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Some Hardy's inequalities on conformable fractional calculus

  • Ghada AlNemer , Samir H. Saker , Gehad M. Ashry , Mohammed Zakarya , Haytham M. Rezk EMAIL logo and Mohammed R. Kenawy
Published/Copyright: September 26, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, we will demonstrate some Hardy’s inequalities by utilizing Hölder inequality, integration by parts, and chain rule of the conformable fractional calculus. When α = 1 , we can obtain some of classical Copson’s and Hardy’s inequalities. In the end, we will obtain an application of Hardy’s inequality utilizing the conformable fractional calculus.

Keywords: Hardy’s inequality; chain rule; conformable fractional calculus; Hölder’s inequality; Jensen’s inequality
MSC 2010: 26D10; 26D15; 34N05; 47B38; 26A33; 39A12

1 Introduction

In 1920, Hardy [1] proved that

(1) ∑ λ = 1 ∞ 1 λ ∑ j = 1 λ w ( j ) s ≤ s s − 1 s ∑ λ = 1 ∞ w s ( λ ) ,

for a positive sequence w ( λ ) for all λ ≥ 1 , s > 1 , and ( s ⁄ ( s − 1 ) ) s is the best constant we can obtain.

In 1925, Hardy [2] proved the integral inequality

(2) ∫ 0 ∞ 1 η ∫ 0 η q ( γ ) d γ s d η ≤ s s − 1 s ∫ 0 ∞ q s ( η ) d η , s > 1 ,

where q is an integrable positive function over ( 0 , η ) and q s is convergent and integrable on ( 0 , ∞ ) such that ( s ⁄ ( s − 1 ) ) s is the best constant we can obtain.

In 1927, Copson [3] proved that if ∫ η ∞ ( q ( γ ) ⁄ γ ) d γ converges for η > 0 , then

(3) ∫ 0 ∞ ∫ η ∞ q ( γ ) γ d γ s d η ≤ s s ∫ 0 ∞ q s ( η ) d η , s > 1 .

In 1928, Hardy [4] generalized inequality (2) and deduced that if s > 1 , q is an integrable and positive function on ( 0 , η ) such that q s is integrable and convergent over ( 0 , ∞ ) , then

(4) ∫ 0 ∞ η − ω ∫ 0 η q ( γ ) d γ s d η ≤ s ω − 1 s ∫ 0 ∞ η s − ω q s ( η ) d η , for ω > 1 ,

and

(5) ∫ 0 ∞ η − ω ∫ η ∞ q ( γ ) d γ s d η ≤ s 1 − ω s ∫ 0 ∞ η s − ω q s ( η ) d η , for 0 < ω ≤ 1 .

The constants ( s ⁄ ( ω − 1 ) ) s and ( s ⁄ ( 1 − ω ) ) s in (4) and (5) are the best constant we can obtain.

In 1964, Levinson [5] used Jensen’s inequality that proved in [6] to obtain an extension of Hardy’s inequality (2) as follows.

Let s > 1 , h ˆ ( η ) > 0 , ν ( η ) > 0 such that η > 0 , ϕ ( u ) be a positive convex real-valued function for u ∈ ( 0 , ∞ ) , Φ ( η ) = ∫ 0 η ν ( γ ) d γ and exists a constant £ > 0 such that

(6) s − 1 + ν ′ ( η ) Φ ( η ) ν 2 ( η ) ≥ s £ , for η > 0 .

Then

(7) ∫ 0 ∞ ϕ 1 Φ ( η ) ∫ 0 η ν ( γ ) h ˆ ( γ ) d γ d η ≤ £ s ∫ 0 ∞ ϕ ( h ˆ ( η ) ) d η .

In 1976, Copson [7] proved that if

ϒ ( γ ) = ∫ 0 γ ν ( η ) d η and Ψ ( γ ) = ∫ 0 γ ν ( η ) u ( η ) d η ,

then

(8) ∫ 0 ∞ ν ( γ ) ϒ h ( γ ) Ψ k ( γ ) d γ ≤ k h − 1 k ∫ 0 ∞ ν ( γ ) ϒ h − k ( γ ) u k ( γ ) d γ ,

for h > 1 , k ≥ 1 and if Ψ ¯ ( γ ) = ∫ γ ∞ ν ( η ) u ( η ) d η , then

(9) ∫ 0 ∞ ν ( γ ) ϒ h ( γ ) Ψ ¯ k ( γ ) d γ ≤ k 1 − h k ∫ 0 ∞ ν ( γ ) ϒ h − k ( γ ) u k ( γ ) d γ ,

for 0 ≤ h < 1 and k ≥ 1 .

In 1999, Yang and Hwang [8] extended Levinson inequality (7) and showed that, if

Ψ ( η ) = ∫ 0 η ν ( γ ) τ ( γ ) h ˆ ( γ ) d γ and Φ ( η ) = ∫ 0 η ν ( γ ) τ ( γ ) d γ ,

where the functions ν ( η ) , τ ( η ) , and h ˆ ( η ) are nonnegative, and there is £ > 0 be a constant such that

s − 1 + τ ′ ( η ) Φ ( η ) τ 2 ( η ) ν ( η ) ≥ s £ , for η > 0 and s > 1 ,

then

(10) ∫ 0 ∞ ν ( η ) Ψ ( η ) Φ ( η ) s d η ≤ £ s ∫ 0 ∞ ν ( η ) h ˆ s ( η ) d η .

Recently, many authors have made generalizations of fractional inequalities by using the conformable calculus like Chebyshev’s inequality [9], Hermite-Hadamard’s inequality [10–13], Opial’s inequality [14–16], and Hardy’s inequality [17–19].

In our article, we will deduce some fractional Hardy’s inequalities by using integration by parts, Jensen’s inequality, Hölder’s inequality, and chain rule.

The article is divided into to three sections, the first section includes fundamentals of fractional calculus, the second section includes our main results and the third section includes an application of Hardy’s inequality in conformable fractional calculus.

2 Preliminaries and basic lemmas

This section includes definitions and lemmas, which are fundamentals of conformable fractional calculus [20,21].

Definition 1

The conformable fractional derivative of order α for a function θ : [ 0 , ∞ ) → R is defined by

(11) D α θ ( v ) = lim ε → 0 θ ( v + ε v 1 − α ) − θ ( v ) ε , v > 0 and α ∈ ( 0 , 1 ] .

If lim v → 0 + θ ( v ) exists and θ is α -differentiable on ( 0 , a ) for a > 0 , then we can define D α θ ( 0 ) = lim v → 0 + D α θ ( v ) .

Definition 2

The conformable fractional integral of order α for a function θ : [ 0 , ∞ ) → R is defined by

(12) I α a θ ( v ) = ∫ a v θ ( γ ) d α γ = ∫ a v γ α − 1 θ ( γ ) d γ , for α ∈ ( 0 , 1 ] .

The following gives the fundamentals of α -derivative of functions.

Theorem 1

Assume that θ and Ω are α -differentiable, then for α ∈ ( 0 , 1 ] , we obtain

  1. For all a , b ∈ R , we have

    (13) D α ( a θ + b Ω ) ( v ) = a D α θ ( v ) + b D α Ω ( v ) .

  2. The product rule

    (14) D α ( θ Ω ) ( v ) = θ D α Ω ( v ) + Ω D α θ ( v ) .

  3. The quotient rule

    (15) D α θ Ω ( v ) = Ω D α θ ( v ) − θ D α Ω ( v ) Ω 2 .

  4. For a function θ , we have

    (16) D α θ ( v ) = v 1 − α d θ d v .

  5. If s ∈ R , then D α ( v s ) = s v s − α .

  6. If θ ( v ) = ξ be a constant function, then D α θ ( v ) = 0 .

Lemma 1

(Chain rule) Let Ω be α -differentiable with respect to v and θ be differentiable with respect to Ω . Then

(17) D α θ ( Ω ( v ) ) = Ω α − 1 ( v ) ⋅ ( D α θ ) ( Ω ( v ) ) ⋅ D α Ω ( v ) .

Lemma 2

(Integration by parts) Let θ and Ω be α -differentiable with respect to v on [ 0 , ∞ ) . Then

(18) ∫ 0 ∞ ( D α θ ( v ) ) Ω ( v ) d α v = θ ( v ) Ω ( v ) ∣ 0 ∞ − ∫ 0 ∞ θ ( v ) ( D α Ω ( v ) ) d α v .

