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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.
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 .
Hardy-type inequalities have crucial importance in the study of function spaces, especially of fractional regularity . 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 . The familiar Hardy inequality (both in the discrete and continuous settings) as presented in  has been enormously studied and used as a model for the inquisition of more general integral inequalities [4, 5].
In , Hardy proved the discrete inequality,where for . Also, in , 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.
Since the discovery of these two inequalities, various papers which deal with new proofs, generalizations, and extensions have appeared in the literature, [6, 7, 10–12] are referred to readers. During the last decades, these inequalities were extended, a time scale version of weighted Hardy-type inequality is proved in , and some preliminary dynamic inequalities  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 , Hardy–Sobolev inequalities on hypersurfaces for Euclidean space , improved Hardy inequalities with exact remainder terms , Hardy inequalities with double singular weights , and Hardy inequalities for class functions in one-dimensional fractional Orlicz–Sobolev spaces . A new approach for the fractional integral operator in time scales with variable exponent Lebesgue spaces is presented in  and -dimensional integral-type inequalities via time scale calculus is studied in .
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 as Delta 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 ,
The Riemann–Liouville derivative with the lower integration limit would be 
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  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 , . 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
If and (, Chap: 1), then
2.3.1. Riemann–Liouville Delta Fractional Integral
2.3.2. Fubini’s Theorem on Time Scale
Consider and . Then,
It can be found in .
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 ) 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  (Theorem 1) and by assuming , (28) becomes  (Theorem 2.5), and result of  (Theorem 2.5) refines the result given in  (Theorem 2.4). Furthermore, respective inequality is obtained in fractional calculus (, Theorem 2.3) by assuming and .
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 , 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  (Theorem 2). By assuming , we obtain the inequality (41) in time scales (, Theorem 2.1), when is nonnegative in  (Theorem 2.1), then  (Theorem 2.1) gives the refinement of  (Theorem 3.2). Moreover, the inequality (41) is obtained in fractional calculus  by assuming and .
Proof. Use in Theorem 2.
Remark 4. In particular, if in Corollary 4.2, we get the following identity:
Proof. Use and in Theorem 2.
5. Inequalities with Special Kernels
Proof. The statement follows from Corollary 4 by using .