1 Introduction

The renowned discrete Hardy inequality [1] states that:

Theorem 1.1

If \(\{a_{n}\}\) is a nonnegative real sequence and \(p>1\), then

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\bigg (\frac{1}{n}\sum \limits _{k=1}^{n}a_{k}\bigg )^{p}\le \Big ( \frac{p}{p-1}\Big )^{p}\sum \limits _{n=1}^{\infty }a_{n}^{p},\quad p>1. \end{aligned}$$
(1.1)

Hardy discovered this inequality while attempting to sketch an easier proof of Hilbert’s inequality for double series which was known at that time.

Using the calculus of variations, Hardy himself in [2] gave the following integral analogous of inequality (1.1).

Theorem 1.2

If f is a nonnegative integrable function over a finite interval (0, x) such that \(f\in L^{p}(0,\infty )\) and \(p>1\), then

$$\begin{aligned} \int \limits _{0}^{\infty }\left( \frac{1}{x}\int \limits _{0}^{x}f(t)dt\right) ^{p}dx\le \left( \frac{p}{p-1}\right) ^{p}\int \limits _{0}^{\infty }f^{p}(x)dx. \end{aligned}$$
(1.2)

It’s worthy to mention that inequalities (1.1) and (1.2) are sharp in the sense that the constant \((\frac{p}{(p-1)})^{p}\) can not be replaced by a smaller one.

In [3], Hardy and Littlewood extended inequality (1.1), and obtained the following two discrete inequalities.

Theorem 1.3

Let \(\{a_n\}\) be a nonnegative real sequence.

(i):

If \(p>1\) and \(\gamma >1\), then

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\frac{1}{n^\gamma }\bigg (\sum \limits _{k=1}^{n}a_k\bigg )^p \le K(p,\gamma )\sum \limits _{n=1}^{\infty }\frac{1}{n^{\gamma -p}}a_n^p. \end{aligned}$$
(1.3)
(ii):

If \(p>1\) and \(\gamma <1\), then

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\frac{1}{n^\gamma }\bigg (\sum \limits _{k=n}^{\infty }a_k\bigg )^p \le K(p,\gamma )\sum \limits _{n=1}^{\infty }\frac{1}{n^{\gamma -p}} a_n^p, \end{aligned}$$
(1.4)

where \(K(p,\gamma )\) in inequalities (1.3) and (1.4) is a nonnegative constant depends on p and \(\gamma \).

In [2], Hardy established the continuous versions of inequalities (1.3) and (1.4) as follows:

Theorem 1.4

Let f be a nonnegative integrable function on \((0,\infty )\).

(i):

If \(p>1\) and \(m>1\), then

$$\begin{aligned} \int \limits _{0}^{\infty }\frac{1}{x^m}\bigg (\int \limits _{0}^{x}f(t)dt\bigg )^pdx \le \Big ( \frac{p}{m-1}\Big )^p\int \limits _{0}^{\infty }\frac{1}{x^{m-p}}f^p(x)dx. \end{aligned}$$
(1.5)
(ii):

If \(p>1\) and \(m<1\), then

$$\begin{aligned} \int \limits _{0}^{\infty }\frac{1}{x^m}\bigg (\int \limits _{x}^{\infty }f(t)dt\bigg )^pdx \le \Big (\frac{p}{1-m}\Big )^p\int \limits _{0}^{\infty }\frac{1}{x^{m-p}}f^p(x)dx. \end{aligned}$$
(1.6)

The reverse of inequality (1.2) was proven by Hardy and Littlewood in [3]. Their result is written as:

Theorem 1.5

If \(0<p<1\) and f is a nonnegative integrable function on \((x,\infty )\) such that \(f\in L^p(0,\infty )\), then

$$\begin{aligned} \int \limits _{0}^{\infty }\left( \frac{1}{x}\int \limits _{x}^{\infty }f(t)dt\right) ^p dx \ge \Big (\frac{p}{1-p}\Big )^p\int \limits _{0}^{\infty }f^p(x)dx. \end{aligned}$$
(1.7)

In the same paper [3], the authors proved the following sharp inequality.

Theorem 1.6

If \(p>1\) and f is a nonnegative integrable function on \((x,\infty )\) such that \(f\in L^p(0,\infty )\), then

$$\begin{aligned} \int \limits _{0}^{\infty }\bigg (\int \limits _{x}^{\infty }\frac{f(t)}{t}dt\bigg )^p dx \le p^p\int \limits _{0}^{\infty }f^p(x)dx, \end{aligned}$$
(1.8)

which by a trivial transformation can be written as

$$\begin{aligned} \int \limits _{0}^{\infty }\bigg (\int \limits _{x}^{\infty }f(t)dt\bigg )^p dx \le p^p\int \limits _{0}^{\infty }(xf(x))^pdx. \end{aligned}$$
(1.9)

The discrete version of inequality (1.9) was given in [4] as follows:

Theorem 1.7

If \(\{a_n\}\) is a nonnegative real sequence and \(p>1\), then

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\bigg (\sum \limits _{k=n}^{\infty }a_k\bigg )^p \le p^p\sum \limits _{n=1}^{\infty }(na_n)^p. \end{aligned}$$
(1.10)

Hardy [5] generalized (1.1) and proved the following result.