Lemma 3

(Hölder inequality) Let 0 < α ≤ 1 and θ , Ω : [ 0 , ∞ ] → R . Then

(19) ∫ 0 ∞ ∣ θ ( v ) Ω ( v ) ∣ d α v ≤ ∫ 0 ∞ ∣ θ ( v ) ∣ λ d α v 1 λ ∫ 0 ∞ ∣ Ω ( v ) ∣ ε d α v 1 ε ,

for 1 λ + 1 ε = 1 and λ > 1 .

Lemma 4

(Jensen’s inequality) Let Ω ∈ C ( [ ω , ε ] , R ) be convex, ω , ε ∈ R , a ∈ [ 0 , ∞ ) , h ˆ ∈ C ( [ a , ∞ ) , ( ω , ε ) ) , and z ∈ C ( [ a , ∞ ) , R ) such that I α a ∣ k ( γ ) ∣ = ∫ a γ ∣ k ( v ) ∣ d α v = ∫ a γ v α − 1 ∣ k ( v ) ∣ d v > 0 . Then

(20) Ω I α a [ ∣ k ( γ ) ∣ h ˆ ( γ ) ] I α a ∣ k ( γ ) ∣ ≤ I α a [ ∣ k ( γ ) ∣ Ω ( h ˆ ( γ ) ) ] I α a ∣ k ( γ ) ∣ .

3 Main results

In this section, we will establish some formula of Yang and Hwang’s and Pachpatte’s inequalities, then we will prove some of Copson’s and Hardy’s inequalities from our results.

Theorem 2

Let ν , τ , h ˆ be nonnegative functions, τ be an increasing function on [ 0 , ∞ ) and 1 < δ ≤ s . Define

(21) Ψ ( σ ) = ∫ 0 σ ν ( γ ) τ ( γ ) h ˆ ( γ ) d α γ and Φ ( σ ) = ∫ 0 σ ν ( γ ) τ ( γ ) d α γ .

Let there is a positive constant K with

(22) δ − 1 + D α τ ( σ ) Φ ( σ ) ν ( σ ) τ 2 ( σ ) = s K , for σ ∈ [ 0 , ∞ ) .

Then

(23) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ≤ K s ∫ 0 ∞ Φ s − δ ( σ ) ν ( σ ) h ˆ s ( σ ) d α σ .

Proof

By using integration by parts (18) on the term

∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ,

with Ω ( σ ) = Ψ s ( σ ) τ ( σ ) and D α θ ( σ ) = ν ( σ ) τ ( σ ) Φ δ ( σ ) , then

(24) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = ∫ 0 ∞ ν ( σ ) τ ( σ ) Φ δ ( σ ) Ψ s ( σ ) τ ( σ ) d α σ = θ ( σ ) Ψ s ( σ ) τ ( σ ) 0 ∞ + ∫ 0 ∞ ( − θ ( σ ) ) D α Ψ s ( σ ) τ ( σ ) d α σ .

Since θ ( σ ) = − ∫ σ ∞ ν ( γ ) τ ( γ ) Φ δ ( γ ) d α γ , Ψ ( 0 ) = 0 , and θ ( ∞ ) = 0 , we obtain

(25) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = ∫ 0 ∞ ( − θ ( σ ) ) D α Ψ s ( σ ) τ ( σ ) d α σ .

Since D α Φ ( σ ) = ν ( σ ) τ ( σ ) > 0 , then by using chain rule (17), we obtain

D α Φ 1 − δ ( σ ) = Φ α − 1 ( σ ) ⋅ ( D α Φ 1 − δ ( σ ) ) ( Φ ( σ ) ) ⋅ D α Φ ( σ ) = Φ α − 1 ( σ ) ⋅ ( 1 − δ ) Φ 1 − δ − α ( σ ) ⋅ D α Φ ( σ ) = − ( δ − 1 ) Φ − δ ( σ ) D α Φ ( σ ) = − ( δ − 1 ) ν ( σ ) τ ( σ ) Φ − δ ( σ ) ,

which can be written as follows:

(26) ν ( σ ) τ ( σ ) Φ − δ ( σ ) = − 1 δ − 1 D α Φ 1 − δ ( σ ) .

Therefore,

(27) − θ ( σ ) = ∫ σ ∞ ν ( γ ) τ ( γ ) Φ δ ( γ ) d α γ = − 1 δ − 1 ∫ σ ∞ D α Φ 1 − δ ( γ ) d α γ = 1 δ − 1 Φ 1 − δ ( σ ) .

By combining (25) and (27), we have

(28) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = 1 δ − 1 ∫ 0 ∞ Φ 1 − δ ( σ ) D α Ψ s ( σ ) τ ( σ ) d α σ .

By using the quotient rule (15), we obtain

(29) D α Ψ s ( σ ) τ ( σ ) = τ ( σ ) D α Ψ s ( σ ) − Ψ s ( σ ) D α τ ( σ ) τ 2 ( σ ) .

By using the chain rule (17) once more, we obtain

(30) D α Ψ s ( σ ) = s Ψ s − α ( σ ) Ψ α − 1 ( σ ) D α Ψ ( σ ) = s Ψ s − 1 ( σ ) D α Ψ ( σ ) .

As D α Ψ ( σ ) = ν ( σ ) τ ( σ ) h ˆ ( σ ) ≥ 0 , then

(31) D α Ψ s ( σ ) = s ν ( σ ) τ ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) .

According to (29) and (31), we find

(32) D α Ψ s ( σ ) τ ( σ ) = s ν ( σ ) τ 2 ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) τ 2 ( σ ) − Ψ s ( σ ) D α τ ( σ ) τ 2 ( σ ) = s ν ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) − Ψ s ( σ ) D α τ ( σ ) τ 2 ( σ ) .

From (28) and (32), we obtain

∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = s δ − 1 ∫ 0 ∞ Φ 1 − δ ( σ ) ν ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) d α σ − 1 δ − 1 ∫ 0 ∞ Ψ s ( σ ) D α τ ( σ ) Φ δ − 1 ( σ ) τ 2 ( σ ) d α σ .

Therefore,

(33) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) δ − 1 + Φ ( σ ) D α τ ( σ ) ν ( σ ) τ 2 ( σ ) d α σ = s ∫ 0 ∞ Φ 1 − δ ( σ ) ν ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) d α σ .

By using (22) and Hölder’s inequality (19) with indices λ = s ⁄ ( s − 1 ) and ε = s , we see that

∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = K ∫ 0 ∞ ν ( σ ) Φ 1 − δ ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) d α σ = K ∫ 0 ∞ ν s − 1 s ( σ ) Φ − δ ( s − 1 ) s ( σ ) Ψ s − 1 ( σ ) Φ δ ( s − 1 ) s ( σ ) Φ 1 − δ ( σ ) h ˆ ( σ ) ν 1 s ( σ ) d α σ ≤ K ∫ 0 ∞ ν ( σ ) Φ − δ ( σ ) Ψ s ( σ ) d α σ s − 1 s ∫ 0 ∞ Φ s − δ ( σ ) h ˆ s ( σ ) ν ( σ ) d α σ 1 s .

Hence,

∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ≤ K s ∫ 0 ∞ Φ s − δ ( σ ) h ˆ s ( σ ) ν ( σ ) d α σ .

Corollary 1

In Theorem 2, if τ ( σ ) = 1 , then we obtain

∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ≤ s δ − 1 s ∫ 0 ∞ Φ s − δ ( σ ) h ˆ s ( σ ) ν ( σ ) d α σ ,

which is Copson’s inequality of Theorem 3 in [22]. Also, if α = 1 and ν ( σ ) = 1 , then we obtain

∫ 0 ∞ 1 σ δ Ψ s ( σ ) d σ ≤ s δ − 1 s ∫ 0 ∞ σ s − δ ( σ ) h ˆ s ( σ ) d σ ,

which is inequality of Hardy-Littlewood in Theorem 330 in [23] such that Φ ( ∞ ) < ∞ and δ > 1 .

Corollary 2

In Theorem 2, if α = 1 , then we obtain

(34) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d σ ≤ K s ∫ 0 ∞ Φ s − δ ( σ ) h ˆ s ( σ ) ν ( σ ) d σ ,

which is inequality (7) due to Levinson, where

δ − 1 + τ ′ ( σ ) Φ ( σ ) ν ( σ ) τ 2 ( σ ) = s K , for σ > 0

and

Ψ ( σ ) = ∫ 0 σ ν ( γ ) τ ( γ ) h ˆ ( γ ) d γ and Φ ( σ ) = ∫ 0 σ ν ( γ ) τ ( γ ) d γ .

