Mathematical Problems in Engineering

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Volume 2021 |Article ID 9939468 | https://doi.org/10.1155/2021/9939468

Usama Hanif, Ammara Nosheen, Rabia Bibi, Khuram Ali Khan, Hamid Reza Moradi, "Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals", Mathematical Problems in Engineering, vol. 2021, Article ID 9939468, 14 pages, 2021. https://doi.org/10.1155/2021/9939468

Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

Academic Editor: Erhan Set
Received21 Mar 2021
Accepted06 May 2021
Published28 May 2021

Abstract

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

1. Introduction

The study of Hardy inequalities has gained huge attention in the literature, and now it became a major field in applied and pure mathematics. The Hardy inequality has a long history and many variants. Together with the Sobolev inequalities, it is one of the most frequently used inequalities in the analysis. Firstly, Hardy inequality was discovered to simplify the proof of another inequality. It was then studied in its own right and acquired several useful variants, and it eventually turned out to be extremely useful in the theory of partial differential equations. Except for the direct application of Hardy’s inequality to the Schrodinger operator, other useful variants have been successfully developed for applications in other areas of physics [1].

Hardy-type inequalities have crucial importance in the study of function spaces, especially of fractional regularity [2]. Another fundamental consequence is the trace theory of weighted Sobolev spaces; in turn, weighted Sobolev spaces are useful in the regularity theory of the superposition operators [2]. The familiar Hardy inequality (both in the discrete and continuous settings) as presented in [3] has been enormously studied and used as a model for the inquisition of more general integral inequalities [4, 5].

In [6], Hardy proved the discrete inequality,where for . Also, in [7], Hardy proved the continuous inequality, using the calculus of variation, which states that for nonnegative, integrable function over any finite interval and is integrable for ,where the constant is best possible.

Hardy inequality (2) has been generalized by G. H. Hardy himself in [8]. There he showed that for any integral function on ,

Littlewood and Hardy [9] established the discrete version of (3) and (4).

Since the discovery of these two inequalities, various papers which deal with new proofs, generalizations, and extensions have appeared in the literature, [6, 7, 1012] are referred to readers. During the last decades, these inequalities were extended, a time scale version of weighted Hardy-type inequality is proved in [11], and some preliminary dynamic inequalities [13] are proved in time scales calculus. In [14, 15], some classical inequalities are proved for isotonic linear functionals and superquadratic functions.

Recently, many new Hardy inequalities have been proved, including Hardy and Rellich inequalities for Bessel pairs [16], Hardy–Sobolev inequalities on hypersurfaces for Euclidean space [17], improved Hardy inequalities with exact remainder terms [18], Hardy inequalities with double singular weights [19], and Hardy inequalities for class functions in one-dimensional fractional Orlicz–Sobolev spaces [20]. A new approach for the fractional integral operator in time scales with variable exponent Lebesgue spaces is presented in [21] and -dimensional integral-type inequalities via time scale calculus is studied in [22].

2. Preliminaries

2.1. Some Basics on Time Scales

A time scale is a closed subset of the real line , and its common notation is . A time scale may or may not be connected. Forward jump operator and backward jump operator are respectively defined as

In general, and . The forward and backward graininess functions are, respectively, defined by

If , then is said to be right scattered. If , then is said to be left scattered. Points that are left-scattered and right scattered at the same time are called isolated. If and , then is called the right-dense and if and , then is called left-dense. Points that are left-dense and right-dense at the same time are called dense.Rd-continuous function: a function is called rd-continuous if it is continuous at right-dense points in and its left-sided limits exist at left-dense points in .Notation: if is a time scale, then the set is defined asDelta derivatives: take a function . The delta derivative (Hilger derivative) exists iff, for every , there exists a neighborhood of such thatfor all , .

2.2. Some Principles of Fractional Calculus
2.2.1. Riemann–Liouville Derivative

Riemann–Liouville derivative is the generalization of the usual derivative. It is based on Cauchy’s formula for calculating iterated integrals, if the first integral of a function, which must be equal to deriving it to , is as follows:

The calculation of the second can be simplified by interchanging the integration order,

This method can be applied repeatedly, resulting in the following formula for calculating iterated integrals,

Now, this can be easily generalized to noninteger values, in what is the Riemann–Liouville derivative [23],

The Riemann–Liouville derivative with the lower integration limit would be [23]

2.2.2. Riemann–Liouville Integral

The Riemann–Liouville integral relates with a real function , another same kind of function for each value of the parameter . The integral is a generalization of the repeated antiderivative of of order real number .