Theorem 1.8

If \(p>1\), \(a_n>0\), \(\lambda _n>0\), for \(n\ge 1\) and \(\Lambda _n = \sum \nolimits _{k=1}^{n}\lambda _n\), then

$$\begin{aligned} \sum \limits _{n=1}^{\infty }\frac{\lambda _n}{\Lambda ^p_n}\bigg (\sum \limits _{k=1}^{n} \lambda _k a_k\bigg )^p \le \Big (\frac{p}{p-1}\Big )^p\sum \limits _{n=1}^{\infty } \lambda _n a^p_n. \end{aligned}$$
(1.11)

The study of Hardy-type inequalities have attracted many researchers and several refinements and extensions have been done to the previous results, we refer the readers to the papers [2, 5,6,7,8,9,10,11,12,13], the books [4, 14,15,16] and the references cited therein.The theory of time scales, which has recently received a lot of attention, was initiated by Stefan Hilger in his PhD thesis in order to unify discrete and continuous analysis [17]. The general idea is to prove a result for a dynamic equation or a dynamic inequality where the domain of the unknown function is so-called time scale \({\mathbb {T}}\), which may be an arbitrary closed subset of the real line \({\mathbb {R}}\) see [18, 19]. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [20]), i.e., when \( {\mathbb {T}} = {\mathbb {R}}\), \({\mathbb {T}} = {\mathbb {Z}}\) and \({\mathbb {T}} = \overline{q^{{\mathbb {Z}}}} = \{q^z: z \in {\mathbb {Z}}\}\cup \{0\}\) where \(q > 1 \) respectively. In books [18, 19], Bohner and Peterson introduce most basic concepts and definitions related with the theory of time scales. During the past decade a number of dynamic inequalities has been established by some authors who were motivated by some applications of dynamic equations, for example, when studying the behavior of solutions of certain class of dynamic equations on a time scale \( {\mathbb {T}}\), see [21,22,23,24,25,26,27] and the references cited therein.

In [28], Řehák has given the time scales version of Hardy inequality (1.2) as follows:

Theorem 1.9

Let \({\mathbb {T}}\) be a time scale, and \(f\in C_{rd}([a,\infty )_{\mathbb {T}},[0,\infty ))\), \( \Lambda (t)=\int \limits _{a}^{t}f(s)\Delta s\), for \(t\in [a,\infty )_{\mathbb {T}}\).

$$\begin{aligned} \int \limits _{a}^{\infty }\bigg (\frac{\Lambda ^\sigma (t)}{\sigma (t)-a}\bigg )^p\Delta t < \Big (\frac{p}{p-1}\Big )^p\int \limits _{a}^{\infty }f^p(t)\Delta t, \quad p>1, \end{aligned}$$
(1.12)

unless \(f\equiv 0\).

Furthermore, if \(\mu (t)/t\rightarrow 0\) as \(t\rightarrow \infty \), then inequality (1.12) is sharp.

In [29], Saker and O’Regan established a generalization of Řehák’s result in the following form.

Theorem 1.10

Let \(a\in [0,\infty )_{\mathbb {T}}\) and define, for \(t\in [0,\infty )_{\mathbb {T}}\),

$$\begin{aligned} \Phi (t):=\int \limits _{a}^{t}\lambda (s)g(s)\Delta s \qquad \text {and} \qquad \Lambda (t):=\int \limits _{a}^{t}\lambda (s)\Delta s. \end{aligned}$$

If \(p\ge q>1\), then

$$\begin{aligned} \int \limits \limits _{a}^{\infty }\lambda (t)\frac{\big (\Phi ^\sigma (t)\big )^p}{\big (\Lambda ^\sigma (t)\big )^q}\Delta t\le \Big (\frac{p}{q-1}\Big )^p \int \limits \limits _{a}^{\infty }\lambda (t)\frac{\big (\Lambda ^\sigma (t)\big )^{q(p-1)}}{\big (\Lambda (t)\big )^{p(q-1)}}g^p(t)\Delta t. \end{aligned}$$
(1.13)

In [30], Ozkan and Yildirim gave the following result among many other results.

Theorem 1.11

Let \(a\in [0,\infty )_{\mathbb {T}}\) and \(u, \lambda \in C_{rd}\big ([a,b]_{\mathbb {T}},{\mathbb {R}}_+\big )\) such that the delta integral \(\displaystyle \int \limits \limits _{x}^{b}\frac{u(t)}{(t-a)(\sigma (t)-a)}\Delta t\) converges. If \(f\in C_{rd}\big ([a,b]_{\mathbb {T}},(\alpha ,\beta )\big )\) where \(\alpha \), \(\beta \in {\mathbb {R}}\) and \(\Phi \in C\big ((\alpha ,\beta ),{\mathbb {R}}\big )\) is convex, then

$$\begin{aligned} \int \limits \limits _{a}^{b}\frac{u(x)}{x-a}\Phi \left( \frac{\int \limits \limits _{a}^{\sigma (x)}f(t)\Delta t}{\sigma (x)-a}\right) \Delta x \le \int \limits \limits _{a}^{b}\Phi \big (f(x)\big )\left( \int \limits _{x}^{b}\frac{u(t)}{(t-a)(\sigma (t)-a)}\Delta t\right) \Delta x. \end{aligned}$$

For more results on Hardy-type inequalities on time scales we refer to [31,32,33,34,35,36, 36,37,38, 38,39,40] and references therein.

In this paper, we prove some generalizations of Hardy-type dynamic inequalities that were given recently by Ozkan and Yildirim in [30]. The obtained results extend some known Hardy-type integral inequalities, unify and extend some continuous inequalities and their corresponding discrete analogues. The paper is arranged as follows: In Sect. 2, some basic concepts of the calculus on time scales and useful lemmas are introduced. In Sect. 3, we state and prove the main results. In Sect. 4, we stated the conclusion.

2 Preliminaries and Lemmas on Time Scales

First we presented some preliminaries on calculus of time scales and some universal symbols used in this article. Throughout the paper \({\mathbb {R}}\) and \({\mathbb {Z}}\) denote the set of all real numbers and the set of all integers, respectively.

