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Some Hermite–Hadamard and Opial dynamic inequalities on time scales

Abstract

In this article, we are interested in some well-known dynamic inequalities on time scales. For this reason, we will prove some new Hermite–Hadamard (H-H) and Opial dynamic inequalities on time scales. The main results here will be derived via the dynamic integration by parts and chain rule formulas on time scales. In addition, we will extend and unify the inequalities for the convex functions.

Introduction

In 1893, the H-H inequality was established for a convex function on a given interval \([d_{1},d_{2}]\) in [1]:

$$\begin{aligned} \wp \biggl(\frac{d_{1}+d_{2}}{2} \biggr)&\leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp (x) \,\mathrm{d}x\leq \frac{ \wp (d_{1})+ \wp (d_{2})}{2}. \end{aligned}$$
(1.1)

The study of the H-H inequality have been attracted the attention of many scholars. In recent years, many refinements, generalizations, and extensions have been made to the inequality (1.1); we advise the interested reader to visit the published papers [28] and the references cited therein.

After the H-H inequality and in 1960, Opial [9] established another important integral inequality, called in the literature Opial’s integral inequality, which is as follows:

$$ \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s}) \wp '(\mathbf{s}) \bigr\vert \,\mathrm{d}\mathbf{s} \leq \frac{\mu }{4} \int _{0}^{\mu } \bigl( \wp '( \mathbf{s}) \bigr)^{2} \,\mathrm{d}\mathbf{s}, $$
(1.2)

where \(\wp (\mathbf{s})\in C^{1}[0,\mu ]\) with \(\wp (0)= \wp (\mu )=0\) and \(\wp (\mathbf{s})>0\) for \(\mathbf{s}\in (0,\mu )\). A best possibility here is \(\frac{\mu }{4}\). Inequality (1.2) with their extensions play a great role in analysis and its applications. The interest in inequality (1.2) comes from their mathematical structure. Many results concerning the generalizations and extensions of this inequality have been established; see [1017].

The inequalities (1.1) and (1.2) have been proved not only for the ordinary order but also for various fractional models, for example, the Riemann–Liouville model, the Atangana–Baleanu model, the tempered fractional model, the Caputo–Fabrizo model, and the conformable model; see [1822] and the references therein.

The use of dynamic system to study the continuous and discrete times is well studied, especially for the real-world modeling issues. It is better to check if structures can be given that encourage us in integrating all dynamic systems at the same time to derive a superior and perspective comprehension of the contrasts between continuous and discrete domains. In fact, constructing a correlation between discrete and continuous situations is the primary aim of dynamic equations on time scales. It is well known that the theory of time scales was originated by Hilger in his Ph.D. thesis [23]. After that, this setting was evolved by many researchers, for more details refer to [24, 25].

Over the recent couple of years, there has been growing interest in the study of dynamic inequalities on time scales and this has become an important field in applied and pure mathematics; see for details [2530].

This article is devoted to establishing some dynamic H-H and Opial inequalities on time scales. The obtained inequalities will extend some known integral inequalities, and extend and unify some continuous inequalities.

The article consists of five sections. Section 1 is for the introduction. In Sect. 2 we present basic concepts and preliminaries of time scale notations, and in Sect. 3 we discuss and derive some dynamic inequalities of H-H on time scales. Opial dynamic inequalities will be discussed in Sect. 4. Section 5 concludes the article finally.

Preliminaries

This section deals with recalling time scale notation and basic lemmas on Steffensen inequalities on time scales. Let R be the set of real numbers, then a time scale \(\mathrm{T_{0}}\) is a nonempty and closed subset of R. For \(\iota \in \mathrm{T_{0}}\), the forward and backward jump operators \(\sigma , \rho :\mathrm{T_{0}}\to \mathrm{R}\) are, respectively, defined by

$$\begin{aligned} \sigma (\iota )=\inf \{n\in \mathrm{T_{0}}: n>\iota \} \quad \text{and}\quad \rho (\iota )=\sup \{n\in \mathrm{T_{0}}: n< \iota \}. \end{aligned}$$

We define the graininess function \(\wp : \mathrm{T_{0}}\to [0, \infty )\) by \(\wp (\iota )=\sigma (\iota )-\iota \). An element \(\iota \in \mathrm{T_{0}}\) is said to be left-dense if \(\rho (\iota )=\iota \) and left-scattered if \(\rho (\iota )<\iota \), and right-dense if \(\sigma (\iota )=\iota \) and right-scattered if \(\sigma (\iota )>\iota \). The set \(\mathrm{T_{0}}^{k}\) is defined to be \(\mathrm{T_{0}}\) if it has a left-scattered maximum \(\wp _{2}\), then \(\mathrm{T_{0}}^{k}=\mathrm{T_{0}}-\{\wp _{2}\}\) otherwise \(\mathrm{T_{0}}^{k}=\mathrm{T_{0}}\). For further information on these notions we refer the reader to Refs. [24, 25].

