(y₁,...,y_n) n

∏

1 β(x)

(y_i−a_i)^−β(x)

×

1−

∏

i=1

yi−ai

l_i−a_{i β(x)}

Δy1...Δyn+I_o. (3.6)

By combining the inequalities (3.5) with (3.6) the inequality (3.3) follows so that the proof is complete.

The next result concerns the dual version of Theorem3.1when the Hardy operator H: f(x1,..,xn)−→ 1

∏n

i=1(σ(x_i)−a_i) _σ(x1)

a1

... ^σ(x1)

a1

f(y1,...,yn)Δy1...Δyn

is replaced by the dual Hardy operator H^∗:f(x1,..,x_n)−→

∏

i=1(σ(xi)−a_i) ^∞

σ(x1)... ^∞

σ(xn)

f(y1,...,y_n)Δy1...Δyn

∏n

i=1(σ(yi)−ai)(yi−a_i). Our next main result concerning the dual Hardy operatorH^∗reads:

THEOREM3.2. Let 0a<b<∞. Let the functions p,β ^:[a,b)T→R be de- fined, respectively, by

p(x) =

po, 0xb,

p₁, x>b, ,β(x) =

βo, 0xb,

β1, x>b. (3.7)

Moreover, assume that po,p₁∈R\{0} are such that po1,p₁1 or po1,p₁<0 or p_o<0,p₁1or p_o<0,p₁<0.If f:[a,l]→Ris non-negativeΔ-integrable and

f∈C_rd([a,l],R)for which _∞

l1

...

_∞ ln

f^p1(y₁,...,y_n) n

∏

(y_i−a_i)^β(y) β(y)

1−

∏

i=1

l_i−a_i yi−ai

_β(y)

× Δy₁...Δy_n

∏n

i=1(σ(y_i)−a_i)(y_i−a_i)<∞, (3.8) then

_∞ l₁ ...

_∞ ln

⎛

⎜⎜

⎝

∏

i=1(σ(x_i)−a_i) _∞

σ(x1)...

_∞ σ(xn)

⎛

⎜⎜

⎝ f(y₁,...,y_n)

∏n

i=1(σ(yi)−a_i)(yi−a_i)

⎞

⎟⎟

⎠Δy₁...Δy_n

⎞

⎟⎟

⎠

p(x)

× n

∏

Δx1...Δxn

∏n

i=1(σ(xi)−a_i)

_∞ l1 ...

_∞ ln

f^p1(y₁,...,y_n) n

∏

(yi−a_i)^β(y) β(y)

1−

∏

i=1

l_i−a_i y_i−a_i

_β(y)

× Δy1...Δyn

∏n

i=1(σ(yi)−ai)(yi−ai)+I_o. (3.9)

where Io=0 if lb (so thatβ(x) =βoand p(x) =po) and

I_o= ^∞

l1

... ^∞

ln

f^p1(y₁,...,y_n)

∏n

i=1(σ(y_i)−a_i)(y_i−a_i)

∏

1 β1

(y_i−a_i)^β1−(l_i−a_i)^β1

Δy₁...Δy_n

− ^∞

l1

... ^∞

ln

f^po(y₁,...,y_n)

∏n

i=1(σ(y_i)−a_i)(y_i−a_i)

∏

1 βo

(yi−ai)^βo−(li−ai)^βo

Δy1...Δyn.

(3.10) If0<p(x)1, then (3.9) holds in the reverse direction.

Proof. Letbl. Applying Jensen’s inequality (see Remark2.3) and Lemma2.1, we obtain that

_∞ l1 ...

_∞ ln

⎛

⎜⎜

⎝

∏

i=1(σ(x_i)−a_i) _∞

σ(x1)...

_∞ σ(xn)

⎛

⎜⎜

⎝ f(y1,...,y_n)

∏n

i=1(σ(yi)−a_i)(yi−a_i)

⎞

⎟⎟

⎠Δy₁...Δy_n

⎞

⎟⎟

⎠

p(x)

× n

∏

Δx₁...Δx_n

∏n

i=1(σ(x_i)−a_i)

^∞

l1

... ^∞

ln

_∞

σ(x1)... ^∞

σ(xn)

⎛

⎜⎜

⎝f^p1(y1,...,y_n)Δy1...Δyn

∏n

i=1(σ(yi)−ai)(yi−ai)

⎞

⎟⎟

⎠

× n

∏

Δx₁...Δx_n

^∞

l1

... ^∞

ln

f^p1(y₁,...,y_n)

∏n

i=1(σ(yi)−a_i)(yi−a_i)

⎡

⎣ ^y1

l1

... ^yn

ln

n

∏

Δx₁...Δx_n

⎤

⎦

HARDY-TYPE INEQUALITIES ON TIME SCALES 735

×Δy1...Δyn

^∞

l1

... ^∞

ln

f^p1(y₁,...,y_n) n

∏

(y_i−a_i)^β(y) β(y)

1−

∏

i=1

l_i−a_i yi−ai

_β(y)

× Δy₁...Δy_n

∏n

i=1(σ(yi)−ai)(yi−a_i). (3.11)

Finally, letbl.Also the proof of this case is completely analogous to the correspond- ing part of the proof of Theorem3.1so we leave out the details.

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(Received November 12, 2018) O. O. Fabelurin

Department of Mathematics Obafemi Awolowo University Ile-Ife, Osun State, Nigeria e-mail:[email protected] J. A. Oguntuase Department of Mathematics Federal University of Agriculture P.M.B. 2240, Abeokuta, Ogun State, Nigeria e-mail:[email protected] L.-E. Persson UiT, The Artic University of Norway Campus Narvik, Lodve Langes Gate 2, 8514 Narvik, Norway e-mail:[email protected]

Journal of Mathematical Inequalities www.ele-math.com

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