A time scale \({\mathbb {T}}\) is an arbitrary nonempty closed subset of the real numbers. We assume throughout that \({\mathbb {T}}\) has the topology that it inherits from the standard topology on the real numbers \({\mathbb {R}}\). We define the forward jump operator \(\sigma : {\mathbb {T}}\rightarrow {\mathbb {T}}\) by

$$\begin{aligned} \sigma (t):=\inf \{s\in {\mathbb {T}}: s>t\}, \qquad t\in {\mathbb {T}}, \end{aligned}$$
(2.1)

and the backward jump operator \(\rho : {\mathbb {T}}:\rightarrow {\mathbb {T}}\) is defined by

$$\begin{aligned} \rho (t):=\sup \{s\in {\mathbb {T}}: s<t\}, \qquad t\in {\mathbb {T}}. \end{aligned}$$
(2.2)

In the previous two definitions, we set \(\inf \emptyset =\sup {\mathbb {T}}\) (i.e., if t is the minimum of \({\mathbb {T}}\), then \(\sigma (t)=t\)) and \( \sup \emptyset =\inf {\mathbb {T}}\) (i.e., if t is the maximum of t, then \( \rho (t)=t\)), where \(\emptyset \) is the empty set.

A point \(t \in {\mathbb {T}}\) with \(\inf {\mathbb {T}}< t < \sup {\mathbb {T}}\), is said to be right-dense if \(\sigma (t) = t\), left-dense if \(\rho (t) = t\), right-scattered if \(\sigma (t)>t\), and left-scattered if \(\rho (t)<t\). Points that are simultaneously right-dense and left-dense are called dense points, and points that are simultaneously right-scattered and left-scattered are called isolated points.

A function \(g: {\mathbb {T}}\rightarrow {\mathbb {R}}\) is said to be right-dense continuous (rd-continuous) if g is continuous at all right-dense points in \({\mathbb {T}}\), and at all the left-dense points in \({\mathbb {T}}\), the left-sided limits of g exist. The set of all such rd-continuous functions is denoted by \(C_{rd}({\mathbb {T}})\). A function \(f: {\mathbb {T}}\rightarrow {\mathbb {R}}\) is said to be left-dense continuous (ld-continuous) if f is continuous at all the left-dense points in \({\mathbb {T}}\), and at all the right-dense points in \({\mathbb {T}}\), the right-sided limits of f exist. The set of all such ld-continuous functions is denoted by \(C_{ld}({\mathbb {T}})\).

The forward and backward graininess functions \(\mu \) and \(\nu \) for a time scale \({\mathbb {T}}\) are defined by \(\mu (t):= \sigma (t)-t\) and \( \nu (t):=t-\rho (t)\), respectively.

The set \({\mathbb {T}}^{\kappa }\)is introduced as follows: If \({\mathbb {T}}\) has a left–scattered maximum \(t_1\), then \({\mathbb {T}}^{\kappa } = {\mathbb {T}}-\{t_1\} \), otherwise \({\mathbb {T}}^{\kappa } = {\mathbb {T}}\).

The interval [a, b] in \({\mathbb {T}}\) is defined by

$$\begin{aligned}{}[a,b]_{\mathbb {T}}=\{t\in {\mathbb {T}}:a\le t\le b\}. \end{aligned}$$

We define the open intervals and half-closed intervals similarly.

Let \(f: {\mathbb {T}} \rightarrow {\mathbb {R}}\) be a real valued function on a time scale \({\mathbb {T}}\). Then, for \(t\in {\mathbb {T}}^{\kappa }\), we define \( f^\Delta (t)\) to be the number (if it exists) with the property that given any \(\varepsilon > 0\) there is a neighborhood U of t such that, for all \( s\in U\), we have

$$\begin{aligned} |[f(\sigma (t))-f(s)]-f^\Delta (t)[\sigma (t)-s]|\le \varepsilon |\sigma (t)-s|. \end{aligned}$$

In this case, we say that f is delta differentiable on \({\mathbb {T}} ^{\kappa } \) provided \(f^\Delta (t)\) exists for all \(t\in {\mathbb {T}}_{\kappa }\).

For \(f: {\mathbb {T}} \rightarrow {\mathbb {R}}\), we define the function \( f^\sigma : {\mathbb {T}} \rightarrow {\mathbb {R}}\) by \(f^\sigma =f \circ \sigma \), that is \(f^{\sigma }(t)=f(\sigma (t))\), for all \(t\in {\mathbb {T}}\). Similarly, we define the function \(f^{\rho }: {\mathbb {T}} \rightarrow {\mathbb {R}}\) by \( f^{\rho }=f\circ \rho \), that is \(f^{\rho }(t)=f(\rho (t))\), for all \(t\in {\mathbb {T}}\). A time scale \({\mathbb {T}}\) is said to be regular if the following two conditions are satisfied simultaneously: (1) \( \sigma (\rho (t))=t \), and (2) \(\rho (\sigma (t))=t\), for all \(t\in {\mathbb {T}}\).

We frequently use the following important relations between time scales calculus on \({\mathbb {T}}\) and continuous calculus on \({\mathbb {R}}\), and discrete calculus on \({\mathbb {Z}}\). Note that:

(i):

If \({\mathbb {T}}={\mathbb {R}}\), then

$$\begin{aligned} \sigma (t)=t,\ \mu (t)=0,\ f^\Delta (t)=f^{\prime }(t),\ \int \limits _{a}^{b}f(t)\Delta t=\int \limits _{a}^{b}f(t)dt. \end{aligned}$$
(2.3)
(ii):

If \({\mathbb {T}}={\mathbb {Z}}\), then

$$\begin{aligned} \sigma (t)=t+1,\ \mu (t)=1,\ f^\Delta (t)=\Delta f(t),\ \int \limits _{a}^{b}f(t)\Delta t=\sum \limits _{t=a}^{b-1}f(t). \end{aligned}$$
(2.4)