Definition 2.1

([25])

Assume that \(\wp : \mathrm{T_{0}}\to \mathrm{R}\) is a real-valued function. Then we say is \(\mathcal{RD}\)-continuous on R if its left limit is finite at any left-dense point of \(\mathrm{T_{0}}\) and it is continuous on every right-dense point of \(\mathrm{T_{0}}\).

Definition 2.2

([25])

Assume that \(\wp : \mathrm{T_{0}}\to \mathrm{R}\) is a real-valued function. Then we say is \(\mathcal{LD}\)-continuous on R if its right limit is finite at any right-dense point of \(\mathrm{T_{0}}\) and it is continuous on every left-dense point of \(\mathrm{T_{0}}\).

Theorem 2.1

([25])

Let \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\). Let \(\wp _{1}, \wp _{2}: [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be Δ-integrable functions such that \(\wp _{1}\) of one sign and decreasing and \(0\leq \wp _{2}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\). Also, suppose that \(\vartheta _{1}, \vartheta _{2}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\) such that

$$\begin{aligned} & d_{2}-\vartheta _{1}\leq \int _{d_{1}}^{d_{2}}\wp _{2}(\mathbf{s}) \Delta \mathbf{s}\leq \vartheta _{2}-d_{1}, \quad \textit{if } \wp _{1}( \mathbf{s})>0, \forall \mathbf{s}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}, \end{aligned}$$
(2.1)
$$\begin{aligned} &\vartheta _{2}-d_{1}\leq \int _{d_{1}}^{d_{2}}\wp _{2}(\mathbf{s}) \Delta \mathbf{s}\leq d_{2}-\vartheta _{1}, \quad \textit{if }\wp _{1}( \mathbf{s})< 0, \forall \mathbf{s}\in [d_{1},d_{2}]_{ \mathrm{T_{0}}}, \end{aligned}$$
(2.2)

then

$$ \int _{\vartheta _{1}}^{d_{2}}\wp _{1}(\mathbf{s}) \Delta \mathbf{s} \leq \int _{d_{1}}^{d_{2}}\wp _{1}(\mathbf{s}) \wp _{2}(\mathbf{s}) \Delta \mathbf{s}\leq \int _{d_{1}}^{\vartheta _{2}}\wp _{1}( \mathbf{s}) \Delta \mathbf{s}. $$
(2.3)

Theorem 2.2

([25])

Let \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\). Let \(\wp _{1}, \wp _{2}: [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be -integrable functions such that \(\wp _{1}\) is of one sign and decreasing and \(0\leq \wp _{2}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\). Also, suppose that \(\vartheta _{1}, \vartheta _{2}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\) such that

$$\begin{aligned} d_{2}-\vartheta _{1} &\leq \int _{d_{1}}^{d_{2}}\wp _{2}(\mathbf{s}) \nabla \mathbf{s}\leq \vartheta _{2}-d_{1}, \quad \textit{if } \wp _{1}( \mathbf{s})>0, \forall \mathbf{s}\in [d_{1},d_{2}]_{ \mathrm{T_{0}}}, \\ \vartheta _{2}-d_{1} &\leq \int _{d_{1}}^{d_{2}}\wp _{2}(\mathbf{s}) \nabla \mathbf{s}\leq d_{2}-\vartheta _{1}, \quad \textit{if } \wp _{1}( \mathbf{s})< 0, \forall \mathbf{s}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}, \end{aligned}$$

then

$$ \int _{\vartheta _{1}}^{d_{2}}\wp _{1}(\mathbf{s}) \nabla \mathbf{s} \leq \int _{d_{1}}^{d_{2}}\wp _{1}(\mathbf{s}) \wp _{2}(\mathbf{s}) \nabla \mathbf{s}\leq \int _{d_{1}}^{\vartheta _{2}}\wp _{1}( \mathbf{s}) \nabla \mathbf{s}. $$
(2.4)

Theorem 2.3

([25])