Let be a finite interval of . The left and right Riemann–Liouville fractional integrals and of order are defined in [23] byrespectively, provided the right-hand sides are pointwise defined on . When , definitions (14) and (15) coincide with the nth integrals of the form,

2.3. Some Principles of Delta Fractional Calculus

For , one has that (see [24], . 256)where is the Gamma function. If time scale is considered to be , then the coordinate-wise rd-continuous functions are given bysuch that and

Furthermore, for , it is assumed that

In the case and for , one obtainsand alsosatisfying (19). Furthermore, it is observed that for and by using (17), one getswhich is satisfying (20).

If and ([25], Chap: 1), then

2.3.1. Riemann–Liouville Delta Fractional Integral

For , the -Riemann–Liouville type fractional integral is defined in [26] asfor and . Notice that , where (Lebesgue -integrable functions on , [27]).

2.3.2. Fubini’s Theorem on Time Scale

Consider and . Then,

It can be found in [28].

2.4. Superquadratic Function

Definition 1. A function is called superquadratic 29 provided that, for all , there exists a constant such thatfor all . We say that is subquadratic if is a superquadratic function.

The following lemma (see [29]) shows that every positive superquadratic function is also a convex function.

Lemma 1. Let be a superquadratic function with as in Definition 1. Then,(i).(ii)If , then , where is differentiable at .(iii)If , then is convex and .

3. Jensen Inequality for Superquadratic Functions via Delta Fractional Integrals

Theorem 1. Let where and let . If with and is rd-continuous, then for every continuous superquadratic function ,where

Proof. Let be a superquadratic function and . According to (27), there is a constant such thatSince is rd-continuous,is well defined. The function is also rd-continuous, so we may apply (30) with and (31) to obtainIntegrating from to , we get

Remark 1. Since is nonnegative superquadratic function, therefore (by Lemma 1) a convex one too, the result of (28) refines the result given in [30] (Theorem 1) and by assuming , (28) becomes [31] (Theorem 2.5), and result of [31] (Theorem 2.5) refines the result given in [31] (Theorem 2.4). Furthermore, respective inequality is obtained in fractional calculus ([29], Theorem 2.3) by assuming and .

Example 1. If , then , , andBy substituting (28) and (34), it becomes

Example 2. If , then . In this case,By substituting (36) in (28), one gets

4. Hardy Inequalities for Superquadratic Functions with General Kernels

Theorem 2. Assume where , , and and where , and such that and . Definewhere is a -measurable function, is a nonnegative kernel, andLet be -integrable function. DenoteFurther assume , , and . If is superquadratic, thenholds for all -integrable such that . If is subquadratic, then inequality (41) is reversed.

Proof. ConsiderBy using Jensen’s inequality (28),By using Fubini’s theorem,

Remark 2. If the function is subquadratic, then Jensen’s inequality for superquadratic functions on delta fractional integrals is reversed, which implies, according to the conclusions made above, that inequality sign in (41) is reversed.
In [30], authors prove the analogous results for convex functions; now, it is easy to see that when in (41) is nonnegative, then inequality (41) gives the refinement of [30] (Theorem 2). By assuming , we obtain the inequality (41) in time scales ([32], Theorem 2.1), when is nonnegative in [32] (Theorem 2.1), then [32] (Theorem 2.1) gives the refinement of [27] (Theorem 3.2). Moreover, the inequality (41) is obtained in fractional calculus [33] by assuming and .

Corollary 1. Assume (38)–(40). If , thenholds for all -integrable such that .
If , then (45) holds in reverse direction.

Proof. Use in Theorem 2.

Remark 3. We obtain the respective results in time scales ([32], Corollary 2.3) by assuming in (45).

Remark 4. In particular, if in Corollary 4.2, we get the following identity:

Corollary 2. Assume (38)–(40), thenholds for all -integrable such that with

Proof. Use and in Theorem 2.

Remark 5. By choosing in (47), the respective result is obtained in [32] (Corollary 2.5).

Corollary 3. Assume (38)–(40) and further if , then

Proof. If , then , . In this case,Substituting (50) and (51) in (41), one gets

5. Inequalities with Special Kernels

Corollary 4. Assume (38)–(40) with kernel such thatthenholds for all -integrable such that .

Proof. By using (53) in (38)–(40), respectively,We use and (55)–(57) in Corollary 4.2 to get

Corollary 5. Assume with and (55)–(57) hold, then