Inequalities of Hardy Type via Superquadratic Functions with General Kernels and Measures for Several Variables on Time Scales

Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Mathematics Department, College of Science, Jouf University, Sakaka, Saudi Arabia Department of Mathematical Science, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, 61413 Abha, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, 71524 Assiut, Egypt


Introduction
In [1], Hardy claimed this fundamental inequality and proved it: where 1 < q < ∞, g ≥ 0, and ðq/ðq − 1ÞÞ q are sharp. They have emerged in the literature since the discovery of (1) numerous papers concerned with new arguments, generalizations, and extensions. One of the most common generalizations for (1) is the disparity of Pólya-Knopp's inequality (see [2]), which is In [3], Kaijser et al. signalized that both (1) and (2) are special states of the Hardy-Knopp's inequality: where Θ ∈ Cðð0, ∞Þ, ℝÞ is a convex function.
Motivated by the above results, our major aim in this paper is to deduce few nouveau 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 Preliminaries. In Inequalities with General Kernel, we obtain the extensions to the general kernel of Hardy-type inequality. In Inequalities with Specific Time Scales, we extend the latest results from Inequalities with General Kernel to several specific time scales. In Inequalities with Specific Time Scales, we discuss several particular cases of Hardy-type inequality by choosing such special kernels. In Inequalities with Specific Kernels, we derive enhanced forms of certain well-known Hardy-Hilberttype inequalities.

Preliminaries
In this section, we will present some fundamental concepts and effects to integrals of time scales and for multivariate superquadratic functions which will be useful to deduce our major results. Let ℝ m be the Euclidean space, θ ≔ ðθ 1 , θ 2 , ⋯, θ m Þ ∈ ℝ m , η ≔ ðη 1 , η 2 , ⋯, η m Þ ∈ ℝ m , and gðtÞ ≔ ðg 1 ðtÞ, g 2 ðtÞ, ⋯, g m ðtÞÞ be the function defined on θ ⊂ ℝ m . Throughout this supplement, we utilize the following notations: Also, θ ≤ ηðθ < ηÞ means that Now, we arraign the definition and few essential properties of superquadratic functions that premised in [27].
If −Θ is a superquadratic, then Θ is a subquadratic, and the reverse inequality of (37) is available.
In the following, we recall a couple of beneficial examples of a superquadratic function.
Example 1. By [2], Example 1, the power function Θ : ½0, ∞Þ ⟶ ℝ, defined by ΘðθÞ ≔ θ p , is called a superquadratic if p ≥ 2 and a subquadratic if 1 < p ≤ 2 (it is also readily seen that if 0 < p ≤ 1 then θ p is a subquadratic function). Since the sum of superquadratic functions is also superquadratic, then is a superquadratic on K m for each p ≥ 2.
The following lemma shows that nonnegative superquadratic functions are indeed convex functions.

Lemma 2.
Suppose that Θ is a superquadratic with cðθÞ ≔ ðc 1 ðθÞ, c 2 ðθÞ, ⋯, c n ðθÞÞ as in Definition 1. Then In the following, we recall the inequality of Minkowski and the inequality of Jensen for superquadratic functions on time scales which are utilized in the proof of the essential results. The following definitions and theorems are referred from [28,29]. Let T i , 1 ≤ i ≤ m be time scales, and is called an m-dimensional time scale. Consider E to be Δ -measurable subplot of Λ m and g : E ⟶ ℝ a Δ-measurable function; then, the corresponding Δ-integral named Lebesgue Δ-integral is denoted by where μ Δ is a σ-additive Lebesgue Δ-measure on Λ m . Also, if gðtÞ ≔ ðg 1 ðtÞ, g 2 ðtÞ, ⋯, g m ðtÞÞ is an m-tuple of functions such that g 1 , g 2 , ⋯, g m are Lebesgue Δ-integrable on E, then Ð E gdμ Δ denotes the m-tuple: i.e., Δ-integral acts on each component of g.
is available provided all integrals in (43) exist. If 0 < q < 1 and is available, then the sign of (43) is reversed.
5 Journal of Function Spaces holds for all functions g such that gðEÞ ⊂ K m . If Θ is a subquadratic, then (46) is reversed.

Inequalities with General Kernel
In this section, we get the Hardy inequality for several variables via multivariate superquadratic functions. Before presenting the results, we labeled the following hypothesis.
is available for g : Ω 2 ⟶ ℝ m that is a nonnegative Δμ 2 -integrable function such that gðΩ 2 Þ ⊂ K m and A l g : If Θ is subquadratic and 0 < λ < 1, then (49) is reversed.
Proof. We begin with an explicit identity By applying the refined Jensen inequality (46) on (51), we find Then, since λ ≥ 1 and Θ ≥ 0, we get Furthermost, by utilizing the famous inequality of Bernoulli, it ensues that the L. H. S. of (53) became that is, we get Multiplying (55) by ξðθÞ and integrating it over Ω 1 with respect to Δμ 1 ðθÞ, we have Applying the inequality of Minkowski on the R. H. S. of (56), we get 6 Journal of Function Spaces Finally, substituting (57) into (56) and utilizing the definition (48) of the weight function ω, we get which is (49). If Θ is subquadratic and 0 < λ < 1, the corresponding results can be obtained similarly. Remark 8. As a special case of Theorem 5 when T = ℝ and m = 1, we have the inequality (19).

Corollary 9.
Given that ξ and ðA l gÞðθÞ are as in Theorem 5 and ω ≥ 0, then, since Θ ≥ 0 and superquadratic, the second term on the L. H. S. of (49) is nonnegative and the integral inequality is valid.
In the following, we labeled some specific superquadratic functions starting with power functions.
Proof. We get the result from Theorem 5 by putting in (49).
Proof. We get the result from Theorem 5 by putting in (49) is valid, where A l g i is defined as in (63). If 0 < λ < 1, then (69) is reversed.
Proof. We get the result from Theorem 5 by putting in (49).

Journal of Function Spaces
Proof. We get the result from Theorem 5 by putting in (49) with the assumption 0 ln 0 = 0.
is valid, where A l g i is defined as in (63). If 0 < λ < 1, then (73) is reversed.
Proof. We get the result from Theorem 5 by taking in (49).

Inequalities with Specific Kernels
In this section, we find some consequential inequalities of the Hardy type by selecting specific kernels and weight functions.

Some Particular Cases
In this section, we obtain a popularization and a refinement of the classical inequality of the Hardy-Hilbert type (16) If λ ≥ 1 and p ≥ 2, then and the operator ðA l gÞðθÞ in this case is defined as Utilizing ðA l gÞðθÞ in (62), we obtain Hence,