Let \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\). Let \(\wp _{1}, \wp _{2}: [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be Δ-integrable functions such that \(\wp _{1}\) of one sign and decreasing and \(0\leq \wp _{2}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\). Suppose that

$$\begin{aligned} \lambda :=\int _{d_{1}}^{d_{2}}\wp _{2}(\mathbf{s}) \Delta \mathbf{s} \quad \textit{provided that } d_{2}-\lambda , d_{1}+ \lambda \in \mathrm{T_{0}}, \end{aligned}$$

then

$$ \int _{d_{2}-\lambda }^{d_{2}}\wp _{1}(\mathbf{s}) \Delta \mathbf{s} \leq \int _{d_{1}}^{d_{2}}\wp _{1}(\mathbf{s}) \wp _{2}(\mathbf{s}) \Delta \mathbf{s}\leq \int _{d_{1}}^{d_{1}+\lambda }\wp _{1}( \mathbf{s}) \Delta \mathbf{s}. $$
(2.5)

Theorem 2.4

([25])

Let \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\). Let \(\wp _{1}, \wp _{2}: [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be -integrable functions such that \(\wp _{1}\) of one sign and decreasing and \(0\leq \wp _{2}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\). Suppose that

$$\begin{aligned} \lambda :=\int _{d_{1}}^{d_{2}}\wp _{2}(\mathbf{s}) \nabla \mathbf{s} \quad \textit{such that } d_{2}-\lambda , d_{1}+\lambda \in \mathrm{T_{0}}, \end{aligned}$$

then

$$ \int _{d_{2}-\lambda }^{d_{2}}\wp _{1}(\mathbf{s}) \nabla \mathbf{s} \leq \int _{d_{1}}^{d_{2}}\wp _{1}(\mathbf{s}) \wp _{2}(\mathbf{s}) \nabla \mathbf{s}\leq \int _{d_{1}}^{d_{1}+\lambda }\wp _{1}( \mathbf{s}) \nabla \mathbf{s}. $$
(2.6)

Theorem 2.5

(Δ-integration by parts [24, 25])

Let \(\wp _{1}, \wp _{2}: [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp _{1}, \wp _{2}\in \mathbb{C}_{\mathcal{RD}}\) and \(d_{1}, d_{2}\in \mathrm{T_{0}}\). Then the integration by parts in the sense of Δ is formulated as follows:

$$\begin{aligned} \int _{d_{1}}^{d_{2}}\wp _{1}(\mathbf{s})\wp _{2}^{\Delta }(\mathbf{s}) \Delta \mathbf{s}=\wp _{1}(\mathbf{s})\wp _{2}(\mathbf{s})|_{d_{1}}^{d_{2}}- \int _{d_{1}}^{d_{2}}\wp _{1}^{\Delta }( \mathbf{s})\wp _{2}^{\sigma }( \mathbf{s})\Delta \mathbf{s}, \end{aligned}$$
(2.7)

Theorem 2.6

(-integration by parts [17, 24, 25])

Let \(\wp _{1}, \wp _{2}: [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp _{1}, \wp _{2}\in \mathbb{C}_{\mathcal{LD}}\) and \(d_{1}, d_{2}\in \mathrm{T_{0}}\). Then the integration by parts in the sense of is formulated as follows:

$$\begin{aligned} \int _{d_{1}}^{d_{2}}\wp _{1}(\mathbf{s})\wp _{2}^{\nabla }(\mathbf{s}) \nabla \mathbf{s}=\wp _{1}(\mathbf{s})\wp _{2}(\mathbf{s})|_{d_{1}}^{d_{2}}- \int _{d_{1}}^{d_{2}}\wp _{1}^{\nabla }( \mathbf{s})\wp _{2}^{\rho }( \mathbf{s})\nabla \mathbf{s}. \end{aligned}$$
(2.8)

Definition 2.3

([31])

Let \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\). A function \(\wp :\mathrm{T_{0}}\to \mathbb{R}\) is said to be convex on \(\mathrm{T_{0}}\), if

$$\begin{aligned} \wp \bigl(\vartheta d_{1}+(1-\vartheta )d_{2}\bigr)\leq \vartheta \wp (d_{1})+(1- \vartheta )\wp (d_{2}) \end{aligned}$$

holds for each \(\vartheta \in \mathrm{T_{0}}_{[d_{1},d_{2}]}\subseteq [0,1]\).