It is known that (see [19]) if \(g \in C_{rd}(T)\), then the definite integral \(G(t):=\int \limits _{t_0}^{t} g(s)\Delta s\) exists, \(t_0 \in {\mathbb {T}}\), and satisfies \(G^\Delta (t) = g(t)\), \(t \in {\mathbb {T}}\). The improper integral is defined as

$$\begin{aligned} \int \limits _{a}^{\infty }f(t)\Delta t=\lim _{b\rightarrow \infty }\int \limits _{a}^{b}f(t)\Delta t. \end{aligned}$$

Lemma 2.1

(Fubini’s Theorem on Time Scales, see [41]) Let f be bounded and \(\Delta \)-integrable over \(R=[a,b)\times [c,d)\) and suppose that the single integrals

$$\begin{aligned} I(t)=\int \limits _{c}^{d}f(t,s)\Delta s \qquad \text {and} \qquad K(s)=\int \limits _{a}^{b}f(t,s)\Delta t \end{aligned}$$

exist for each \(t\in [a,b)\) and for each \(s\in [c,d)\), respectively. Then the iterated integrals

$$\begin{aligned} \int \limits _{a}^{b}\Delta t\int \limits _{c}^{d}f(t,s)\Delta s \qquad \text {and} \qquad \int \limits _{c}^{d}\Delta s\int \limits _{a}^{b}f(t,s)\Delta t \end{aligned}$$

exist and the equality

$$\begin{aligned} \int \limits _{a}^{b}\Delta t\int \limits _{c}^{d}f(t,s)\Delta s=\int \limits _{c}^{d}\Delta s\int \limits _{a}^{b}f(t,s)\Delta t \end{aligned}$$

holds.

Lemma 2.2

(Dynamic Jensen’s Inequality, see[25]) Suppose that \(a,b\in {\mathbb {T}}\) with \(a<b\) and \(-\infty \le \alpha \le \beta \le \infty \). Further, let \(f\in C_{rd}\big ([a,b]_{\mathbb {T}},(\alpha ,\beta )\big )\) and \(h\in C_{rd}\big ([a,b]_{\mathbb {T}},{\mathbb {R}}_+\big )\). If \(\Phi \in C\big ((\alpha ,\beta ),{\mathbb {R}}_+\big )\) is convex, then

$$\begin{aligned} \Phi \left( \frac{\int \limits \limits \limits _{a}^{b}h(s)f(s)\Delta s}{\int \limits \limits ^{b}_{a}h(s)\Delta s}\right) \le \int \limits \limits _{a}^{b}\frac{h(s)\Phi \big (f(s)\big )\Delta s}{\int \limits \limits ^{b}_{a}h(s)\Delta s}. \end{aligned}$$
(2.5)

3 Main Results

We need the following lemma, which gives a two-dimensional dynamic Jensen’s inequality, in the proof of our main results.

Lemma 3.1

Suppose that \(a,b,c,d\in {\mathbb {T}}\) with \(a<b\) and \(c<d\). Further, let \(f\in C_{rd}\big ([a,b]_{\mathbb {T}}\times [c,d]_{\mathbb {T}},(\alpha ,\beta )\big )\), \(g\in C_{rd}\big ([c,d]_{\mathbb {T}},{\mathbb {R}}_+\big )\) and \(h\in C_{rd}\big ([a,b]_{\mathbb {T}},{\mathbb {R}}_+\big )\). If \(\Phi \in C\big ((\alpha ,\beta ),{\mathbb {R}}_+\big )\) is convex, where C is the space of all continuous functions, then

$$\begin{aligned} \Phi \left( \frac{\int \limits \limits _{a}^{b}\int \limits \limits _{c}^{d}g(t)h(s)f(t,s)\Delta t\Delta s}{\int \limits ^{b}_{a}\int \limits ^{d}_{c}g(t)h(s)\Delta t\Delta s}\right) \le \int \limits _{a}^{b}\int \limits _{c}^{d}\frac{g(t)h(s)\Phi \big (f(t,s)\big )\Delta t \Delta s}{\int \limits ^{b}_{a}\int \limits ^{d}_{c}g(t)h(s)\Delta t\Delta s}. \end{aligned}$$
(3.1)

Proof

This lemma is a direct extension of the [25, Theorem 2.2.6]. \(\square \)

Remark 3.2

In Lemma 3.1, if we put \(g(t)=h(t)=1\), we get [30, Theorem 3.1].

Theorem 3.3

Let \(a\in [0,\infty )_{\mathbb {T}}\) and \(u, \lambda \in C_{rd}\big ([a,b)_{\mathbb {T}},{\mathbb {R}}_+\big )\) such that the delta integral \(\displaystyle \int \limits _{x}^{b}\frac{\lambda (t)u(t)}{\int \limits ^{t}_{a}\lambda (x)\Delta x\int \limits ^{\sigma (t)}_{a}\lambda (x)\Delta x}\Delta t\) converges. If \(f\in C_{rd}\big ([a,b)_{\mathbb {T}},(\alpha ,\beta )\big )\) and \(\Phi \in C\big ((\alpha ,\beta ),{\mathbb {R}}\big )\) is convex, then

$$\begin{aligned}{} & {} \int \limits _{a}^{b}\frac{\lambda (x)u(x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\Phi \left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) \Delta x \le \int \limits _{a}^{b}\lambda (x)\Phi \big (f(x)\big )\nonumber \\{} & {} \qquad \quad \left( \int _{x}^{b}\frac{\lambda (t)u(t)}{\int \limits ^{t}_{a}\lambda (x)\Delta x\int \limits ^{\sigma (t)}_{a}\lambda (x)\Delta x}\Delta t\right) \Delta x. \end{aligned}$$
(3.2)

Proof

Employing the dynamic Jensen inequality (2.5) and Fubini’s theorem (Lemma 2.1) on time scales, we obtain

$$\begin{aligned} \int \limits _{a}^{b}\frac{\lambda (x)u(x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\Phi \left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) \Delta x\le & {} \int \limits _{a}^{b}\frac{\lambda (x)u(x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\bigg (\int \limits _{a}^{\sigma (x)}\lambda (t)\Phi \big (f(t)\big )\Delta t\bigg )\Delta x\\= & {} \int \limits _{a}^{b}\lambda (t)\Phi \big (f(t)\big )\left( \int \limits _{t}^{b}\frac{\lambda (x)u(x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\Delta x\right) \Delta t, \end{aligned}$$

which is our desired result. \(\square \)

Below, we present various applications of Theorem 3.3.

