On Refinements of Multidimensional Inequalities of Hardy-Type via Superquadratic and Subquadratic Functions
By utilizing the peculiarities of superquadratic and subquadratic functions, we give the extensions for multidimensional inequalities of Hardy-type with general kernel. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality to prove the essential results in this paper. The performance of the superquadratic functions is reliable and effective to obtain new dynamic inequalities on time scales. By utilizing special kernels, we also acquire numerous examples and implementations of the related inequalities.
In , Hardy proved that if , over interval and ; then,where the constant is sharp. By rewriting (1) with rather than and taking the limit as , we acquire the limiting case of Hardy’s inequality known as the inequality of Pólya-Knopp (), that is,
In , Kaijser et al. applied Fubini’s theorem and Jensen’s inequality to establish an invitingly popularization of (1). Particularly, it was evidenced that if and , such thatand , is a convex function and is defined by
Then, the inequality,holds for any non-negative integrable function such that where is defined by
As a popularization of (6), Krulic et al.  have demonstrated that if and are measure spaces with positive -finite measures, and are measurable functions such that is a -integrable function for , is defined asand is defined bywhere . If is a convex function on , then the inequality,holds for any non-negative -integrable function such that where is defined by
Moreover, if is a non-negative superquadratic function, thenholds for any non-negative -integrable function where is defined by (11).
In , the researchers demonstrated some Hardy-type inequalities with a general kernel. They have determined that if and are two time scale measure spaces with positive -finite measures, is such that is a convex function and such thatthenholds for all non-negative -integrable such that .
In , the authors improved the inequality (16) by replacing the function by an -tuple of functions.such as are -integrable on as follows: let , are non-negative such that is a -integrable function for , and the function are defined by (14), (15), respectively. Then, for a convex function over a convex set , the integral inequality,holds for all -integrable functions such that .
In , the authors derived some inequalities of Hardy-type by utilizing the concept of superquadratic functions. Particularly, they proved that if and are measurable functions such that is a -integrable function for , and the function are defined by (14) and (15), respectively. Let be a non-negative superquadratic function. Then, the inequality,holds for all non-negative -integrable function , where is defined by
Moreover, in [10, 11], the authors generalized (19) and proved that if and are measurable functions such that is a -integrable function for , and the function are defined by (14) and (15), respectively. Let be a non-negative superquadratic function. Then, the inequalityholds for all -integrable functions such that , where is defined by
In , the authors deduced several generalizations of (19) on time scales. They proved that if , and are measurable functions such that is a -integrable function for , is defined by (14) and the function be defined by
Let be a non-negative superquadratic function. Then, the inequality,holds for all non-negative -integrable function where is defined by (20).
Another development of Hardy-type inequality (24) has been made by Saker et al.  as follows: let and be non-negative functions such that is a -integrable function for , is defined byand is defined bywhere . If is a non-negative superquadratic function, then the inequality,holds for all non-negative -integrable function where is defined by
Motivated by the previous results, our major aim in this paper is to deduce several generalizations of general Hardy-type inequalities for multivariate superquadratic functions that involve more general kernels on arbitrary time scales.
The paper is governed as follows: We remember some basic notions, definitions, and results of multivariate superquadratic functions on time scales in Section 2. In Section 3, we prove some new refined dynamic inequalities of Hardy’s type with non-negative kernel by utilizing the peculiarities of superquadratic (or subquadratic) functions. In Section 4, we discuss several particular cases of Hardy-type inequality by choosing such special kernels. Eventually, in Section 5, we give more implementations of our obtained results on particular time scales.
In this section, we will introduce some fundamental concepts and effects to integrals of time scales and for multivariate superquadratic functions which will be useful to deduce our major results.
Before introducing the main results for multidimensional inequalities, it is necessary to present some further definitions. Firstly, suppose , be the Euclidean space. Letbe the function that is defined on . We utilize the following notations:also, for , we write if component wise , , the relations , are defined analogously, is the null vector and . Correspondingly, for , , we define and the n-cells , and are defined similarly. Furthermore, the subsets and in are defined by
In particular,where . Correspondingly, means the set , and .
Now, we arraign the definition and some essential properties of superquadratic functions that are premised in .
Definition 2. (see ). A function is superadditive provided for all . If the reverse inequality holds, then is said to be subadditive.
Lemma 1 (see ). Suppose is continuously differentiable and . If is superadditive or is nondecreasing, then is superquadratic.
In the next, we recall a couple of beneficial examples of a superquadratic function.