Dynamic H-H inequalities

Theorem 3.1

Let \(\wp : [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be convex and monotonic and \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\). Suppose that also \(\vartheta _{1}, \vartheta _{2}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\), then we have

$$\begin{aligned} \wp \biggl(\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \frac{2 \wp (d_{1})- \wp (\vartheta _{1})- \wp (\vartheta _{2})+2 \wp (d_{2})}{2}, \end{aligned}$$
(3.1)

such that \(\frac{d_{1}+d_{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2}\in \mathrm{T_{0}}\) and

$$\begin{aligned} \vartheta _{1}, \vartheta _{2}\geq \frac{d_{1}+3d_{2}}{4}, \quad \textit{if $ \wp $ is decreasing}, \\ \vartheta _{1}, \vartheta _{2}\leq \frac{d_{1}+3d_{2}}{4}, \quad \textit{if $ \wp $ is increasing}. \end{aligned}$$

Proof

Suppose that is decreasing and convex. It follows that \(\wp ^{\Delta }\leq 0\). Set \(\Psi _{1} :=- \wp ^{\Delta }\), then it is clear that \(\Psi _{1}\) is decreasing and \(\Psi _{1}\geq 0\). If we choose \(\Psi _{2}(\mathbf{s}):=\frac{2(d_{2}-\mathbf{s})}{d_{2}-d_{1}}\), we see that \(0\leq \Psi _{2}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [\frac{d_{1}+d_{2}}{2},d_{2} ]\). Now, by making use of inequality (2.1) with \(\wp _{2}(\mathbf{s})=\Psi _{2}(\mathbf{s})= \frac{2(d_{2}-\mathbf{s})}{d_{2}-d_{1}}\), we get

$$\begin{aligned} d_{2}-\vartheta _{1} &\leq \frac{d_{2}-d_{1}}{4} \leq \vartheta _{2}- \frac{d_{1}+d_{2}}{2}. \end{aligned}$$

This implies that \(\vartheta _{1}, \vartheta _{2}\geq \frac{d_{1}+3d_{2}}{4}\). Thus, \(\Psi _{1}\) and \(\Psi _{2}\) satisfy the hypotheses in Theorem 2.1 and therefore

$$ \int _{\vartheta _{1}}^{d_{2}}\Psi _{1}(\mathbf{s}) \Delta \mathbf{s} \leq \int _{\frac{d_{1}+d_{2}}{2}}^{d_{2}}\Psi _{1}(\mathbf{s}) \Psi _{2}(\mathbf{s})\Delta \mathbf{s} \leq \int _{ \frac{d_{1}+d_{2}}{2}}^{\vartheta _{2}}\Psi _{1}(\mathbf{s}) \Delta \mathbf{s}. $$
(3.2)

By using Δ-integration by parts (Theorem 2.5), we have

$$\begin{aligned} \int _{\frac{d_{1}+d_{2}}{2}}^{d_{2}}\Psi _{1}(\mathbf{s}) \Psi _{2}( \mathbf{s})\Delta \mathbf{s}= \wp \biggl( \frac{d_{1}+d_{2}}{2} \biggr) -\frac{2}{d_{2}-d_{1}} \int _{\frac{d_{1}+d_{2}}{2}}^{d_{2}} \wp ^{ \sigma }(\mathbf{s}) \Delta \mathbf{s}. \end{aligned}$$

Then, by making use of the above and the fact \(\int _{x_{1}}^{x_{2}}\Psi _{1}(\mathbf{s})\Delta \mathbf{s}= \wp (x_{1})- \wp (x_{2})\) in the inequality (3.2), we get

$$\begin{aligned} \wp (\vartheta _{1})- \wp (d_{2}) \leq \wp \biggl( \frac{d_{1}+d_{2}}{2} \biggr) -\frac{2}{d_{2}-d_{1}} \int _{ \frac{d_{1}+d_{2}}{2}}^{d_{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \wp \biggl(\frac{d_{1}+d_{2}}{2} \biggr)- \wp ( \vartheta _{2}). \end{aligned}$$

This simplifies to

$$\begin{aligned} \wp (\vartheta _{2}) \leq \frac{2}{d_{2}-d_{1}} \int _{ \frac{d_{1}+d_{2}}{2}}^{d_{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \frac{ \wp (d_{1})+ \wp (d_{2})}{2}+ \wp (d_{2})- \wp ( \vartheta _{1}), \end{aligned}$$
(3.3)

where we used the convexity of to get \(\wp (\frac{d_{1}+d_{2}}{2} )\leq \frac{ \wp (d_{1})+ \wp (d_{2})}{2}\).

