Multidimensional Hardy-type inequalities on time scales with variable exponents

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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 inequality for superquadratic and subquadratic functions of several variables.

1. Introduction

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

∑

∞ n=1

1 n

∑

n k=1

a_k

_p

p

p− 1

_{p ∞}

n=1

∑

a_n^p. (1.1)

Furthermore, G. H. Hardy [9] announced (without proof) that if p> 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 inequality (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 last five 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

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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. ˇReh´ak [19] who in 2005 derived 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, where a> 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) 0 is 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 functions f :Ω2→ Rⁿ such that f(Ω2) ⊂ U ⊂ Rⁿ.

In a recent paper, Oguntuase and Persson [18] presented a number of Hardy-type inequalities on time scales using superquadraticity technique which is based on the application 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 involving more general kernels on arbitrary time scales.

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

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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 lineR with the topology of the subspace R. Examples of time scales are the real numbersR and the discrete time scale Z. Since a time scale T may or may not be connected, we need the concept of jump operators. For t∈ T, we define the forward jump operatorσ^:T → T by

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 that t is right-scattered and ifρ(t) < t we say that t is left-scattered.

The points that are both right-scattered and left-scattered are called isolated. Ifσ(t) = t, then t 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 = R by f^σ(t) = f (σ(t)) for all t ∈ 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 → R is 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, when T = R, σ(t) = t + 1, μ(t) = 1, f^Δ= Δ f ,

_b

a f(t)Δt =^b−1_t=a

∑

^f(t), when T = Z.

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, then φ ^isλΔ−integrable on Λ,ϕ ^isμΔ-integrable onΩ and

ΩΔx

Λf(x,y)Δy =

ΛΔy

Ωf(x,y)Δx. (2.1)

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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→ R be 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 functions f :Ω2→ U, where fj(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) fn(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 0 ai< li ∞.

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

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

k(x,y) =

1 if a y <σ(x) l,

0 otherwise, (2.3)

that is

k(x1,..,xn,y1,...,yn) =

1 if ai yi<σ(xi) li,i = 1,...,n

0 otherwise, (2.4)

(A4.) Φ(u) = u^p, p > 1.

REMARK2.3. Under the assumptions (A1- A4), for m= 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(xi) − ai)

_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.

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LEMMA2.4. Letβ > 0 and a,b,l ∈ T be such that 0 a < b l.

(i) Ifβ > 1, then

_l

b(s − a)^{β −1}Δs 1 β

(t − a)^β− (b − a)^β

^l

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

_l

b(s − a)^{β −1}Δs 1 β

(l − a)^β− (b − a)^β

_l

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

(t − a)^β_Δ

=β ¹

0 [h(σ(t) − a) + (1 − h)(t − a)]^{β −1}dh

β ¹

0 [h(t − a) + (1 − h)(t − a)]^{β −1}dh

=β(t − a)^{β −1}. Integrating, we obtain

_l

b(t − a)^{β −1}Δt 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 −1}dh

β ¹

0 [h(σ(t) − a) + (1 − h)(σ(t) − a)]^{β −1}dh

=β(σ(t) − a)^{β −1}, yielding

1 β

(l − a)^β− (b − a)^β

^l

b(σ(t) − a)^{β −1}Δt. (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. If 0 xi yi, for 1 i n. Then

∏

n i=1

(yi− xi)

∏

ⁿ

i=1

y_i−

∏

ⁿ

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.

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3. Multidimensional Hardy-type inequalities for convex functions on time scales Our first main result reads:

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

p(x) =

p_o, 0 x b,

p₁, x> b, and β(x) =

βo, 0 x b,

β1, x> b. (3.1) Moreover, assume that po, p1∈ R\{0} are such that po 1, p1 1 or po 1, p1< 0 or po< 0, p1 1 or po< 0, p1< 0. If f : [a,l] → R is non-negative Δ-integrable and

f∈ Crd([a,l],R) for which

_b₁ a1 ...