Remark

In (34), if δ = s > 1 , then we have the inequality (10).

Remark

In (34), if τ ( σ ) = 1 , then we have inequality of Copson (8).

Remark

In (34), if ν ( σ ) = τ ( σ ) = 1 , then we have inequality of Hardy (4).

Remark

In (34), if ν ( σ ) = τ ( σ ) = 1 and δ = s , then we have the classical inequality of Hardy (2).

Theorem 3

Let 0 ≤ δ < 1 , s > 1 , τ ( σ ) be an increasing function on [ 0 , ∞ ) , and

(35) Ψ ( σ ) = ∫ σ ∞ ν ( γ ) τ ( γ ) h ˆ ( γ ) d α γ and Φ ( σ ) = ∫ 0 σ ν ( γ ) τ ( γ ) d α γ .

There is a positive constant K with

(36) 1 − δ − D α τ ( σ ) Φ ( σ ) ν ( σ ) τ 2 ( σ ) = s K , for σ ∈ [ 0 , ∞ ) .

Then

(37) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ≤ K s ∫ 0 ∞ Φ s − δ ( σ ) ν ( σ ) h ˆ s ( σ ) d α σ .

Proof

By using integration by parts (18) on the term

∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ,

such that θ ( σ ) = Ψ s ( σ ) τ ( σ ) and D α Ω ( σ ) = ν ( σ ) τ ( σ ) Φ δ ( σ ) , then

(38) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = ∫ 0 ∞ ν ( σ ) τ ( σ ) Φ δ ( σ ) Ψ s ( σ ) τ ( σ ) d α σ = Ω ( σ ) Ψ s ( σ ) τ ( σ ) 0 ∞ + ∫ 0 ∞ Ω ( σ ) − D α Ψ s ( σ ) τ ( σ ) d α σ .

Since Ω ( σ ) = ∫ 0 σ ν ( s ) τ ( s ) Φ δ ( s ) d α s , Ψ ( ∞ ) = 0 and Ω ( 0 ) = 0 , we obtain

(39) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = ∫ 0 ∞ Ω ( σ ) − D α Ψ s ( σ ) τ ( σ ) d α σ .

As D α Φ ( σ ) = ν ( σ ) τ ( σ ) > 0 , using the chain rule (17), we obtain

D α Φ 1 − δ ( σ ) = Φ α − 1 ( σ ) ⋅ ( D α Φ 1 − δ ( σ ) ) ( Φ ( σ ) ) ⋅ D α Φ ( σ ) = Φ α − 1 ( σ ) ⋅ ( 1 − δ ) Φ 1 − δ − α ( σ ) ⋅ D α Φ ( σ ) = ( 1 − δ ) ν ( σ ) τ ( σ ) Φ − δ ( σ ) ,

which can be written as follows:

(40) ν ( σ ) τ ( σ ) Φ − δ ( σ ) = 1 1 − δ D α Φ 1 − δ ( σ ) .

Therefore,

(41) Ω ( σ ) = ∫ 0 σ ν ( s ) τ ( s ) Φ δ ( s ) d α s = 1 1 − δ ∫ 0 σ D α ( Φ 1 − δ ( s ) ) d α s = 1 1 − δ Φ 1 − δ ( σ ) .

By combining (39) and (41), we have

(42) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = 1 1 − δ ∫ 0 ∞ Φ 1 − δ ( σ ) − D α Ψ s ( σ ) τ ( σ ) d α σ .

By using the quotient rule (15), we obtain

(43) − D α Ψ s ( σ ) τ ( σ ) = − τ ( σ ) D α Ψ s ( σ ) + Ψ s ( σ ) D α τ ( σ ) τ 2 ( σ ) .

By using the chain rule (17) once more, we obtain

− D α Ψ s ( σ ) = − s Ψ s − α ( σ ) Ψ α − 1 ( σ ) D α Ψ ( σ ) = − s Ψ s − 1 ( σ ) D α Ψ ( σ ) .

Since D α Ψ ( σ ) = − ν ( σ ) τ ( σ ) h ˆ ( σ ) ≤ 0 , then

(44) − D α Ψ s ( σ ) = s ν ( σ ) τ ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) .

By combining (43) and (44), we find

(45) − D α Ψ s ( σ ) τ ( σ ) = s ν ( σ ) τ 2 ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) τ 2 ( σ ) + Ψ s ( σ ) D α τ ( σ ) τ 2 ( σ ) = s ν ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) + Ψ s ( σ ) D α τ ( σ ) τ 2 ( σ ) .

From (42) and (45), we obtain

(46) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = s 1 − δ ∫ 0 ∞ Φ 1 − δ ( σ ) ν ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) d α σ + 1 1 − δ ∫ 0 ∞ Ψ s ( σ ) D α τ ( σ ) Φ δ − 1 ( σ ) τ 2 ( σ ) d α σ .

Then

(47) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) 1 − δ − Φ ( σ ) D α τ ( σ ) ν ( σ ) τ 2 ( σ ) d α σ = s ∫ 0 ∞ Φ 1 − δ ( σ ) ν ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) d α σ .

By employing (36) and Hölder’s inequality (19) with indices λ = s ⁄ ( s − 1 ) and ε = s , we see that

∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ = K ∫ 0 ∞ ν ( σ ) Φ 1 − δ ( σ ) h ˆ ( σ ) Ψ s − 1 ( σ ) d α σ = K ∫ 0 ∞ ν s − 1 s ( σ ) Φ − δ ( s − 1 ) s ( σ ) Ψ s − 1 ( σ ) Φ δ ( s − 1 ) s ( σ ) Φ 1 − δ ( σ ) h ˆ ( σ ) ν 1 s ( σ ) d α σ ≤ K ∫ 0 ∞ ν ( σ ) Φ − δ ( σ ) Ψ s ( σ ) d α σ s − 1 s ∫ 0 ∞ Φ s − δ ( σ ) h ˆ s ( σ ) ν ( σ ) d α σ 1 s .

Hence,

□ ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ≤ K s ∫ 0 ∞ ν ( σ ) h ˆ s ( σ ) Φ s − δ ( σ ) d α σ .

Corollary 3

In Theorem 3, if τ ( σ ) = 1 , then we obtain

(48) ∫ 0 ∞ ν ( σ ) Φ δ ( σ ) Ψ s ( σ ) d α σ ≤ s 1 − δ s ∫ 0 ∞ ν ( σ ) h ˆ s ( σ ) Φ s − δ ( σ ) d α σ ,

which is Copson’s inequality of Theorem 4 in [22].

Corollary 4

In Theorem 3, if α = 1 , then we obtain inequality of Copson (9), where

1 − δ − τ ′ ( σ ) Φ ( σ ) ν ( σ ) τ 2 ( σ ) = s K , σ ∈ [ 0 , ∞ ) ,

and

Ψ ( σ ) = ∫ σ ∞ ν ( γ ) τ ( γ ) h ˆ ( γ ) d γ and Φ ( σ ) = ∫ 0 σ ν ( γ ) τ ( γ ) d γ .

Theorem 4

Let s > 1 , f ( w ) be a positive nondecreasing convex function such that w > 0 ,

Φ ( σ ) = ∫ 0 σ ν ( γ ) d α γ , Ψ ( σ ) = ∫ 0 σ ν ( γ ) h ˆ ( γ ) d α γ ,

(49) Λ ( σ ) = ∫ 0 σ ν ( γ ) f ( h ˆ ( γ ) ) d α γ and Θ ( σ ) = Λ ( σ ) Φ ( σ ) .

Then

(50) ∫ 0 ∞ ν ( σ ) f s Ψ ( σ ) Φ ( σ ) d α σ ≤ s s − 1 ∫ 0 σ ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) d α σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) f s ( h ˆ ( σ ) ) d α σ .

Proof

By using (49), we can write

(51) ν ( σ ) f ( h ˆ ( σ ) ) = D α ( Θ ( σ ) Φ ( σ ) ) = Φ ( σ ) D α Θ ( σ ) + Θ ( σ ) D α Φ ( σ ) .

