Abstract

By utilizing the precepts related to the punctuation of time scales , we present some nouveau forms in quotients for Hardy’s and related inequalities on time scales. In particular, some recent results for the Pólya-Knopp, Hardy-Hilbert, and Hardy-Littlewood-Pólya-type inequalities are presented.

1. Introduction

In [1], Hardy stated and proved the following integral inequality: where is a positive function and the constant is the best ever. We rewrite (1) with the function rather than , and by assuming limit we acquired the limiting instance of the inequality of Hardy known as the inequality of Pólya-Knopp (see [2]), that is,

Lately, Kaijser et al. in [3] pointed out that both (1) and (2) are just special states of the much more general inequality of Hardy-Knopp for positive function : where is a convex function. A popularization of inequality (3) with two weight functions is proved in [4]. Particularly, it was proved that if , is a nonnegative function and is a convex on , then the inequality is available for all integrable functions , such that and is defined by

Using the inequality of Jensen for convex functions and the theorem of Fubini, Kaijser et al. [5] established an inviting popularization of (1). Particularly, they proved that if and are nonnegative functions such that , for and then where is a convex function and

For further popularization of (7), Krulić et al. [6] proved that if and are two measure spaces with positive -finite measures and which are nonnegative measurable functions such that , , , and is defined by then the inequality is available for all measurable functions such that , where is a convex function and is defined by

In [7], Iqbal et al. checked some new weighted Hardy-type inequalities on and measure spaces with positive -finite measures by replacing by and by , where are measurable functions in (10) as follows: where , is a convex function, is a nonnegative measurable function, and are defined by

In [8], the author founded the time scale version of (1). Particularly, he proved that if , is a nonnegative function and exists, then

In [9], the authors proved the time scale version of (4), which is given by where is a convex function, is a nonnegative function, and 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 and are two time scale measure spaces, and , which are nonnegative measurable functions such that and is defined by then the inequality is available for all -integrable such that and is a convex function. For development of dynamic inequalities on time scale calculus, we refer the reader to articles [11–18].

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 Hardy-type 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.

2. 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 is called a time scale which is denoted by For , if and , then the forward jump operator and the backward jump operator are defined as respectively. The -derivative of at is the number that enjoys the property that for all , there exists a neighborhood of such that

Furthermore, is called a delta differentiable on if it is delta differentiable at every . Similarly, for , we define the -derivative of at as the number that enjoys the property that for all , there exists a neighborhood of such that

Moreover, is called a nabla differentiable on if it is nabla differentiable at every . For , the delta integral of is defined as

Similarly, for the nabla integral of is defined as

Now, let be differentiable on in the and senses. For , where , the diamond- dynamic derivative is defined by

The diamond- derivative debases to the standard -derivative for or the standard -derivative for

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 and are two finite-dimensional time scale measure spaces, and let , , and be positive functions on , , and respectively. If then the inequality is available for all integrals in (26). If and are available, then (26) is reversed. For in addition with (27), if is available, then again (26) is reversed.

Theorem 2. Let and . Suppose that and are nonnegative with If is convex, then

3. Inequalities with General Kernels

In this section, we state and prove our major results. Before presenting the results, we labeled the following hypotheses.  (H1) and are two time scale measure spaces (H2) is a positive measurable kernel and (H3) is a -integrable andwhere

In what follows, we will prove the foundation theorem that will be the decisive step in establishing our major result.

Theorem 3. Assume (H1)–(H3).
If is a positive convex, then the following inequality is available for all nonnegative -integrable function such that , where is defined by

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 , it follows that Finally, elevating (36) to the th power, we acquired (33).

Remark 4. For the Lebesgue scale measures , , and , inequality (32) reduces to inequality (19) premised in Section 1. Choosing in Theorem 3, then inequality (32) reduces to inequality (10).

In what follows, we labeled few particular convex functions starting with power functions.

Corollary 5. Suppose the supposition of Theorem 3 be gratified only with for . Then, inequality (32) yields the following result:

Remark 6. For the Lebesgue scale measures , , and , Corollary 5 coincides with Corollary 3.3 in [10].

Now, considering Theorem 3 with and , we get the following inviting result.

Corollary 7. Suppose the supposition of Theorem 3 is gratified only with . If , then inequality (32) yields the following result:

Remark 8. For the Lebesgue scale measures , , and , Corollary 7 coincides with Corollary 3.4 in [10]. Also, in the special case , Remark 8 coincides with Corollary 3.5 in [10].