Remark 3.4

If we put \(\lambda (t)=1\) in Theorem 3.3, then we recapture Theorem 1.11.

Corollary 3.5

In Theorem 3.3, if \(u(x)=1\) and b is finite, then inequality (3.2) reads

$$\begin{aligned}{} & {} \int \limits _{a}^{b}\frac{\lambda (x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\Phi \left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) \Delta x\nonumber \\{} & {} \quad \le \int \limits _{a}^{b}\lambda (x)\Phi \big (f(x)\big )\left( \int _{x}^{b}\frac{\lambda (t)}{\int \limits ^{t}_{a}\lambda (s)\Delta s\int \limits ^{\sigma (t)}_{a}\lambda (s)\Delta s}\Delta t\right) \Delta x\nonumber \\{} & {} \quad =\int \limits _{a}^{b}\lambda (x)\Phi \big (f(x)\big )\left( \int _{x}^{b}\left[ \frac{-1}{\int \limits ^{t}_{a}\lambda (s)\Delta s}\right] ^{\Delta }\Delta t\right) \Delta x\nonumber \\{} & {} \quad = \int \limits _{a}^{b}\lambda (x)\Phi \big (f(x)\big )\left( \frac{1}{\int \limits ^{x}_{a}\lambda (s)\Delta s}-\frac{1}{\int \limits ^{b}_{a}\lambda (s)\Delta s}\right) \Delta x, \end{aligned}$$
(3.3)

while for \(b\rightarrow \infty \) it becomes

$$\begin{aligned} \int \limits _{a}^{\infty }\frac{\lambda (x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\Phi \left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) \Delta x \le \int \limits _{a}^{\infty }\frac{\lambda (x)\Phi \big (f(x)\big )}{\int \limits ^{x}_{a}\lambda (s)\Delta s}\Delta x. \end{aligned}$$
(3.4)

Corollary 3.6

If we take \(\Phi (u)=u^p\), where \(p>1\) is a constant, then inequalities (3.3) and (3.4) respectively be

$$\begin{aligned}{} & {} \int \limits _{a}^{b}\frac{\lambda (x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) ^p\Delta x \le \int \limits _{a}^{b}\lambda (x)f^p(x)\left( \frac{1}{\int \limits ^{x}_{a}\lambda (s)\Delta s}-\frac{1}{\int \limits ^{b}_{a}\lambda (s)\Delta s}\right) \Delta x, \\{} & {} \quad \qquad \int \limits _{a}^{\infty }\frac{\lambda (x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) ^p\Delta x \le \int \limits _{a}^{\infty }\frac{\lambda (x)f^p(x)}{\int \limits ^{x}_{a}\lambda (s)\Delta s}\Delta x. \end{aligned}$$

Remark 3.7

If we put \(\lambda (t)=1\) in Corollary 3.5 and 3.6, then we recapture Corollary 2.1 and Corollary 2.2 in [30] respectively, which is a time scale version of Hardy’s inequality.

Corollary 3.8

If we take \(\Phi (u)=\exp (u)\) and replace f by \(\ln f\), then inequalities (3.3) and (3.4) respectively be

$$\begin{aligned}{} & {} \int \limits _{a}^{b}\frac{\lambda (x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\exp \left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)\ln f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) \Delta x \le \int \limits _{a}^{b}\lambda (x)f(x)\left( \frac{1}{\int \limits ^{x}_{a}\lambda (s)\Delta s}-\frac{1}{\int \limits ^{b}_{a}\lambda (s)\Delta s}\right) \Delta x, \\{} & {} \quad \qquad \int \limits _{a}^{\infty }\frac{\lambda (x)}{\int \limits ^{x}_{a}\lambda (t)\Delta t}\exp \left( \frac{\int \limits _{a}^{\sigma (x)}\lambda (t)\ln f(t)\Delta t}{\int \limits ^{\sigma (x)}_{a}\lambda (t)\Delta t}\right) \Delta x \le \int \limits _{a}^{\infty }\frac{\lambda (x)f(x)}{\int \limits ^{x}_{a}\lambda (s)\Delta s}\Delta x. \end{aligned}$$

Remark 3.9

If we put \(\lambda (t)=1\) in Corollary 3.8. then we recapture Corollary 2.3 in [30].