Then, we obtain

(52) ν ( σ ) Θ s ( σ ) − s s − 1 ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) = ν ( σ ) Θ s ( σ ) − s s − 1 Θ s − 1 ( σ ) [ Φ ( σ ) D α Θ ( σ ) + Θ ( σ ) D α Φ ( σ ) ] = ν ( σ ) Θ s ( σ ) − s s − 1 Θ s − 1 ( σ ) Φ ( σ ) D α Θ ( σ ) − s s − 1 Θ s ( σ ) D α Φ ( σ ) = ν ( σ ) Θ s ( σ ) − s s − 1 Θ s − 1 ( σ ) Φ ( σ ) D α Θ ( σ ) − s s − 1 Θ s ( σ ) ν ( σ ) = − 1 s − 1 Θ s ( σ ) ν ( σ ) − s s − 1 Θ s − 1 ( σ ) Φ ( σ ) D α Θ ( σ ) .

By using the quotient rule (15), we obtain

(53) D α Θ ( σ ) = D α Λ ( σ ) Φ ( σ ) = Φ ( σ ) D α Λ ( σ ) − Λ ( σ ) D α Φ ( σ ) Φ 2 ( σ ) = Φ ( σ ) ν ( σ ) f ( h ˆ ( σ ) ) − Λ ( σ ) ν ( σ ) Φ 2 ( σ ) = ν ( σ ) Φ 2 ( σ ) f ( h ˆ ( σ ) ) ∫ 0 σ ν ( γ ) d α γ − ∫ 0 σ ν ( γ ) f ( h ˆ ( γ ) ) d α γ .

Since f is nondecreasing function, then

∫ 0 σ ν ( γ ) f ( h ˆ ( γ ) ) d α γ ≤ f ( h ˆ ( σ ) ) ∫ 0 σ ν ( γ ) d α γ .

Therefore, D α Θ ( σ ) > 0 . According to chain rule (17), we obtain

D α Θ s ( σ ) = s Θ s − α ( σ ) Θ α − 1 ( σ ) D α Θ ( σ ) = s Θ s − 1 ( σ ) D α Θ ( σ ) ,

which can be written as follows:

(54) Θ s − 1 ( σ ) D α Θ ( σ ) = 1 s D α Θ s ( σ ) .

By combining (52) and (54), we obtain

(55) ν ( σ ) Θ s ( σ ) − s s − 1 ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) = − 1 s − 1 Θ s ( σ ) ν ( σ ) − 1 s − 1 Φ ( σ ) D α Θ s ( σ ) = − 1 s − 1 D α ( Φ ( σ ) Θ s ( σ ) ) .

By integrating both sides of (55) from 0 to z , we have

∫ 0 z ν ( σ ) Θ s ( σ ) d α σ − s s − 1 ∫ 0 z ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) d α σ = − 1 s − 1 Φ ( σ ) Θ s ( σ ) ≤ 0 .

Therefore,

∫ 0 z ν ( σ ) Θ s ( σ ) d α σ ≤ s s − 1 ∫ 0 z ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) d α σ = s s − 1 ∫ 0 z ν s − 1 s ( σ ) Θ s − 1 ( σ ) ν 1 s ( σ ) f ( h ˆ ( σ ) ) d α σ .

By using (49) and inequality of Hölder (19) with indices λ = s ⁄ ( s − 1 ) and ε = s , we obtain

∫ 0 z ν ( σ ) Θ s ( σ ) d α σ ≤ s s − 1 ∫ 0 z ν ( σ ) Θ s ( σ ) d α σ s − 1 s ∫ 0 z ν ( σ ) f s ( h ˆ ( σ ) ) d α σ 1 s .

Hence,

(56) ∫ 0 z ν ( σ ) Λ ( σ ) Φ ( σ ) s d α σ ≤ s s − 1 s ∫ 0 z ν ( σ ) f s ( h ˆ ( σ ) ) d α σ .

By using Jensen’s inequality (20) with replace Ω by f and k by ν , we obtain

(57) f Ψ ( σ ) Φ ( σ ) = f ∫ 0 σ ν ( γ ) h ˆ ( γ ) d α γ ∫ 0 σ ν ( γ ) d α γ ≤ ∫ 0 σ ν ( γ ) f ( h ˆ ( γ ) ) d α γ ∫ 0 σ ν ( γ ) d α γ = Λ ( σ ) Φ ( σ ) .

By substituting (57) into (56), we find

∫ 0 z ν ( σ ) f s Ψ ( σ ) Φ ( σ ) d α σ ≤ ∫ 0 z ν ( σ ) Λ ( σ ) Φ ( σ ) s d α σ ≤ s s − 1 s ∫ 0 z ν ( σ ) f s ( h ˆ ( σ ) ) d α σ .

By assuming that z → ∞ , then we have

□ ∫ 0 ∞ ν ( σ ) f s Ψ ( σ ) Φ ( σ ) d α σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) f s ( h ˆ ( σ ) ) d α σ .

Corollary 5

In Theorem 4, if f ( w ) = w , then we obtain

(58) ∫ 0 ∞ ν ( σ ) Ψ ( σ ) Φ ( σ ) s d α σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) h ˆ s ( σ ) d α σ .

If α = 1 , then we obtain

(59) ∫ 0 ∞ ν ( σ ) Ψ ( σ ) Φ ( σ ) s d σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) h ˆ s ( σ ) d σ ,

where

Φ ( σ ) = ∫ 0 σ ν ( γ ) d γ and Ψ ( σ ) = ∫ 0 σ ν ( γ ) h ˆ ( γ ) d γ ,

which is an extension of Hardy’s inequality.

Theorem 5

Let s > 1 , f ( w ) be a positive convex nonincreasing function such that w > 0 ,

Φ ( σ ) = ∫ σ ∞ ν ( γ ) d α γ , Ψ ( σ ) = ∫ σ ∞ ν ( γ ) h ˆ ( γ ) d α γ ,

(60) Λ ( σ ) = ∫ σ ∞ ν ( γ ) f ( h ˆ ( γ ) ) d α γ and Θ ( σ ) = Λ ( σ ) Φ ( σ ) .

Then

(61) ∫ 0 ∞ ν ( σ ) f s Ψ ( σ ) Φ ( σ ) d α σ ≤ s s − 1 ∫ 0 ∞ ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) d α σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) f s ( h ˆ ( σ ) ) d α σ .

Proof

From (60), we can write

(62) ν ( σ ) f ( h ˆ ( σ ) ) = − D α ( Θ ( σ ) Φ ( σ ) ) = − ( Θ ( σ ) D α Φ ( σ ) + Φ ( σ ) D α Θ ( σ ) ) = − ( − ν ( σ ) Θ ( σ ) + Φ ( σ ) D α Θ ( σ ) ) = ν ( σ ) Θ ( σ ) − Φ ( σ ) D α Θ ( σ ) .

Then, we obtain

(63) ν ( σ ) Θ s ( σ ) − s s − 1 ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) = ν ( σ ) Θ s ( σ ) − s s − 1 Θ s − 1 ( σ ) [ ν ( σ ) Θ ( σ ) − Φ ( σ ) D α Θ ( σ ) ] = ν ( σ ) Θ s ( σ ) − s s − 1 Θ s − 1 ( σ ) ν ( σ ) Θ ( σ ) + s s − 1 Θ s − 1 ( σ ) Φ ( σ ) D α Θ ( σ ) = ν ( σ ) Θ s ( σ ) − s s − 1 Θ s ( σ ) ν ( σ ) + s s − 1 Θ s − 1 ( σ ) Φ ( σ ) D α Θ ( σ ) = − 1 s − 1 ν ( σ ) Θ s ( σ ) + s s − 1 Θ s − 1 ( σ ) Φ ( σ ) D α Θ ( σ ) .

According to the quotient rule (15), we obtain

(64) D α Θ ( σ ) = D α Λ ( σ ) Φ ( σ ) = Φ ( σ ) D α Λ ( σ ) − Λ ( σ ) D α Φ ( σ ) Φ 2 ( σ ) = Φ ( σ ) ( − ν ( σ ) f ( h ˆ ( σ ) ) ) − Λ ( σ ) ( − ν ( σ ) ) Φ 2 ( σ ) = − ν ( σ ) Φ 2 ( σ ) f ( h ˆ ( σ ) ) ∫ σ ∞ ν ( γ ) d α γ − ∫ σ ∞ ν ( γ ) f ( h ˆ ( γ ) ) d α γ .