_b_n an

f^p(x)(y1,...,yn)

n

∏

i=1

1 β(x)

(yi− ai)^{−β (x)}

×

1−

∏

ⁿ

i=1

y_i− ai

l_i− ai

_{β (x)}

Δy1...Δyn< ∞,

(3.2)

then

_l₁ a1

... ^lⁿ

an

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi) − ai)

_σ(x₁₎ a1

... ^σ(x¹⁾

a1

f(y1,...,yn)Δy1...Δyn

⎞

⎟⎟

⎠

p(x)

×

∏

ⁿ

i=1(σ(xi) − ai)^{−β (x)}Δx1...Δxn

_b₁ a1

...

_b_n an

f^p(x)(y1,...,yn)

n

∏

i=1

1 β(x)

(yi− ai)^{−β (x)}

×

1−

∏

ⁿ

i=1

y_i− ai

l_i− ai

_{β (x)}

Δy1...Δyn+ Io, (3.3)

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

I_o= ^b¹

a1

... ^bⁿ

an

f^p¹(y1,...,yn)

∏

ⁿ

i=1

1 β1

(yi− ai)^−β¹− (li− ai)^−β¹

Δy1...Δyn

− ^b^o

a1

... ^bⁿ

an

f^p^o(y1,...,yn)

∏

ⁿ

i=1

1 βo

(yi− ai)^−β^o− (li− ai)^−β^o

Δy1...Δyn. (3.4) If 0< p(x) 1, then (3.3) holds in the reverse direction.

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Proof. Let b l. By applying Jensen’s inequality (see Remark2.3), Lemma2.1 and Lemma2.4, we find that

_l₁ a₁ ...

_l_n an

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi) − ai)

_σ(x₁₎ a₁ ...

_σ(x₁₎

a₁ f(y1,...,yn)Δy1...Δyn

⎞

⎟⎟

⎠

p(x)

×

∏

ⁿ

i=1(σ(xi) − ai)^{−β (x)}Δx1...Δxn

^l¹

a1

... ^lⁿ

an

⎡

⎢⎢

⎣ 1

∏n

i=1(σ(xi) − ai)

_σ(x₁₎ a1

... ^σ(x¹⁾

a1

f^p^o(y1,...,yn)Δy1...Δyn

⎤

⎥⎥

⎦

×

∏

ⁿ

i=1(σ(xi) − ai)^−β^oΔx1...Δxn

_l₁ a₁ ...

_l_n an

f^p^o(y1,...,yn)

l1

y₁ ...

_l_n yn

∏

n

i=1(σ(xi) − ai)^−(β^o⁺¹⁾Δx1...Δxn

× Δy1...Δyn

^b¹

a1

... ^bⁿ

an

f^p(x)(y1,...,yn)

n

∏

i=1

1 β(x)

(yi− ai)^{−β (x)}

×

1−

∏

ⁿ

i=1

yi− ai

l_i− ai

_{β (x)}

Δy1...Δyn.

Hence, (3.3) is proved for this case.

Next, let b l. By applying Jensen’s inequality (see Remark2.3) and Lemma2.1, we find 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)

×

∏

ⁿ

i=1(σ(xi) − ai)^{−β (x}ⁱ⁾Δx1...Δxn

^b¹

a1

... ^bⁿ

an

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi) − ai)

_σ(x₁₎ a1

... ^σ(x¹⁾

a1

f(y1,...,yn)Δy1...Δyn

⎞

⎟⎟

⎠

po

×

∏

ⁿ

i=1(σ(xi) − ai)^−β^oΔx1...Δxn

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+ ^l¹

b1

... ^lⁿ

bn

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi) − ai)

_b₁ a1

... ^bⁿ

an

f(y1,...,yn)Δy1...Δyn

⎞

⎟⎟

⎠

p1

×

∏

ⁿ

i=1(σ(xi) − ai)^−β¹Δx1...Δxn

+ ^l¹

b1

... ^lⁿ

bn

⎛

⎜⎜

⎝ 1

∏n

i=1(σ(xi) − ai)