Corollary 3.10

If \({\mathbb {T}}={\mathbb {R}}\) in Theorem 3.3, then, using relations (2.3), inequality (3.2) reduces to

$$\begin{aligned} \int \limits _{a}^{b}\frac{\lambda (x)u(x)}{\int \limits ^{x}_{a}\lambda (t)dt}\Phi \left( \frac{\int \limits _{a}^{x}\lambda (t)f(t)dt}{\int \limits ^{x}_{a}\lambda (t)dt}\right) dx \le \int \limits _{a}^{b}\lambda (x)\Phi \big (f(x)\big )\left( \int _{x}^{b}\frac{\lambda (t)u(t)}{\bigg (\int \limits ^{t}_{a}\lambda (x)dx\bigg )^{2} }dt\right) dx. \end{aligned}$$

Corollary 3.11

If \({\mathbb {T}}={\mathbb {Z}}\) in Theorem 3.3, then, using relations (2.4), inequality (3.2) reduces to

$$\begin{aligned} \sum \limits _{x=a}^{b-1}\frac{\lambda (x)u(x)}{\sum \nolimits _{t=a}^{x-1}\lambda (t)} \Phi \left( \frac{\sum \nolimits _{t=a}^{x}\lambda (t)f(t)}{\sum \nolimits ^{x}_{t=a}\lambda (t)}\right) \le \sum \limits _{x=a}^{b-1}\lambda (x)\Phi \big (f(x)\big )\left( \sum _{t=x}^{b-1}\frac{\lambda (t)u(t)}{\sum \nolimits ^{t}_{x=a}\lambda (x) \sum \nolimits ^{t-1}_{x=a}\lambda (x)}\right) . \end{aligned}$$

Theorem 3.12

Suppose that a, b, c, \(d\in [0,\infty )_{\mathbb {T}}\), \(f\in C_{rd}\big ([a,b)_{\mathbb {T}}\times [c,d)_{\mathbb {T}},{\mathbb {R}}\big )\), \(g\in C_{rd}\big ([c,d)_{\mathbb {T}},{\mathbb {R}}_+\big )\) and \(h\in C_{rd}\big ([a,b)_{\mathbb {T}},{\mathbb {R}}_+\big )\). If \(\Phi \in C\big ((\alpha ,\beta ),{\mathbb {R}}_+\big )\) is convex, then

$$\begin{aligned}{} & {} \int \limits \limits _{a}^{b}\int \limits \limits _{c}^{d}\frac{g(x)h(y)}{\int \limits _{a}^{x}\int \limits _{c}^{y}g(t)h(s)\Delta t \Delta s}\Phi \left( \frac{\int \limits \limits _{a}^{\sigma (x)}\int \limits \limits _{c}^{\sigma (y)}g(t)h(s)f(t,s)\Delta t\Delta s}{\int \limits ^{\sigma (x)}_{a}\int \limits ^{\sigma (y)}_{c}g(t)h(s)\Delta t\Delta s}\right) \Delta y\Delta x\nonumber \\{} & {} \qquad \le \int \limits _{a}^{b}\int \limits \limits _{c}^{d}g(t)h(s)\Phi \big (f(t,s)\big )\left( \frac{1}{\int \limits ^{s}_{a}h(\tau )\Delta \tau }-\frac{1}{\int \limits ^{b}_{a}h(\tau )\Delta \tau }\right) \left( \frac{1}{\int \limits ^{t}_{c}g(\tau )\Delta \tau }-\frac{1}{\int \limits ^{d}_{c}g(\tau )\Delta \tau }\right) \Delta t\Delta s.\nonumber \\ \end{aligned}$$
(3.5)

Proof

By using two-dimensional dynamic Jensen inequality (3.1) and Fubini’s theorem on time scales, we get

$$\begin{aligned}{} & {} \int \limits \limits _{a}^{b}\int \limits \limits _{c}^{d}\frac{g(x)h(y)}{\int \limits _{a}^{x}\int \limits _{c}^{y}g(t)h(s)\Delta t \Delta s} \Phi \left( \frac{\int \limits \limits _{a}^{\sigma (x)}\int \limits \limits _{c}^{\sigma (y)}g(t)h(s)f(t,s)\Delta t\Delta s}{\int \limits ^{\sigma (x)}_{a}\int \limits ^{\sigma (y)}_{c}g(t)h(s)\Delta t\Delta s}\right) \Delta y\Delta x\\{} & {} \quad \le \int \limits _{a}^{b}\int \limits _{c}^{d}\frac{g(x)h(y)}{\int \limits ^{x}_{a}\int \limits ^{y}_{c}g(t)h(s)\Delta t\Delta s\int \limits ^{\sigma (x)}_{a}\int \limits ^{\sigma (y)}_{c}g(t)h(s)\Delta t\Delta s}\\{} & {} \qquad \times \left( \int \limits _{a}^{\sigma (x)}\int \limits _{c}^{\sigma (y)}g(t)h(s)\Phi \big (f(t,s)\big )\Delta t \Delta s\right) \Delta y\Delta x\\{} & {} \quad =\int \limits _{a}^{b}\int \limits _{c}^{d}g(t)h(s)\Phi \big (f(t,s)\big )\left( \int \limits _{s}^{b}\int \limits _{t}^{d} \frac{g(x)h(y)\Delta y\Delta x}{\int \limits _{a}^{x}\int \limits _{c}^{y}g(t)h(s)\Delta t \Delta s\int \limits ^{\sigma (x)}_{a}\int \limits ^{\sigma (y)}_{c}g(t)h(s)\Delta t\Delta s}\right) \Delta t \Delta s\\{} & {} \quad =\int \limits _{a}^{b}\int \limits \limits _{c}^{d}g(t)h(s)\Phi \big (f(t,s)\big )\left( \frac{1}{\int \limits ^{s}_{a} h(\tau )\Delta \tau }-\frac{1}{\int \limits ^{b}_{a}h(\tau )\Delta \tau }\right) \\{} & {} \qquad \times \left( \frac{1}{\int \limits ^{t}_{c}g(\tau )\Delta \tau }-\frac{1}{\int \limits ^{d}_{c}g(\tau )\Delta \tau }\right) \Delta t\Delta s. \end{aligned}$$

This concludes the proof. \(\square \)

Remark 3.13

If we put \(g(t)=h(t)=1\) in Theorem 3.12, then we recapture [30, Theorem 3.2].

