Dynamic Inequalities in Quotients with General Kernels and Measures

Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni Suef, Egypt Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt Mathematics Department, College of Science, Jouf University, Sakaka (2014), Saudi Arabia Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, 61413 Abha, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, 71524 Assiut, Egypt

In [9], the authors proved the time scale version of (4), which is given by where Φ ∈ Cððc 1 , d 1 Þ, ℝÞ is a convex function, λ ∈ C rd ð½a 1 , b 1 T , ℝÞ is a nonnegative function, and υ is defined by In [10], the authors outstretched a number of Hardy-type inequalities with certain kernels on time scale. Namely, they proved that if ðΩ 1 , ∑ 1 , μ 1 Þ and ðΩ 2 , ∑ 2 , μ 2 Þ are two time scale measure spaces, ζ : Ω 1 → ℝ + and k : Ω 1 × Ω 2 → ℝ + , which are nonnegative measurable functions such that Journal of Function Spaces and υ is defined by then the inequality is available for all Δμ 2 -integrable f : Ω 2 → ℝ such that f ðΩ 2 Þ ⊂ I and Φ ∈ CðI, ℝÞ is a convex function. For development of dynamic inequalities on time scale calculus, we refer the reader to articles [11][12][13][14][15][16][17][18]. The article is regimented as follows. In Section 2, we recall the precepts related to the punctuation of time scales. In Section 3, we prove our results and give some remarks. Particularly, we prove a general dynamic weighted Hardytype inequality with a nonnegative kernel. In Section 4, we critique a few particular states of the obtained inequalities, related to power and exponential functions and to the most simplest shapes of kernels.

Preliminaries
In this section, we will premise some fundamental precepts and effects on time scales which will be beneficial for deducing our major results. The following definitions and theorems are referred from [19,20].
A nonempty arbitrary locked subplot of the real numbers ℝ is called a time scale which is denoted by T : For τ ∈ T , if inf Φ = sup T and sup Φ = inf T , then the forward jump operator σ : T → T and the backward jump operator ρ : T → T are defined as respectively. The Δ-derivative of χ : T → ℝ at τ ∈ T k = T /ðρ ðsup T Þ, sup T is the number that enjoys the property that for all ε > 0, there exists a neighborhood U ⊂ T of τ ∈ T k such that Furthermore, χ is called a delta differentiable on T k if it is delta differentiable at every τ ∈ T k . Similarly, for τ ∈ T k , we define the ∇-derivative of χ : T → ℝ at τ ∈ T k = T /ðinf T , σ ðinf T Þ as the number that enjoys the property that for all ε > 0, there exists a neighborhood V ⊂ T of τ ∈ T k such that Moreover, χ is called a nabla differentiable on T k if it is nabla differentiable at every τ ∈ T k . For θ, t ∈ T k , the delta integral of χ Δ is defined as Similarly, for θ, t ∈ T k , the nabla integral of χ ∇ is defined as Now, let χðτÞ be differentiable on T in the Δ and ∇ senses.
The diamond-α derivative debases to the standard Δ -derivative for α = 1 or the standard ∇-derivative for α = 0: Next, we recall the inequality of Minkowski and the inequality of Jensen on time scales which are utilized in the proof of the major results. Theorem 1. Suppose ðΩ, M, μ Δ Þ and ðΛ, L, λ Δ Þ are two finite-dimensional time scale measure spaces, and let u, v, and g be positive functions on Ω, Λ, and Ω × Λ, respectively. If ξ ≥ 1, then the inequality is available for all integrals in (26). If 0 < ξ < 1 and is available, then again (26) is reversed.

Inequalities with General Kernels
In this section, we state and prove our major results. Before presenting the results, we labeled the following hypotheses.
(H1) ðΩ 1 , Σ 1 , μ 1 Þ and ðΩ 2 , Σ 2 , μ 2 Þ are two time scale measure spaces In what follows, we will prove the foundation theorem that will be the decisive step in establishing our major result. If Φ ∈ CðI, ℝÞ is a positive convex, then the following inequality is available for all nonnegative Proof. We begin with an evident identity Utilizing the inequality of Jensen (29) and the theorem of Minkowski (2) on (34), we find that Taking into computation definition (31) of υ, it follows that Finally, elevating (36) to the ξ * th power, we acquired (33). In what follows, we labeled few particular convex functions starting with power functions.
Now, by using a special substitution, we obtain our central result; that is, if we replace kðθ, ηÞ by kðθ, ηÞf 2 ðηÞ and f by f 1 /f 2 , where f j : Ω 2 → ℝðj = 1, 2Þ are measurable functions, we obtain these results. Theorem 9. Assume (H1) and (H2) and υ be defined on Ω 2 by where 0 < ξ ≤ ξ * < ∞: If Φ ∈ CðI, ℝÞ is a positive convex, then the following inequality is available for all measurable functions f j : Ω 2 → ℝðj = 1, 2Þ and As a special case of Theorem 9 for ξ = ξ * , we get the next corollary. Also, we note that the function Φ need not to be positive.

Corollary 11. Assume (H1) and (H2) and υ be defined on Ω 2 by
If Φ ∈ CðI, ℝÞ is a positive convex, then the following inequality is available for all measurable functions f j : Ω 2 → ℝðj = 1, 2Þ and g j is defined by (41).
Remark 21. If we set α = 1, then the delta version form of inequality (50) takes the form which coincided with inequality (8) in [21].

Remark 22.
If we set α = 0, then the nabla version form of inequality (50) takes the form which coincided with inequality (9) in [21].

Inequalities with Special Kernels
The next theorem states the general result for Hardy's inequality in quotient.
Remark 30. If we put in Theorem 29 T = ℝ, e = 0, and l = ∞, we see that σðθÞ = θ and then (77) takes the form where which is the same result due to Iqbal et al. (Corollary 2.6, [7]).