Example 1. (see ). The power function that is defined as is called superquadratic if and subquadratic if (it is also readily seen that if then is a subquadratic function). Since the sum of superquadratic functions is also superquadratic, the function is defined as follows:which is superquadratic on for each (as shown in , Example 2) and the function is defined bywhich is superquadratic on for each (see , Example 3).
Example 2 (see ). Examples 4, 5, and 6 By utilizing the same argument as in Example 4, the functions that defineare superquadratic.
The following lemma shows that non-negative superquadratic functions are indeed convex functions:
Lemma 2 (see ). Let be a superquadratic with that is defined as in Definition 1. Then,(i) and ;(ii)If and , then whenever exists for some index at ; where means that the gradient of the function and denotes the partial derivative of over the i-th variable;(iii)If , then is convex and .The following definitions and theorems are referred from [26, 27]. Let , be time scales, andwhich is called an n-dimensional time scale. Let E be -measurable subset of and be a -measurable function. Then, the corresponding -integral is called Lebesgue -integral and is denoted bywhere is a -additive Lebesgue -measure on . Also, ifwhich is an m-tuple of functions in n-variables such that are Lebesgue -integrable on E, then denotes the m-tuple.i.e., -integral acts on each component of .
Particularly, if is an arbitrary time scale and includes only isolated points, thenwhere is referred to as the graininess function on time scale.
Lemma 3 (Minkowski’s inequality ). Let and be two finite-dimensional time scale measures spaces. Suppose that , and that are defined on and , respectively. If , thenprovided that all integrals in (23) exist. If ,then (42) is reversed. For , if in addition to (43) andthen (42) is again reversed.
In , (Corollary 5.1), Bibi get the following generalization of Jensen’s inequality and the converse of it for superquadratic functions.
Theorem 1 (Jensen’s inequality). Let be a superquadratic function and be -integrable function for and . Then, for every -integrable functions such that and , are -integrable, we have
If is subquadratic, then (45) is reversed.
3. Main Results
In this section, we prove multidimensional Hardy-type inequalities with general kernels on time scales. Before proceeding with results, we introduce the following notations: and are two time scale delta measure spaces with positive -finite measures. be a non-negative so that is -integrable function for and be defined by is -integrable and the function be defined by where .
Theorem 2. Weassume are satisfied. If is non-negative superquadratic function, thenholds for all -integrable functions such that , where is defined byIf and is subquadratic, then (48) is reversed.
Proof. We begin with the following identity:By applying (45) on (50), we findSince and , we haveFurthermore, by employing the Bernoulli inequality ,It follows that the L. H. S of (52) is not less thanthat is,By multiplying (55) with and integrating it over with respect to , we getApplying (42) on the R. H. S of (56), we getFinally, substituting (57) into (56) and using the definition (47) of the weight function , we haveThis proves (48). The proof of the case in which and is subquadratic is similar; the only difference is that the inequality sign in (48) is reversed. The proof is complete.
Corollary 1. Given that and are as in Theorem 2 and is measurable function. Since is superquadratic function, then the second term on the L. H. S of (48) is non-negative and the integral inequalityholds.
Remark 5. If we rewrite (48) with , or and , then we get
Remark 7. In Remark 5, since , then the second term on the L. H. S. of (60) is non-negative and (60) reduces to the weighted Hardy-type inequality of the formwhich is a refinement of general Hardy-type inequality established in  [Remark 2.1.4] and .
As a specific case of Theorem 2 when for , we get the next result.
Remark 8. Clearly, for , inequality (62) in Corollary 2 coincides with inequality (46) which is , (Corollary 2.1).
In fact, the function is not superquadratic but by working with the superquadratic function (see Lemma 3) and replacing by in Theorem 2, we obtain the next multidimensional version of the Pólya–Knopp type inequality.
Remark 9. For , inequality (64) in Corollary 3 coincides with inequality (48) which is , (Corollary 2.2).
In the next results, we also suppose the following hypothesis:
Let , , for every , where is an arbitrary time scale and let , .
Theorem 3. Weassume and are satisfied. Suppose such thatwhere . If is non-negative superquadratic function, thenholds for all -integrable functions such that , where is defined byIf is subquadratic function and , then (48) is reversed.
Proof. We get the result from Theorem 2 by taking
4. Inequalities with Special Kernels
In this section, we get some consequential inequalities of Hardy-type by selecting special kernels.
Theorem 4. Suppose andsuch that is -integrable function and . If is non-negative superquadratic function, thenholds for all -integrable functions such that , where is defined byIf is subquadratic function and , then (71) is reversed.
Proof. We get the result from Theorem 2 by taking
We have the following in this case:, .
Corollary 4. Assuming , we define