Since f is nonincreasing function, then

∫ σ ∞ ν ( γ ) f ( h ˆ ( γ ) ) d α γ ≤ f ( h ˆ ( σ ) ) ∫ σ ∞ ν ( γ ) d α γ .

Therefore, D α Θ ( σ ) < 0 . Using chain rule (17), we obtain

D α Θ s ( σ ) = s Θ s − α ( σ ) Θ α − 1 ( σ ) D α Θ ( σ ) = s Θ s − 1 ( σ ) D α Θ ( σ ) ,

which can be written as follows:

(65) Θ s − 1 ( σ ) D α Θ ( σ ) = 1 s D α Θ s ( σ ) .

By combining (63) and (65), we obtain

(66) ν ( σ ) Θ s ( σ ) − s s − 1 ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) = − 1 s − 1 ν ( σ ) Θ s ( σ ) + 1 s − 1 Φ ( σ ) D α Θ s ( σ ) = 1 s − 1 ( − ν ( σ ) Θ s ( σ ) + Φ ( σ ) D α Θ s ( σ ) ) = 1 s − 1 ( Θ s ( σ ) D α Φ ( σ ) + Φ ( σ ) D α Θ s ( σ ) ) = − 1 s − 1 D α ( Φ ( σ ) Θ s ( σ ) ) .

By integrating both sides of (66) from 0 to z , we have

∫ 0 z ν ( σ ) Θ s ( σ ) d α σ − s s − 1 ∫ 0 z ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) d α σ = 1 s − 1 [ Φ ( σ ) Θ s ( σ ) ∣ 0 z = 1 s − 1 ( Φ ( z ) Θ s ( z ) − Φ ( 0 ) Θ s ( 0 ) ) ≤ Φ ( 0 ) s − 1 ( Θ s ( z ) − Θ s ( 0 ) ) ≤ 0 ,

where D α Θ ( σ ) < 0 . Therefore,

∫ 0 z ν ( σ ) Θ s ( σ ) d α σ ≤ s s − 1 ∫ 0 z ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) d α σ = s s − 1 ∫ 0 z ν s − 1 s ( σ ) Θ s − 1 ( σ ) ν 1 s ( σ ) f ( h ˆ ( σ ) ) d α σ .

Assume that z → ∞ , then we have

∫ 0 ∞ ν ( σ ) Θ s ( σ ) d α σ ≤ s s − 1 ∫ 0 ∞ ν ( σ ) f ( h ˆ ( σ ) ) Θ s − 1 ( σ ) d α σ = s s − 1 ∫ 0 ∞ ν s − 1 s ( σ ) Θ s − 1 ( σ ) ν 1 s ( σ ) f ( h ˆ ( σ ) ) d α σ .

By using (60) and inequality of Hölder (19) with indices λ = s ⁄ ( s − 1 ) and ε = s , we obtain

∫ 0 ∞ ν ( σ ) Θ s ( σ ) d α σ ≤ s s − 1 ∫ 0 ∞ ν ( σ ) Θ s ( σ ) d α σ s − 1 s ∫ 0 ∞ ν ( σ ) f s ( h ˆ ( σ ) ) d α σ 1 s .

Hence,

(67) ∫ 0 ∞ ν ( σ ) Λ ( σ ) Φ ( σ ) s d α σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) f s ( h ˆ ( σ ) ) d α σ .

By using Jensen’s inequality (20) with replacing Ω by f and k by ν , we see that

(68) f Ψ ( σ ) Φ ( σ ) = f ∫ σ ∞ ν ( γ ) h ˆ ( γ ) d α γ ∫ σ ∞ ν ( γ ) d α γ ≤ ∫ σ ∞ ν ( γ ) f ( h ˆ ( γ ) ) d α γ ∫ σ ∞ ν ( γ ) d α γ = Λ ( σ ) Φ ( σ ) .

By substituting (68) into (67), we obtain

□ ∫ 0 ∞ ν ( σ ) f s Ψ ( σ ) Φ ( σ ) d α σ ≤ ∫ 0 ∞ ν ( σ ) Λ ( σ ) Φ ( σ ) s d α σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) f s ( h ˆ ( σ ) ) d α σ .

Corollary 6

In Theorem 5, if α = 1 , then we obtain the following Pachpatte’s inequality:

∫ 0 ∞ ν ( σ ) f s Ψ ( σ ) Φ ( σ ) d σ ≤ s s − 1 s ∫ 0 ∞ ν ( σ ) f s ( h ˆ ( σ ) ) d σ ,

where

Φ ( σ ) = ∫ σ ∞ ν ( γ ) d γ and Ψ ( σ ) = ∫ σ ∞ ν ( γ ) h ˆ ( γ ) d γ .

4 Application

First, fractional integral versions of Hardy inequalities in [24], and their new generalizations will be obtained in this section, which is considered as basic items for our main results of higher integrability.

Theorem 6

Let ω ∈ R , Λ : [ 0 , ∞ ) → R be a differentiable convex function and Φ : [ 0 , ω ] → R be a nonnegative nonincreasing function. Then, we obtain the following inequality:

(69) Λ ( 0 ) + ∫ 0 ω Λ ′ ( ρ Φ ( ρ ) ) Φ ( ρ ) d α ρ ≤ Λ ∫ 0 ω Φ ( ρ ) d α ρ .

Proof

Let ρ ∈ [ 0 , ω ] , where Φ is nonincreasing function. Then ρ Φ ( ρ ) ≤ ∫ 0 ρ Φ ( γ ) d α γ . By using the definition Ω ( ρ ) = ∫ 0 ρ Φ ( γ ) d α γ , we have D α Ω ( ρ ) = Φ ( ρ ) ≥ 0 , which leads to Ω , which is an increasing function. Since Λ is convex function, then we obtain that

(70) Λ ′ ( ρ Φ ( ρ ) ) Φ ( ρ ) ≤ Λ ′ ∫ 0 ρ Φ ( γ ) d α γ Φ ( ρ ) = Λ ′ ( Ω ( ρ ) ) Φ ( ρ ) .

By utilizing chain rule (17), we obtain

D α ( Λ ( Ω ( ρ ) ) ) = ( ( D α Λ ( Ω ( ρ ) ) ) ( Ω ( ρ ) ) ) ⋅ Ω α − 1 ( ρ ) ⋅ D α Ω ( ρ ) = Λ ′ ( Ω ( ρ ) ) D α Ω ( ρ ) ,

then

(71) D α ( Λ ( Ω ( ρ ) ) ) = Λ ′ ( Ω ( ρ ) ) D α Ω ( ρ ) = Λ ′ ( Ω ( ρ ) ) Φ ( ρ ) ,

where Ω be an increasing function. From (70) and (71), we find

(72) Λ ′ ( ρ Φ ( ρ ) ) Φ ( ρ ) ≤ D α Λ ( Ω ( ρ ) ) .

By integrating the inequality (72) from 0 to ω , we obtain

□ ∫ 0 ω Λ ′ ( ρ Φ ( ρ ) ) Φ ( ρ ) d α ρ ≤ ∫ 0 ω D α Λ ( Ω ( ρ ) ) d α ρ = Λ ( Ω ( ω ) ) − Λ ( Ω ( 0 ) ) = Λ ( Ω ( ω ) ) − Λ ( 0 ) .

Corollary 7

In Theorem 6, if Λ ( v ) = v s with s ≥ 1 , then we obtain

(73) ∫ 0 ω ρ s − 1 Φ s ( ρ ) d α ρ ≤ 1 s ∫ 0 ω Φ ( ρ ) d α ρ s .

Remark

By utilizing inequality of Hölder (19) on the right-hand side of (73) with indices λ = ( s − 1 ) ⁄ s and ε = 1 ⁄ s , we obtain that

∫ 0 ω ρ s − 1 Φ s ( ρ ) d α ρ ≤ 1 s ∫ 0 ω d α ρ ( s − 1 ) ⁄ s ∫ 0 ω Φ s ( ρ ) d α ρ 1 ⁄ s s = 1 s ∫ 0 ω d α ρ s − 1 ∫ 0 ω Φ s ( ρ ) d α ρ = 1 s ω α α s − 1 ∫ 0 ω Φ s ( ρ ) d α ρ .