On the other hand, if we choose \(\Psi _{3}(\mathbf{s}):=\frac{2(\mathbf{s}-d_{1})}{d_{2}-d_{1}}\), we see that \(0\leq \Psi _{3}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [d_{1},\frac{d_{1}+d_{2}}{2} ]\). Again, by making use of inequality (2.1) for the new \(\Psi _{3}(\mathbf{s})\), we find

$$\begin{aligned} \frac{d_{1}+d_{2}}{2}-\vartheta _{1} &\leq \frac{d_{2}-d_{1}}{4} \leq \vartheta _{2}-d_{1}. \end{aligned}$$

This implies that \(\vartheta _{1}, \vartheta _{2}\geq \frac{3d_{1}+d_{2}}{4}< \frac{d_{1}+3d_{2}}{4}\). Thus, \(\Psi _{1}\) and \(\Psi _{3}\) satisfy the hypotheses in Theorem 2.1. Then, by using the same technique as used above, we can deduce

$$\begin{aligned} \wp (\vartheta _{1}) \leq \frac{2}{d_{2}-d_{1}} \int _{d_{1}}^{ \frac{d_{1}+d_{2}}{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \wp (d_{1})+ \wp \biggl(\frac{d_{1}+d_{2}}{2} \biggr)- \wp ( \vartheta _{2}). \end{aligned}$$

By convexity of , it follows that

$$\begin{aligned} \wp (\vartheta _{1}) \leq \frac{2}{d_{2}-d_{1}} \int _{d_{1}}^{ \frac{d_{1}+d_{2}}{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \wp (d_{1})+\frac{ \wp (d_{1})+ \wp (d_{2})}{2}- \wp ( \vartheta _{2}). \end{aligned}$$
(3.4)

Adding inequalities (3.3) and (3.4) together and simplifying the result we get

$$\begin{aligned} \frac{ \wp (\vartheta _{2})+ \wp (\vartheta _{1})}{2} \leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \frac{2 \wp (d_{1})+2 \wp (d_{2})}{2}- \frac{ \wp (\vartheta _{2})+ \wp (\vartheta _{1})}{2}. \end{aligned}$$

Again, by using the convexity of (we see \(\wp (\frac{\vartheta _{2}+\vartheta _{1}}{2} )\leq \frac{ \wp (\vartheta _{2})+ \wp (\vartheta _{1})}{2}\)) for the last inequality and rearranging the terms, we get the desired result. □

Corollary 3.1

Theorem 3.1 with \(\vartheta _{1}=\vartheta _{2}=\frac{d_{1}+3d_{2}}{4}\) gives the new inequality

$$\begin{aligned} \wp \biggl(\frac{d_{1}+3d_{2}}{4} \biggr) \leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \frac{2 \wp (d_{1})+2 \wp (d_{2})}{2}- \wp \biggl( \frac{d_{1}+3d_{2}}{4} \biggr). \end{aligned}$$
(3.5)

Theorem 3.2

Let \(\wp : [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be convex and monotonic and \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\), then we have

$$\begin{aligned} \wp \biggl(\frac{d_{1}+d_{2}}{2} \biggr) \leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \frac{2 \wp (d_{1})+2 \wp (d_{2})}{2}- \wp \biggl( \frac{d_{1}+d_{2}}{2} \biggr), \end{aligned}$$
(3.6)

such that \(\frac{d_{1}+d_{2}}{2}\in \mathrm{T_{0}}\).

Proof

Suppose that is decreasing and convex. It follows that \(\wp ^{\Delta }\leq 0\). Let \(\Psi _{1} :=- \wp ^{\Delta }\), then we see that \(\Psi _{1}\) is decreasing and \(\Psi _{1}\geq 0\). If we choose \(\Psi _{2}(\mathbf{s}):=\frac{d_{2}-\mathbf{s}}{d_{2}-d_{1}}\), we see that \(0\leq \Psi _{2}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [d_{1},d_{2} ]\). Moreover,

$$\begin{aligned} \lambda :=\int _{d_{1}}^{d_{2}}\Psi _{2}(\mathbf{s}) \Delta \mathbf{s}=\frac{d_{2}-d_{1}}{2}. \end{aligned}$$

It follows that \(d_{1}+\lambda =d_{2}-\lambda =\frac{d_{1}+d_{2}}{2}\in \mathrm{T_{0}}\). Thus, \(\Psi _{1}\) and \(\Psi _{2}\) satisfy the hypotheses in Theorem 2.3 and therefore inequality (2.5) holds true for \(\Psi _{1}=- \wp ^{\Delta }\) and \(\Psi _{2}(\mathbf{s})=\frac{d_{2}-\mathbf{s}}{d_{2}-d_{1}}\).