Corollary 3.14

In Theorem 3.12, if we take \(\Phi (u)=u^p\), where \(p>1\) is a constant, then we have

$$\begin{aligned}{} & {} \int \limits \limits _{a}^{b}\int \limits \limits _{c}^{d}\frac{g(x)h(y)}{\int \limits _{a}^{x}\int \limits _{c}^{y}g(t)h(s)\Delta t \Delta s}\left( \frac{\int \limits \limits _{a}^{\sigma (x)}\int \limits \limits _{c}^{\sigma (y)}g(t)h(s)f(t,s)\Delta t\Delta s}{\int \limits ^{\sigma (x)}_{a}\int \limits ^{\sigma (y)}_{c}g(t)h(s)\Delta t\Delta s}\right) ^p\Delta y\Delta x\\ \nonumber{} & {} \le \int \limits _{a}^{b}\int \limits \limits _{c}^{d}g(t)h(s)f^p(t,s)\left( \frac{1}{\int \limits ^{s}_{a}h(\tau )\Delta \tau }-\frac{1}{\int \limits ^{b}_{a}h(\tau )\Delta \tau }\right) \left( \frac{1}{\int \limits ^{t}_{c}g(\tau )\Delta \tau }-\frac{1}{\int \limits ^{d}_{c}g(\tau )\Delta \tau }\right) \Delta t\Delta s. \end{aligned}$$

Corollary 3.15

In Theorem 3.12, if we take \(\Phi (u)=\exp (u)\) and replace f by \(\ln f\), then we have

$$\begin{aligned}{} & {} \int \limits \limits _{a}^{b}\int \limits \limits _{c}^{d}\frac{g(x)h(y)}{\int \limits _{a}^{x}\int \limits _{c}^{y}g(t)h(s)\Delta t \Delta s}\exp \left( \frac{\int \limits \limits _{a}^{\sigma (x)}\int \limits \limits _{c}^{\sigma (y)}g(t)h(s)\ln f(t,s)\Delta t\Delta s}{\int \limits ^{\sigma (x)}_{a}\int \limits ^{\sigma (y)}_{c}g(t)h(s)\Delta t\Delta s}\right) \Delta y\Delta x\\{} & {} \qquad \le \int \limits _{a}^{b}\int \limits \limits _{c}^{d}g(t)h(s)f(t,s)\left( \frac{1}{\int \limits ^{s}_{a}h(\tau )\Delta \tau }-\frac{1}{\int \limits ^{b}_{a}h(\tau )\Delta \tau }\right) \left( \frac{1}{\int \limits ^{t}_{c}g(\tau )\Delta \tau }-\frac{1}{\int \limits ^{d}_{c}g(\tau )\Delta \tau }\right) \Delta t\Delta s. \end{aligned}$$

Corollary 3.16

If \({\mathbb {T}}={\mathbb {R}}\) in Theorem 3.12, then, using relations (2.3), inequality (3.5) reduces to

$$\begin{aligned}{} & {} \int \limits \limits _{a}^{b}\int \limits \limits _{c}^{d}\frac{g(x)h(y)}{\int \limits _{a}^{x}\int \limits _{c}^{y}g(t)h(s)dtds} \Phi \left( \frac{\int \limits \limits _{a}^{x}\int \limits \limits _{c}^{y}g(t)h(s)f(t,s)dtds}{\int \limits ^{x}_{a}\int \limits ^{y}_{c}g(t)h(s)dtds}\right) dydx\\{} & {} \quad \quad \le \int \limits _{a}^{b}\int \limits \limits _{c}^{d}g(t)h(s)\Phi \big (f(t,s)\big )\left( \frac{1}{\int \limits ^{s}_{a}h(\tau ) d\tau }-\frac{1}{\int \limits ^{b}_{a}h(\tau )d\tau }\right) \left( \frac{1}{\int \limits ^{t}_{c}g(\tau )d\tau } -\frac{1}{\int \limits ^{d}_{c}g(\tau )d\tau }\right) dtds. \end{aligned}$$

Corollary 3.17

If \({\mathbb {T}}={\mathbb {Z}}\) in Theorem 3.12, then, using relations (2.4), inequality (3.5) reduces to

$$\begin{aligned}{} & {} \sum \limits \limits _{x=a}^{b-1}\sum \limits \limits _{y=c}^{d-1}\frac{g(x)h(y)}{\sum \limits _{s=a}^{x-1}\sum \limits _{t=c}^{y-1}g(t)h(s)} \Phi \left( \frac{\sum \limits \limits _{s=a}^{x}\sum \limits \limits _{t=c}^{y}g(t)h(s)f(t,s)}{\sum \limits ^{x}_{s=a}\sum \limits ^{y}_{t=c}g(t)h(s)}\right) \\{} & {} \quad \le \sum \limits _{s=a}^{b-1}\sum \limits \nolimits _{t=c}^{d-1}g(t)h(s)\Phi \big (f(t,s)\big )\\{} & {} \qquad \left( \frac{1}{\sum \limits ^{s-1}_{\tau =a}h(\tau )} -\frac{1}{\sum \limits ^{b-1}_{\tau =a}h(\tau )}\right) \left( \frac{1}{\sum \nolimits ^{t-1}_{\tau =c}g(\tau )} -\frac{1}{\sum \limits ^{d-1}_{\tau =c}g(\tau )}\right) . \end{aligned}$$

Our aim in the following theorem is to establish a dynamic Hardy inequality for several functions.