Remark

In (73), if we replace Φ by ρ ξ − α Φ such that ξ < α ≤ 1 , then we obtain

(74) ∫ 0 ω ρ ( ξ − α + 1 ) s − 1 Φ s ( ρ ) d α ρ ≤ 1 s ∫ 0 ω ρ ξ − α Φ ( ρ ) d α ρ s .

Remark

In (74), if we replace Φ by Φ k and put s = ℓ ⁄ k ≥ 1 , ξ = 1 ⁄ s ≤ 1 such that k ≤ ℓ , where k , ℓ ∈ R + , then we obtain

(75) ∫ 0 ω ρ k ℓ ( 1 − α ) Φ ℓ ( ρ ) d α ρ k ℓ ≤ k ℓ ∫ 0 ω ρ k ℓ − α Φ k ( ρ ) d α ρ .

In the following, we aim to obtain an important formula of Hardy’s inequality, which is an extension of (2). First, we define a nonnegative function y : [ 0 , T ˇ ] → R and G is defined as follows:

(76) G ( ρ ) = 1 ρ ∫ 0 ρ γ 1 − α y ( γ ) d α γ , for γ ∈ [ 0 , T ˇ ] .

Lemma 5

Let ρ ∈ [ 0 , T ˇ ] and y : [ 0 , T ˇ ] → R be a nonnegative nonincreasing function. Then

G ( ρ ) ≤ y ( ρ ) .

Proof

Since y is nonincreasing, then we obtain

□ G ( ρ ) = 1 ρ ∫ 0 ρ γ 1 − α y ( γ ) d α γ ≤ 1 ρ ∫ 0 ρ γ 1 − α y ( ρ ) d α γ = 1 ρ y ( ρ ) ∫ 0 ρ γ 1 − α d α γ = 1 ρ y ( ρ ) ∫ 0 ρ d γ = y ( ρ ) .

Now, we will prove that G is an nonincreasing function using the definition of y .

Lemma 6

If y : [ 0 , T ˇ ] → R is a nonnegative nonincreasing function, then G is also nonincreasing.

Proof

According to the quotient rule (15), we obtain

D α G ( ρ ) = ρ D α ∫ 0 ρ γ 1 − α y ( γ ) d α γ − ∫ 0 ρ γ 1 − α y ( γ ) d α γ D α ρ ρ 2

= ρ 2 − α y ( ρ ) − ρ 1 − α ∫ 0 ρ γ 1 − α y ( γ ) d α γ ρ 2 = ρ 1 − α y ( ρ ) − ρ − α ∫ 0 ρ γ 1 − α y ( γ ) d α γ ρ , ρ ∈ [ 0 , T ˇ ] .

By using Lemma 5, we obtain

□ ρ D α G ( ρ ) = ρ 1 − α y ( ρ ) − ρ − α ∫ 0 ρ γ 1 − α y ( γ ) d α γ = ρ 1 − α ( y ( ρ ) − G ( ρ ) ) ≤ 0 , for ρ ∈ [ 0 , T ˇ ] .

Now, we will obtain a new generalization for inequality of Hardy on conformable fractional calculus which is an extension of (2). To prove this, we will use the following algebraic inequality:

(77) ( Ω + θ ) s ≥ Ω s + s Ω s − 1 θ , where s > 1 or s < 0 ,

which is considered as a variation of the Bernoulli inequality, and it is well defined for all Ω + θ ≥ 0 and Ω ≥ 0 , if s > 1 , or for Ω + θ > 0 and Ω > 0 , if s < 0 . The equality in (77) satisfied if θ = 0 only.

Theorem 7

Let ω ∈ R , 0 < α ≤ 1 , β > 1 , ζ < α ≤ 1 and y be a nonnegative nonincreasing function on [ 0 , ω ] . Then we obtain

(78) β β − ζ β ∫ 0 ω ρ ζ − α y β ( ρ ) d α ρ ≥ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ + β c ζ β − ζ G β ( ω ) ,

where G is defined as in (76).

Proof

Since G is nonincreasing on [ 0 , ω ] , from Lemma 6, and according to the chain rule (17), we obtain

D α ( G β ( ρ ) ) = β G β − α ( ρ ) G α − 1 ( ρ ) D α G ( ρ ) = β G β − 1 ( ρ ) D α G ( ρ ) .

Employing the product rule formula (14), and since ρ G ( ρ ) = ∫ 0 ρ γ 1 − α y ( γ ) d α γ , we obtain

D α ( ρ G ( ρ ) ) = ( D α ρ ) G ( ρ ) + ρ D α G ( ρ ) = ρ 1 − α G ( ρ ) + ρ D α G ( ρ ) = D α ∫ 0 ρ γ 1 − α y ( γ ) d α γ = ρ 1 − α y ( ρ ) .

Therefore,

ρ ζ D α G ( ρ ) = ρ ζ − 1 [ ρ 1 − α y ( ρ ) − ρ 1 − α G ( ρ ) ] = ρ ζ − α [ y ( ρ ) − G ( ρ ) ] .

By using integration by parts (18) on the term

∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ,

such that Ω ( ρ ) = ∫ 0 ρ γ ζ − α d α γ and θ ( ρ ) = G β ( ρ ) , then

∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ = [ Ω ( ρ ) G β ( ρ ) ] 0 ω − ∫ 0 ω Ω ( ρ ) D α ( G β ( ρ ) ) d α ρ = Ω ( ω ) G β ( ω ) − lim ρ → 0 + Ω ( ρ ) G β ( ρ ) − ∫ 0 ω Ω ( ρ ) D α ( G β ( ρ ) ) d α ρ

= ω ζ ζ G β ( ω ) − lim ρ → 0 + Ω ( ρ ) G β ( ρ ) − β ζ ∫ 0 ω ρ ζ ( D α G ( ρ ) ) G β − 1 ( ρ ) d α ρ = ω ζ ζ G β ( ω ) − lim ρ → 0 + Ω ( ρ ) G β ( ρ ) − β ζ ∫ 0 ω ρ ζ − α [ y ( ρ ) − G ( ρ ) ] G β − 1 ( ρ ) d α ρ = ω ζ ζ G β ( ω ) − β ζ ∫ 0 ω ρ ζ − α y ( ρ ) G β − 1 ( ρ ) d α ρ + β ζ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ − lim ρ → 0 + Ω ( ρ ) G β ( ρ ) ,

where

Ω ( ρ ) = ∫ 0 ρ γ ζ − α d α γ = ∫ 0 ρ γ ζ − 1 d γ = γ ζ ζ 0 ρ = ρ ζ ζ .

Also, since

Ω ( ρ ) G β ( ρ ) = ∫ 0 ρ γ ζ − α d α γ 1 ρ ∫ 0 ρ γ 1 − α y ( γ ) d α γ β = 1 ρ β ρ ζ ζ ∫ 0 ρ γ 1 − α y ( γ ) d α γ β ≤ y β ( 0 ) 1 ρ β ρ ζ ζ ∫ 0 ρ γ 1 − α d α γ β = y β ( 0 ) 1 ρ β ρ ζ ζ ρ β = y β ( 0 ) ρ ζ ζ .

Since ζ < α ≤ 1 , then lim ρ → 0 + Ω ( ρ ) G β ( ρ ) = 0 . Hence,

(79) β − ζ ζ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ + ω ζ ζ G β ( ω ) = β ζ ∫ 0 ω ρ ζ − α y ( ρ ) G β − 1 ( ρ ) d α ρ .

By using inequality of Hölder (19) on right hand side of inequality (79) with indices λ = β and ε = β ⁄ ( β − 1 ) , then we obtain

β − ζ ζ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ + ω ζ ζ G β ( ω ) ≤ β ζ ∫ 0 ω ρ ζ − α y β ( ρ ) d α ρ 1 ⁄ β ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β ,

which can be rewritten in the following form:

β ζ β ∫ 0 ω ρ ζ − α y β ( ρ ) d α ρ ≥ β − ζ ζ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β + ω ζ ζ G β ( ω ) ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β β = β − ζ ζ β ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ 1 ⁄ β + ω ζ β − ζ G β ( ω ) ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β β .

Hence,

β β − ζ β ∫ 0 ω ρ ζ − α y β ( ρ ) d α ρ ≥ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ 1 ⁄ β + ω ζ β − ζ G β ( ω ) ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β β .