$$\begin{aligned} \int _{\frac{d_{1}+d_{2}}{2}}^{d_{2}}\Psi _{1}(\mathbf{s}) \Delta \mathbf{s}\leq \int _{d_{1}}^{d_{2}}\Psi _{1}(\mathbf{s}) \Psi _{2}( \mathbf{s})\Delta \mathbf{s}\leq \int _{d_{1}}^{\frac{d_{1}+d_{2}}{2}} \Psi _{1}(\mathbf{s}) \Delta \mathbf{s}. \end{aligned}$$

By making use of integration by parts and the fact \(\int _{x_{1}}^{x_{2}}\Psi _{1}(\mathbf{s})\Delta \mathbf{s}= \wp (x_{1})- \wp (x_{2})\), we can deduce

$$\begin{aligned} \wp \biggl(\frac{d_{1}+d_{2}}{2} \biggr)- \wp (d_{2}) \leq \wp (d_{1})- \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\sigma }(\mathbf{s}) \Delta \mathbf{s} \leq \wp (d_{1})- \wp \biggl( \frac{d_{1}+d_{2}}{2} \biggr), \end{aligned}$$

which rearranges to the desired result. □

The above results can be obtained for the case by using Theorems 2.2 and 2.4, respectively.

Theorem 3.3

Let \(\wp : [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be convex and monotonic and \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\). Suppose that also \(\vartheta _{1}, \vartheta _{2}\in [d_{1},d_{2}]_{\mathrm{T_{0}}}\), then we have

$$\begin{aligned} \wp \biggl(\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\rho }(\mathbf{s}) \nabla \mathbf{s} \leq \frac{2 \wp (d_{1})- \wp (\vartheta _{1})- \wp (\vartheta _{2})+2 \wp (d_{2})}{2}, \end{aligned}$$
(3.7)

such that \(\frac{d_{1}+d_{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2}\in \mathrm{T_{0}}\) and

$$\begin{aligned} \vartheta _{1}, \vartheta _{2}\geq \frac{d_{1}+3d_{2}}{4}, \quad \textit{if $ \wp $ is decreasing}, \\ \vartheta _{1}, \vartheta _{2}\leq \frac{d_{1}+3d_{2}}{4}, \quad \textit{if $ \wp $ is increasing}. \end{aligned}$$

Corollary 3.2

Theorem 3.3 with \(\vartheta _{1}=\vartheta _{2}=\frac{d_{1}+3d_{2}}{4}\) gives the new inequality

$$\begin{aligned} \wp \biggl(\frac{d_{1}+3d_{2}}{4} \biggr) \leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\rho }(\mathbf{s}) \nabla \mathbf{s} \leq \frac{2 \wp (d_{1})+2 \wp (d_{2})}{2}- \wp \biggl( \frac{d_{1}+3d_{2}}{4} \biggr). \end{aligned}$$
(3.8)

Theorem 3.4

Let \(\wp : [d_{1},d_{2}]_{\mathrm{T_{0}}}\to \mathrm{R}\) be convex and monotonic and \(d_{1}, d_{2}\in \mathrm{T_{0}}\) with \(d_{1}< d_{2}\), then we have

$$\begin{aligned} \wp \biggl(\frac{d_{1}+d_{2}}{2} \biggr) \leq \frac{1}{d_{2}-d_{1}} \int _{d_{1}}^{d_{2}} \wp ^{\rho }(\mathbf{s}) \nabla \mathbf{s} \leq \frac{2 \wp (d_{1})+2 \wp (d_{2})}{2}- \wp \biggl( \frac{d_{1}+d_{2}}{2} \biggr), \end{aligned}$$
(3.9)

such that \(\frac{d_{1}+d_{2}}{2}\in \mathrm{T_{0}}\).

Dynamic Opial inequalities

Theorem 4.1

Let \(0, \mu \in \mathrm{T_{0}}\). For a delta differentiable and increasing function \(\wp :[0, \mu ]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp (0)=0\), then

$$\begin{aligned} \bigl\vert \wp (\mu ) \bigr\vert \biggl( \bigl\vert \wp ( \mu ) \bigr\vert - \biggl\vert \wp \biggl(\frac{\mu }{2} \biggr) \biggr\vert \biggr)\leq \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s}) \bigr\vert \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s}\leq \bigl\vert \wp (\mu ) \bigr\vert \biggl\vert \wp \biggl( \frac{\mu }{2} \biggr) \biggr\vert , \end{aligned}$$
(4.1)

with equality when \(\wp (\mathbf{s})=c\mathbf{s}\), for some \(c\in \mathrm{R}\).