Theorem 3.18

Assume that \(a\in [0,\infty )_{\mathbb {T}}\) and \(\lambda , f_1, f_2, \dots , f_n\in C_{rd}\big ([a,\infty )_{\mathbb {T}},{\mathbb {R}}_+\big )\). Define \(\Lambda (t):=\int \nolimits _{a}^{t}\lambda (s)\Delta s\) and \(F_k(t):=\int \nolimits _{a}^{t}\lambda (s)f_k(s)\Delta s\) for \(k=1,2,\dots ,n\). If \(p\ge q>1\), then

$$\begin{aligned}{} & {} \int \nolimits _{a}^{\infty }\lambda (t)\frac{\big (F_1^\sigma (t)F_2^\sigma (t)\dots F_n^\sigma (t)\big )^{p/n}}{\big (\Lambda ^\sigma (t)\big )^q}\Delta t\nonumber \\{} & {} \quad \le \Big (\frac{p}{nq-n}\Big )^p\int \nolimits _{a}^{\infty }\frac{\lambda (t)\big (\Lambda ^\sigma (t)\big )^{q(p-1)}}{\Lambda ^{p(q-1)}(t)} \big (f_1(t)+f_2(t)+\dots +f_n(t)\big )^p\Delta t. \qquad \quad \nonumber \\ \end{aligned}$$
(3.6)

Proof

By utilizing the discrete Jensen inequality, we have

$$\begin{aligned} \big (F_1^\sigma (t)F_2^\sigma (t)\dots F_n^\sigma (t)\big )^{1/n}\le \frac{\sum \nolimits _{k=1}^{n}F_k^\sigma (t)}{n}, \end{aligned}$$

and thus

$$\begin{aligned} \big (F_1^\sigma (t)F_2^\sigma (t)\dots F_n^\sigma (t)\big )^{p/n}\le \frac{\Big (\sum \nolimits _{k=1}^{n}F_k^\sigma (t)\Big )^p}{n^p}. \end{aligned}$$
(3.7)

Multiplying both sides of (3.7) by \(\lambda (t)/\big (\Lambda ^\sigma (t)\big )^q\) and integrating the resulting inequality over t from a to \(\infty \) yield

$$\begin{aligned} \int \nolimits _{a}^{\infty }\lambda (t)\frac{\big (F_1^\sigma (t)F_2^\sigma (t)\dots F_n^\sigma (t)\big )^{p/n}}{\big (\Lambda ^\sigma (t)\big )^q}\Delta t\le \frac{1}{n^p}\int \limits _{a}^{\infty }\lambda (t)\frac{\Big (\sum \nolimits _{k=1}^{n}F_k^\sigma (t)\Big )^p}{\big (\Lambda ^\sigma (t)\big )^q}\Delta t. \end{aligned}$$

Applying inequality (1.13) to the right-hand side of the last inequality implies

$$\begin{aligned}{} & {} \int \limits _{a}^{\infty }\lambda (t)\frac{\big (F_1^\sigma (t)F_2^\sigma (t)\dots F_n^\sigma (t)\big )^{p/n}}{\big (\Lambda ^\sigma (t)\big )^q}\Delta t\\{} & {} \quad \le \Big (\frac{p}{nq-n}\Big )^p\int \limits _{a}^{\infty }\frac{\lambda (t)\big (\Lambda ^\sigma (t)\big )^{q(p-1)}}{\Lambda ^{p(q-1)}(t)} \big (f_1(t)+f_2(t)+\dots +f_n(t)\big )^p\Delta t. \end{aligned}$$

The proof is complete. \(\square \)

Remark 3.19

If we put \(\lambda (t)=1\) in Theorem 3.18, then we recapture [30, Theorem 4.1].

Corollary 3.20

If \({\mathbb {T}}={\mathbb {R}}\) in Theorem 3.18, then, using relation (2.3), inequality (3.6) reduces to

$$\begin{aligned}{} & {} \int \limits _{a}^{\infty }\lambda (t)\frac{\big (F_1(t)F_2(t)\dots F_n(t)\big )^{p/n}}{\Lambda ^q(t)}dt\\{} & {} \quad \le \Big (\frac{p}{nq-n}\Big )^p\int \limits _{a}^{\infty }\lambda (t)\Lambda ^{p-q}(t)\big (f_1(t)+f_2(t)+\dots +f_n(t)\big )^p dt, \end{aligned}$$

where \(\Lambda (t):=\int \limits _{a}^{t}\lambda (s)ds\) and \(F_k(t)=\int \limits _{a}^{t}\lambda (s)f_k(s)ds\) for \(k=1,2,\dots ,n\).

Corollary 3.21

If \({\mathbb {T}}={\mathbb {Z}}\) in Theorem 3.18, then, using relation (2.4), inequality (3.6) reduces to

$$\begin{aligned}{} & {} \sum \limits _{t=a}^{\infty }\lambda (t)\frac{\big (F_1(t+1)F_2(t+1)\dots F_n(t+1)\big )^{p/n}}{\Lambda ^q(t+1)}\\{} & {} \quad \le \Big (\frac{p}{nq-n}\Big )^p\sum \limits _{t=a}^{\infty }\frac{\lambda (t)\big (\Lambda (t+1)\big )^{q(p-1)}}{\Lambda ^{p(q-1)}(t)} \big (f_1(t)+f_2(t)+\dots +f_n(t)\big )^p, \end{aligned}$$

where \(\Lambda (t):=\sum \nolimits _{s=a}^{t-1}\lambda (s)\) and \(F_k(t)=\sum \nolimits _{s=a}^{t-1}\lambda (s)f_k(s)\) for \(k=1,2,\dots ,n\).

4 Conclusion

Hardy-type inequalities have many applications and are subject to strong research: see the books [4, 16, 42] and the recent publications [29, 35, 36, 38, 43]. In this manuscript, by employing the Fubini’s theorem and Jensen inequality on time scales,, several new Hardy-type inequalities are proved. The results extend several dynamic inequalities known in the literature, being new even in the discrete and continuous. As a future work, will try to generalize these inequalities by using \(\alpha \)- conformable calculus.