By employing Bernoulli inequality (77), with

Ω = ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ 1 ⁄ β and θ = ω ζ β − ζ G β ( ω ) ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β ,

then, we obtain that

□ β β − ζ β ∫ 0 ω ρ ζ − α y β ( ρ ) d α ρ ≥ ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ + β ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β ω ζ β − ζ G β ( ω ) ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ ( β − 1 ) ⁄ β = ∫ 0 ω ρ ζ − α G β ( ρ ) d α ρ + β ω ζ β − ζ G β ( ω ) .

Corollary 8

In Theorem 7, if ζ = z ⁄ s and β = z such that s ≥ z > 1 , then for G previously defined in (76), we obtain

(80) ∫ 0 ω ρ z s − α G z ( ρ ) d α ρ + z ω z s z − z s G z ( ω ) ≤ z z − z s z ∫ 0 ω ρ z s − α y z ( ρ ) d α ρ .

Remark

In Corollary 8, if α = 1 , s = z , and G ( ρ ) = 1 ρ ∫ 0 ρ y ( γ ) d γ , then we have an extension of the classical inequality of Hardy (2) to a class of decreasing functions.

Now, we will state some lemmas with their proofs, which are important for obtaining the higher integrability theorem.

Lemma 7

Let a, ω ∈ R , and τ , φ be nonnegative functions defined on [ a , ω ] . Then we have

∫ a ω τ ( γ ) ∫ γ ω φ ( ρ ) d α ρ d α γ = ∫ a ω φ ( γ ) ∫ a γ τ ( ρ ) d α ρ d α γ .

Proof

Let ϒ ( γ ) = ∫ γ ω φ ( ρ ) d α ρ and utilizing integration by parts (18) with Ω ( γ ) = ϒ ( γ ) and D α θ ( γ ) = τ ( γ ) . Then we have

∫ a ω τ ( γ ) ∫ γ ω φ ( ρ ) d α ρ d α γ = ϒ ( γ ) θ ( γ ) ∣ a ω − ∫ a ω θ ( γ ) D α ∫ γ ω φ ( ρ ) d α ρ d α γ .

Since θ ( γ ) = ∫ a γ τ ( ρ ) d α ρ , θ ( a ) = 0 , and ϒ ( ω ) = 0 , then

□ ∫ a ω τ ( γ ) ∫ γ ω φ ( ρ ) d α ρ d α γ = ∫ a ω θ ( γ ) − D α ∫ γ ω φ ( ρ ) d α ρ d α γ = ∫ a ω φ ( γ ) ∫ a γ τ ( ρ ) d α ρ d α γ .

Lemma 8

Let a , ω ∈ R and q be a nonnegative increasing function on [ a , ω ] , then for H ( ρ ) = 1 q ( ρ ) ∫ a ρ τ ( γ ) d α γ and μ < 0 , we obtain

(81) ∫ a ω q μ ( ρ ) ( D α q ( ρ ) ) H ( ρ ) d α ρ = 1 μ q μ ( ω ) ∫ a ω τ ( ρ ) d α ρ − ∫ a ω τ ( ρ ) q μ ( ρ ) d α ρ .

Proof

Since H ( ρ ) = 1 h ˆ ( ρ ) ∫ a ρ τ ( γ ) d α γ , we find

(82) ∫ a ω q μ ( ρ ) ( D α q ( ρ ) ) H ( ρ ) d α ρ = ∫ a ω q μ ( ρ ) ( D α q ( ρ ) ) 1 q ( ρ ) ∫ a ρ τ ( γ ) d α γ d α ρ = ∫ a ω q μ − 1 ( ρ ) ( D α q ( ρ ) ) ∫ a ρ τ ( γ ) d α γ d α ρ = ∫ a ω τ ( ρ ) ∫ ρ ω q μ − 1 ( γ ) ( D α q ( γ ) ) d α γ d α ρ .

According to chain rule (17), we have

(83) D α ( q μ ( γ ) ) = μ q μ − α ( γ ) ⋅ q α − 1 ( γ ) ⋅ D α q ( γ ) = μ q μ − 1 ( γ ) D α q ( γ ) .

From (82) and (83), by using Lemma 7, we see that

□ ∫ a ω q μ ( ρ ) ( D α q ( ρ ) ) H ( ρ ) d α ρ = ∫ a ω τ ( ρ ) 1 μ ∫ ρ ω D α ( q μ ( γ ) ) d α γ d α ρ = 1 μ ∫ a ω τ ( ρ ) [ q μ ( ω ) − q μ ( ρ ) ] d α ρ = 1 μ q μ ( ω ) ∫ a ω τ ( ρ ) d α ρ − ∫ a ω τ ( ρ ) q μ ( ρ ) d α ρ .

Lemma 9

Let s , z ∈ R − { 0 } with z > s > 0 or z < 0 < ω , ℵ be a function, which is defined on [ 0 , 1 ] as follows:

(84) ℵ ( s , z , ρ ) = 1 − K z ( 1 − ρ ) z z − s ρ z s , for K > 1 .

Then the equation

(85) ℵ ( s , z , ρ ) = 0 ,

has just only one solution ρ z and ℵ ( s , z , ρ ) > 0 if and only if ρ ∈ ( ρ z , 1 ] .

Proof

For the following auxiliary function:

q ( ρ ) = ( 1 − ρ ) z z − s ρ z s .

which has the range [ 0 , 1 ] . Now, since q is a nonincreasing in [ 0 , 1 ] , then ρ z = q − 1 ( K − z ) be a unique solution of equation q ( ρ ) = K − z , which is equation of (85) such that K − z ∈ [ 0 , 1 ] . Also, as q is a nonincreasing function,

□ ℵ ( s , z , ρ ) > 0 ⇔ q ( ρ ) < K − z ⇔ ρ > ρ z .

Theorem 8

Let a , ω ∈ R such that a < ω , Φ , and ϒ be two positive functions defined on [ a , ω ] with 0 < b ≤ Φ s ⁄ ϒ z ≤ B < ∞ . Then we have

(86) ∫ a ω Φ ( ρ ) d α ρ 1 s ∫ a ω ϒ ( ρ ) d α ρ 1 z ≤ B b 1 s z ∫ a ω Φ 1 s ( ρ ) ϒ 1 z ( ρ ) d α ρ ,

for all s > 1 and z > 1 such that 1 s + 1 z = 1 .

Proof

According to condition Φ s ⁄ ϒ z ≤ B , we obtain ϒ ≥ B − 1 z Φ s z , then

Φ ϒ ≥ B − 1 z Φ s z + 1 = B − 1 z Φ s + z z = B − 1 z Φ s .

Hence, we obtain

(87) ∫ a ω Φ s ( ρ ) d α ρ 1 s ≤ B 1 s z ∫ a ω Φ ( ρ ) ϒ ( ρ ) d α ρ 1 s .

Similarly, according to condition b ≤ Φ s ⁄ ϒ z , we obtain Φ ≥ b 1 s ϒ z s , then

∫ a ω Φ ( ρ ) ϒ ( ρ ) d α ρ ≥ b 1 s ∫ a ω ϒ z s + 1 ( ρ ) d α ρ = b 1 s ∫ a ω ϒ z ( ρ ) d α ρ .

Hence,

(88) ∫ a ω Φ ( ρ ) ϒ ( ρ ) d α ρ 1 z ≥ b 1 s z ∫ a ω ϒ z ( ρ ) d α ρ 1 z .

From (87) and (88), we obtain

(89) ∫ a ω Φ s ( ρ ) d α ρ 1 s ∫ a ω ϒ z ( ρ ) d α ρ 1 z ≤ B b 1 s z ∫ a ω Φ ( ρ ) ϒ ( ρ ) d α ρ .

By replacing Φ s and ϒ z by Φ and ϒ , we obtain

□ ∫ a ω Φ ( ρ ) d α ρ 1 s ∫ a ω ϒ ( ρ ) d α ρ 1 z ≤ B b 1 s z ∫ a ω Φ 1 s ( ρ ) ϒ 1 z ( ρ ) d α ρ .

Theorem 9

Let a , ω ∈ R such that a < ω , 0 < γ < 1 and two positive functions Φ and ϒ be defined on [ a , ω ] with 0 < b ≤ Φ s ≤ B < ∞ . Then we have

(90) 1 ρ ∫ a ω Φ s ( ρ ) d α ρ 1 s ≤ ω α − a α α − 1 β z b B − 1 β s z 1 ρ ∫ a ω Φ γ ( ρ ) d α ρ 1 γ ,

for s > 1 , z > 1 , s > z , and β = 1 − 1 s 1 γ − 1 s .