Proof

Let \(\Psi _{1}(\mathbf{s}):= \vert \wp ^{\Delta }(\mathbf{s}) \vert \), then \(\Psi _{1}(\mathbf{s})\geq 0\) for all \(\mathbf{s}\in [0, \mu ]\). Taking \(\Psi _{2}(\mathbf{s}):=\frac{| \wp (\mathbf{s})|}{| \wp (\mu )|}\) and since \(\wp (\mathbf{s})\) is an increasing function on \([0, \mu ]\), we see that \(0\leq \Psi _{2}(\mathbf{s})\leq 1\) for each \(\mathbf{s}\in [0, \mu ]\). Thus, \(\Psi _{1}\) and \(\Psi _{2}\) satisfy the hypotheses in Theorem 2.3. Hence

$$\begin{aligned} \int _{\frac{\mu }{2}}^{\mu }\Psi _{1}(\mathbf{s}) \Delta \mathbf{s} \leq \int _{0}^{\mu }\Psi _{1}(\mathbf{s}) \Psi _{2}(\mathbf{s}) \Delta \mathbf{s} \leq \int _{0}^{\frac{\mu }{2}}\Psi _{1}(\mathbf{s}) \Delta \mathbf{s}. \end{aligned}$$

So,

$$\begin{aligned} \int _{\frac{\mu }{2}}^{\mu } \bigl\vert \wp ^{\Delta }( \mathbf{s}) \bigr\vert \Delta \mathbf{s} \leq \frac{1}{ \vert \wp (\mu ) \vert } \int _{0}^{\mu } \bigl\vert \wp ( \mathbf{s}) \bigr\vert \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \leq \int _{0}^{\frac{\mu }{2}} \bigl\vert \wp ^{\Delta }( \mathbf{s}) \bigr\vert \Delta \mathbf{s}. \end{aligned}$$

By making use of integration by parts and the fact

$$ \int _{a_{1}}^{a_{2}}\Psi _{1}(\mathbf{s}) \Delta \mathbf{s}= \bigl\vert \wp (a_{2}) \bigr\vert - \bigl\vert \wp (a_{1}) \bigr\vert , $$

we get

$$\begin{aligned} \bigl\vert \wp (\mu ) \bigr\vert - \biggl\vert \wp \biggl(\frac{\mu }{2} \biggr) \biggr\vert \leq \frac{1}{ \vert \wp (\mu ) \vert } \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s}) \bigr\vert \bigl\vert \wp ^{ \Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s}\leq \biggl\vert \wp \biggl( \frac{\mu }{2} \biggr) \biggr\vert - \bigl\vert \wp (0) \bigr\vert . \end{aligned}$$
(4.2)

Multiplying inequality (4.2) on both sides by the factor \(| \wp (\mu )|>0\) and from the condition \(\wp (0)=0\) we obtain the desired result (4.1). Now, let \(\wp (\mathbf{s})=c\mathbf{s}\) for some \(c\in \mathrm{R}\). Then \(\wp ^{\Delta }(\mathbf{s})=c\) and it is easy to check that equality holds in (4.1). The proof is complete. □

Theorem 4.2

Let \(d_{1}, d_{2}\in \mathrm{T_{0}}\) and \(d_{1}< d_{2}\). Assume that \(\wp , \wp ^{\sigma }, \wp ^{\Delta }\in \mathbb{C}_{\mathcal{RD}} ([d_{1}, d_{2}]_{\mathrm{T_{0}}}, \mathrm{R} )\) and \(p>1\). Then

$$\begin{aligned} \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}} \leq& \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}} \\ & {}+ \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{ \frac{1}{p}}. \end{aligned}$$
(4.3)

Proof

Note

$$\begin{aligned} &\int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s}\\ &\quad = \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p-1} \bigl\vert \wp ( \mathbf{s})+ \wp ^{ \sigma }(\mathbf{s}) \bigr\vert \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \\ &\quad \leq \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }( \mathbf{s}) \bigr\vert ^{p-1} \bigl\vert \wp ( \mathbf{s}) \bigr\vert \bigl\vert \wp ^{ \Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \\ &\qquad{}+ \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p-1} \bigl\vert \wp ^{\sigma }(\mathbf{s}) \bigr\vert \bigl\vert \wp ^{\Delta }( \mathbf{s}) \bigr\vert \Delta \mathbf{s}. \end{aligned}$$