_σ(x₁₎ b1

... ^σ(xⁿ⁾

bn

f(y1,...,yn)Δy1...Δyn

⎞

⎟⎟

⎠

p1

×

∏

ⁿ

i=1(σ(xi) − ai)^−β¹Δx1...Δxn

^b¹

a1

... ^bⁿ

an

f^p^o(y1,...,yn)

b1

y1

... ^bⁿ

yn

∏

n

i=1(σ(xi) − ai)^−β^oΔx1...Δxn

Δy1...Δyn

+ ^b¹

a1

... ^bⁿ

an

f^p¹(y1,...,yn)

l1

b1

... ^lⁿ

bn

∏

n

i=1(σ(xi) − ai)^−β^oΔx1...Δxn

Δy1...Δyn

+

_l₁ b₁ ...

_l_n bn

f^p¹(y1,...,yn)

l1

y₁ ...

_l_n yn

∏

n

i=1(σ(xi) − ai)^−β^oΔx1...Δxn

Δy1...Δyn

:=I. (3.5)

By Lemma2.4and Lemma2.5, we find that

I ^b¹

a1

... ^bⁿ

an

f^p^o(y1,...,yn)

∏

ⁿ

i=1

1 βo

(yi− ai)^−β^o− (bi− ai)^−β^o

Δy1...Δyn

+ ^b¹

a1

... ^bⁿ

an

f^p¹(y1,...,yn)

∏

ⁿ

i=1

1 β1

(bi− ai)^−β¹− (li− ai)^−β¹

Δy1...Δyn

+ ^l¹

b1

... ^lⁿ

bn

f^p¹(y1,...,yn)

∏

ⁿ

i=1

1 β1

(yi− ai)^−β¹− (li− ai)^−β¹

Δy1...Δyn

^b¹

a1

... ^bⁿ

an

f^p^o(y1,...,yn)

n

∏

i=1

1 βo

(yi− ai)^−β^o

1−

∏

ⁿ

i=1

y_i− ai

l_i− ai

_β_o

· Δy1...Δyn

+ ^l¹

b1

... ^lⁿ

bn

f^p¹(y1,...,yn)

n

∏

i=1

1 β1

(yi− ai)^−β¹

1−

∏

ⁿ

i=1

yi− ai

l_i− ai

_β₁

· Δy1...Δyn

+

_b₁ a1

...

_b_n an

f^p¹(y1,...,yn)

∏

ⁿ

i=1

1 β1

(yi− ai)^−β¹− (li− ai)^−β¹

Δy1...Δyn

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−

_b_o a1 ...

_b_n an

f^p^o(y1,...,yn)

∏

ⁿ

i=1

1 βo

(yi− ai)^−β¹− (li− ai)^−β^o

Δy1...Δyn

=

_b₁ a1 ...

_b_n an

f^p(x)(y1,...,yn)

n

∏

i=1

1 β(x)

(yi− ai)^{−β (x)}

×

1−

∏

ⁿ

i=1

yi− ai

l_i− ai

_{β (x)}

Δy1...Δyn+ Io. (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(xi) − ai)

_σ(x₁₎ a1

... ^σ(x¹⁾

a1

f(y1,...,yn)Δy1...Δyn

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

∏

ⁿ

i=1(σ(xi) − ai) ^∞

σ(x1)... ^∞

σ(xn)

f(y1,...,yn)Δy1...Δyn

∏n

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

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

p(x) =

po, 0 x b,

p₁, x> b, ,β(x) =

βo, 0 x b,

β1, x> b. (3.7)

Moreover, assume that po, p1∈ R\{0} are such that po 1, p1 1 or po 1, p1< 0 or p_o< 0, p1 1 or po< 0, p1< 0. If f : [a,l] → R is non-negative Δ-integrable and

f∈ Crd([a,l],R) for which

_∞ l1

...

