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

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.


Introduction
e study of Hardy inequalities has gained huge attention in the literature, and now it became a major field in applied and pure mathematics. e 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]. e 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 a(n) ≥ 0 for n ≥ 1. Also, in [7], Hardy proved the continuous inequality, using the calculus of variation, which states that for nonnegative, integrable function over any finite interval (0, x) and f p is integrable for p > 1, Hardy inequality (2) has been generalized by G. H. Hardy himself in [8]. ere he showed that for any integral function f(x) > 0 on (0, ∞), p > 1, 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,[10][11][12] 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 n-dimensional integral-type inequalities via time scale calculus is studied in [22].

Some Basics on Time Scales.
A time scale is a closed subset of the real line R, and its common notation is T. A time scale T may or may not be connected. Forward jump operator and backward jump operator are respectively defined as In general, σ(ω) ≥ ω and ρ(ω) ≤ ω. e forward and backward graininess functions μ, ]: T ⟶ [0, +∞) are, respectively, defined by If σ(ω) > ω, then ω is said to be right scattered. If ρ(ω) < ω, then ω is said to be left scattered. Points that are left-scattered and right scattered at the same time are called isolated. If ω < supT and σ(ω) � ω, then ω is called the right-dense and if ω > infT and ρ(ω) � ω, then ω is called left-dense. Points that are left-dense and right-dense at the same time are called dense.
Rd-continuous function: a function Z: T ⟶ R is called rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist at left-dense points in T. Notation: if T is a time scale, then the set T k is defined as Delta derivatives: take a function Z: T ⟶ R. e delta derivative (Hilger derivative) Z Δ (ξ) exists iff, for every I > 0, there exists a neighborhood ℵ of ξ such that for all s ∈ ℵ, ξ ∈ T.

Some Principles of Fractional
e calculation of the second can be simplified by interchanging the integration order, is 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], e Riemann-Liouville derivative with the lower integration limit a would be [23]

Riemann-Liouville
Integral. e Riemann-Liouville integral relates with a real function Z: R ⟶ R, another same kind of function I c Z for each value of the parameter c > 0. e integral is a generalization of the repeated antiderivative of Z of order real number c.
Let J � [a, b](− ∞ < a < b < ∞) be a finite interval of R. e left and right Riemann-Liouville fractional integrals respectively, provided the right-hand sides are pointwise defined on [a, b]. When c � n ∈ N + , definitions (14) and (15) coincide with the nth integrals of the form, 2.3. Some Principles of Delta Fractional Calculus. For μ, ] > 0, one has that (see [24], p. 256) where Γ is the Gamma function. If time scale T is considered to be T � T k , then the coordinate-wise rd-continuous functions are given by such that h 0 (ξ, r) � 1, ∀ξ, r ∈ T and Furthermore, for c, β > 1, it is assumed that In the case T � R and for σ(r) � r, one obtains and also satisfying (19). Furthermore, it is observed that for c, β > 1 and by using (17), one gets which is satisfying (20).

Fubini's eorem on Time
It can be found in [28].

Mathematical Problems in Engineering
for all ] ≥ 0. We say that ϕ is subquadratic if − ϕ is a superquadratic function.
e following lemma (see [29]) shows that every positive superquadratic function is also a convex function. Lemma 1. Let Θ be a superquadratic function with C(ω) as in Definition 1. en,
Proof. Consider By using Jensen's inequality (28), 6 Mathematical Problems in Engineering By using Fubini's theorem, □ Remark 2. If the function Θ 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.

Conclusion
In the paper, some dynamic inequalities of Jensen and Hardy for superquadratic functions with Riemann-Liouville delta fractional integrals are generalized. Particular cases of obtained inequalities have been given in time scales [32] by assuming c, β � 1 and respective inequalities in fractional calculus [33] by choosing T � R are also obtained. Furthermore, some discrete fractional inequalities for superquadratic functions are obtained by choosing T � hZ and T � N 2 . Inequalities in quantum discrete calculus are also obtained by choosing T � q Z , which are new, up to the knowledge of authors.

Data Availability
All data generated or analyzed during this study are included in this published article. ere are no experimental data in this article.