Proof

Assume ϒ = 1 in (89), then we have

∫ a ω Φ s ( ρ ) d α ρ 1 s ω α − a α α 1 z ≤ B b 1 s z ∫ a ω Φ ( ρ ) d α ρ = b B − 1 s z ∫ a ω Φ ( ρ ) d α ρ ,

which can be written as follows:

(91) ∫ a ω Φ s ( ρ ) d α ρ 1 s ≤ ω α − a α α − 1 z b B − 1 s z ∫ a ω Φ ( ρ ) d α ρ .

Since β ∈ ( 0 , 1 ) and 1 − β s + β γ = 1 , then by using inequality of Hölder (19) with indices γ = s ⁄ ( 1 − β ) and ε = γ ⁄ β , then we obtain

1 ω ∫ a ω Φ s ( ρ ) d α ρ 1 s ≤ ω α − a α α − 1 z b B − 1 s z 1 ω ∫ a ω Φ ( ρ ) d α ρ .

= ω α − a α α − 1 z b B − 1 s z 1 ω ∫ a ω Φ 1 − β ( ρ ) Φ β ( ρ ) d α ρ ≤ ω α − a α α − 1 z b B − 1 s z 1 ω ∫ a ω Φ s ( ρ ) d α ρ 1 − β s 1 ω ∫ a ω Φ γ ( ρ ) d α ρ β γ .

Hence,

□ 1 ω ∫ a ω Φ s ( ρ ) d α ρ 1 s ≤ ω α − a α α − 1 β z b B − 1 β s z 1 ω ∫ a ω Φ γ ( ρ ) d α ρ 1 γ .

Notice that the inequality (90) of Theorem 9 can be written as follows:

(92) 1 ω ∫ a ω Φ s ( ρ ) d α ρ 1 s ≤ K ˜ 1 ω ∫ a ω Φ γ ( ρ ) d α ρ 1 γ ,

which is called the reverse Hölder’s inequality.

For our main theorem, we will need to prove the next theorem.

Theorem 10

Let ω ∈ R , 0 < α ≤ 1 , 0 < s < z , K ˜ > 1 , Φ : [ 0 , ω ] → R be a nonnegative decreasing satisfying (92)and

(93) ℵ ˜ ( s , z , ζ , K ˜ ) = 1 − ( 1 − ζ ) K ˜ z z z − s ζ z ⁄ s .

Then, for all ζ ∈ ( ζ z , 1 ] such that ζ z is the only root of the equation (84), we obtain the inequality

(94) ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ ≤ ω ζ − 1 ℵ ˜ ( s , z , ζ , K ˜ ) ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ .

Proof

Since the inequality (94) is clearly satisfied when ζ = 1 , we can now assume ζ ∈ ( ζ z , 1 ) . In this case, we define a function Φ : [ 0 , ω ] → R , that satisfies the reverse Hölder’s inequality (92). Therefore, we obtain the relation

(95) G z ( ρ ) ≤ K ˜ z ( G s ( ρ ) ) z s ,

where

G s ( ρ ) = 1 ρ ∫ 0 ρ γ 1 − α Φ s ( γ ) d α γ and G z ( ρ ) = 1 ρ ∫ 0 ρ γ 1 − α Φ z ( γ ) d α γ .

Then by integrating (95) from 0 to ω , we have

(96) ∫ 0 ω ρ ζ − α G z ( ρ ) d α ρ ≤ K ˜ z ∫ 0 ω ρ ζ − α ( G s ( ρ ) ) z ⁄ s d α ρ .

Applying Lemma 8 with μ = ζ − 1 , a = 0 , h ˆ ( ρ ) = ρ , and τ ( ρ ) = ρ 1 − α Φ z ( ρ ) on left-hand side of (96), it follows that

(97) ∫ 0 ω ρ z ζ − α G ( ρ ) d α ρ = 1 ζ − 1 ω ζ − 1 ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ − ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ .

By using Theorem 7 with β = z ⁄ s and y = Φ s to the right-hand side of inequality (96), we have

(98) ∫ 0 ω ρ ζ − α ( G s ( ρ ) ) z ⁄ s d α ρ ≤ z ⁄ s ( z ⁄ s ) − ζ z ⁄ s ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ − ( z ⁄ s ) ω ζ ( z ⁄ s ) − ζ G z ⁄ s ( ω )

≤ z ⁄ s ( z ⁄ s ) − ζ z ⁄ s ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ .

By substituting (98) and (97) into (96), we find

1 ζ − 1 ω ζ − 1 ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ − ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ ≤ K ˜ z z ⁄ s ( z ⁄ s ) − ζ z ⁄ s ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ .

Therefore,

ω ζ − 1 ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ ≥ ( ζ − 1 ) K ˜ z z z − s ζ z ⁄ s ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ + ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ = 1 − ( 1 − ζ ) K ˜ z z z − s ζ z ⁄ s ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ = ℵ ˜ ( s , z , ζ , K ˜ ) ∫ 0 ω ρ ζ − α Φ z ( ρ ) d α ρ .

Finally, from Lemma 9, there exists a unique ζ z ∈ ( 0 , 1 ) such that ℵ ˜ ( s , z , ζ , K ˜ ) > 0 for ζ ∈ ( ζ z , 1 ] . This provides the inequality (94).□

Finally, we are able to state and prove the higher integrability theorem for decreasing functions on conformable calculus.

Theorem 11

Assume that K ˜ > 1 , 0 < s < z , ω ∈ R , and Φ : [ 0 , ω ] → R is a nonnegative decreasing function satisfying (92) such that Φ ∈ L z [ 0 , ω ] . Then Φ ∈ L ℓ [ 0 , ω ] with z ≤ ℓ < z 0 and z 0 is the only solution to equation

(99) ρ ρ − z 1 ⁄ z = K ˜ ρ ρ − s 1 ⁄ s ,

and the following inequality holds

(100) 1 ω ∫ 0 ω ρ z ℓ ( 1 − α ) Φ ℓ ( ρ ) d α ρ z ℓ ≤ z ℓ ℵ ˜ ( s , z , z ⁄ ℓ , K ˜ ) 1 ω ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ ,

where

K ˜ = ω α − a α α − 1 β z b B − 1 β s z .

Proof

According to Theorem 10 with replacing ζ and ζ z by z ⁄ ℓ and z ⁄ z 0 , respectively, we obtain

(101) ∫ 0 ω ρ z ℓ − α Φ z ( ρ ) d α ρ ≤ ω z ℓ − 1 ℵ ˜ ( s , z , z ⁄ ℓ , K ˜ ) ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ , z ≤ ℓ < z 0 .

By using the inequality (75) with k = z to the left-hand side of inequality (101), we find

∫ 0 ω ρ z ℓ ( 1 − α ) Φ ℓ ( ρ ) d α ρ z ℓ ≤ z ω z ℓ − 1 ℓ ℵ ˜ ( s , z , z ⁄ ℓ , K ˜ ) ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ , z ≤ ℓ < z 0 .

Therefore,

1 ω ∫ 0 ω ρ z ℓ ( 1 − α ) Φ ℓ ( ρ ) d α ρ z ℓ ≤ z ℓ ℵ ˜ ( s , z , z ⁄ ℓ , K ˜ ) 1 ω ∫ 0 ω ρ 1 − α Φ z ( ρ ) d α ρ ,

that is the wanted inequality (100). Therefore, z 0 is the only solution to equation

ℵ ˜ ( s , z , z ⁄ ℓ , K ˜ ) = 0 ,

which reduces to (99).□

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through large Research Project under grant number RGP 2/190/45. Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R45), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  1. Funding information: This work was funded by the King Khalid University, grant number RGP 2/190/45 and Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R45).

  2. Author contributions: M.R.K. and G.M.A.: investigation, software, and writing–original draft; S.H.S.: supervision; H.M.R., G.N., and M.Z: writing-review editing and funding. All authors have read and agreed to the published version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2023-06-18
Revised: 2024-01-01
Accepted: 2024-05-17
Published Online: 2024-09-26

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2024-0027
Keywords for this article
Hardy’s inequality; chain rule; conformable fractional calculus; Hölder’s inequality; Jensen’s inequality
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  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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