_∞ ln

f^p¹(y1,...,yn)

n

∏

i=1

(yi− ai)^{β (y)} β(y)

1−

∏

ⁿ

i=1

l_i− ai

yi− ai

_{β (y)}

× Δy1...Δyn

∏n

i=1(σ(yi) − ai)(yi− ai)< ∞, (3.8) then

_∞ l₁ ...

_∞ ln

⎛

⎜⎜

⎝

∏

n

i=1(σ(xi) − ai)

_∞ σ(x1)...

_∞ σ(xn)

⎛

⎜⎜

⎝ f(y1,...,yn)

∏n

i=1(σ(yi) − ai)(yi− ai)

⎞

⎟⎟

⎠Δy¹...Δyn

⎞

⎟⎟

⎠

p(x)

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×

n

∏

i=1(xi− ai)

_{β (x)−1}

Δx1...Δxn

∏n

i=1(σ(xi) − ai)

_∞ l1 ...

_∞ ln

f^p¹(y1,...,yn)

n

∏

i=1

(yi− ai)^{β (y)} β(y)

1−

∏

ⁿ

i=1

l_i− ai

y_i− ai

_{β (y)}

× Δy1...Δyn

∏n

i=1(σ(yi) − ai)(yi− ai)+ Io. (3.9)

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

I_o= ^∞

l1

... ^∞

ln

f^p¹(y1,...,yn)

∏n

∏

n i=1

1 β1

(yi− ai)^β¹− (li− ai)^β¹

Δy1...Δyn

− ^∞

l1

... ^∞

ln

f^p^o(y1,...,yn)

∏n

∏

n i=1

1 βo

(yi− ai)^β^o− (li− ai)^β^o

Δy1...Δyn.

(3.10) If 0< p(x) 1, then (3.9) holds in the reverse direction.

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

_∞ l1 ...

_∞ ln

⎛

⎜⎜

⎝

∏

n

i=1(σ(xi) − ai)

_∞ σ(x1)...

_∞ σ(xn)

⎛

⎜⎜

⎝ f(y1,...,yn)

∏n

⎞

⎟⎟

⎠Δy¹...Δyn

⎞

⎟⎟

⎠

p(x)

×

n

∏

i=1(xi− ai)

_{β (x)−1}

Δx1...Δxn

∏n

i=1(σ(xi) − ai)

^∞

l1

... ^∞

ln

_∞

σ(x1)... ^∞

σ(xn)

⎛

⎜⎜

⎝f^p¹(y1,...,yn)Δy1...Δyn

∏n

⎞

⎟⎟

⎠

×

n

∏

i=1(xi− ai)

_β₁₋₁

Δx1...Δxn

^∞

l1

... ^∞

ln

f^p¹(y1,...,yn)

∏n

⎡

⎣ ^y¹

l1

... ^yⁿ

ln

n

∏

i=1(xi− ai)

_β₁₋₁

Δx1...Δxn

⎤

⎦

(11)

× Δy1...Δyn

^∞

l1

... ^∞

ln

f^p¹(y1,...,yn)

n

∏

i=1

(yi− ai)^{β (y)} β(y)

1−

∏

ⁿ

i=1

l_i− ai

yi− ai

_{β (y)}

× Δy1...Δyn

∏n

i=1(σ(yi) − ai)(yi− ai). (3.11)

Finally, let b l. 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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[18] J. A. OGUNTUASE ANDL. E. PERSSON, Time scales Hardy-type inequalities via superquadraticity, Ann. Funct. Anal. 5 (2014), no.2, 61–73.

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

Department of Mathematics Obafemi Awolowo University Ile-Ife, Osun State, Nigeria e-mail:fabelur@yahoo.com J. A. Oguntuase Department of Mathematics Federal University of Agriculture P.M.B. 2240, Abeokuta, Ogun State, Nigeria e-mail:oguntuase@yahoo.com L.-E. Persson UiT, The Artic University of Norway Campus Narvik, Lodve Langes Gate 2, 8514 Narvik, Norway e-mail:larserik6pers@gmail.com

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