Applying the Hölder inequality, we get

$$\begin{aligned} & \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \\ &\quad \leq \biggl( \int _{d_{1}}^{d_{2}} \bigl( \bigl\vert \wp ( \mathbf{s})+ \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p-1} \bigr)^{q} \bigl\vert \wp ^{ \Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{q}} \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{ \frac{1}{p}} \\ &\qquad {}+ \biggl( \int _{d_{1}}^{d_{2}} \bigl( \bigl\vert \wp ( \mathbf{s})+ \wp ^{ \sigma }(\mathbf{s}) \bigr\vert ^{p-1} \bigr)^{q} \bigl\vert \wp ^{\Delta }( \mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{q}} \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{ \frac{1}{p}} \\ &\quad = \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{q}} \\ &\qquad {}\times \biggl[ \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}}+ \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}} \biggr]. \end{aligned}$$

Therefore

$$\begin{aligned}& \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}} \\& \quad = \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{1- \frac{1}{q}} \\& \quad \leq \biggl[ \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}}+ \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}} \biggr], \end{aligned}$$

which is the desired inequality (4.3). The proof is completed. □

By making use of Theorem 4.1 and the well-known inequality

$$ \vert d_{1}+d_{2} \vert ^{p}\leq 2^{p-1} \bigl( \vert d_{1} \vert ^{p}+ \vert d_{2} \vert ^{p} \bigr), \quad p\geq 1, $$

we can obtain the following result.

Corollary 4.1

Let \(0, \mu \in \mathrm{T_{0}}\). Assume that \(\wp , \wp ^{\sigma }, \wp ^{\Delta }\in \mathbb{C}_{\mathcal{RD}} ([0, \mu ]_{\mathrm{T_{0}}}, \mathrm{R} )\). For a delta differentiable and increasing function \(\wp :[0, \mu ]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp (0)= \wp ^{\sigma }(0)=0\) and \(p\geq 1\), then

$$\begin{aligned} \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s}\leq 2^{p-1} \biggl\vert \wp \biggl(\frac{\mu }{2} \biggr) \biggr\vert \bigl( \bigl\vert \wp (\mu ) \bigr\vert ^{p}+ \bigl\vert \wp ^{\sigma }(\mu ) \bigr\vert ^{p} \bigr). \end{aligned}$$
(4.4)

By making use of Theorems 4.1 and 4.2, we can deduce the following inequality.

Corollary 4.2

Let \(0, \mu \in \mathrm{T_{0}}\). Assume that \(\wp , \wp ^{\sigma }, \wp ^{\Delta }\in \mathbb{C}_{\mathcal{RD}} ([0, \mu ]_{\mathrm{T_{0}}}, \mathrm{R} )\). For a delta differentiable and increasing function \(\wp :[0, \mu ]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp (0)= \wp ^{\sigma }(0)=0\) and \(p>1\), then

$$\begin{aligned} \biggl( \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s})+ \wp ^{\sigma }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\Delta }(\mathbf{s}) \bigr\vert \Delta \mathbf{s} \biggr)^{\frac{1}{p}}\leq \biggl\vert \wp \biggl( \frac{\mu }{2} \biggr) \biggr\vert ^{\frac{1}{p}} \bigl( \bigl\vert \wp (\mu ) \bigr\vert + \bigl\vert \wp ^{\sigma }(\mu ) \bigr\vert \bigr). \end{aligned}$$
(4.5)

The above results can be obtained for the case by using Theorems 2.2 and 2.4, respectively.

Theorem 4.3

Let \(0, \mu \in \mathrm{T_{0}}\). For a nabla differentiable and increasing function \(\wp :[0, \mu ]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp (0)=0\), then

$$\begin{aligned} \bigl\vert \wp (\mu ) \bigr\vert \biggl( \bigl\vert \wp ( \mu ) \bigr\vert - \biggl\vert \wp \biggl(\frac{\mu }{2} \biggr) \biggr\vert \biggr)\leq \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s}) \bigr\vert \bigl\vert \wp ^{\nabla }(\mathbf{s}) \bigr\vert \nabla \mathbf{s}\leq \bigl\vert \wp (\mu ) \bigr\vert \biggl\vert \wp \biggl( \frac{\mu }{2} \biggr) \biggr\vert , \end{aligned}$$
(4.6)

with equality when \(\wp (\mathbf{s})=c\mathbf{s}\), for some \(c\in \mathrm{R}\).

Theorem 4.4

Let \(d_{1}, d_{2}\in \mathrm{T_{0}}\) and \(d_{1}< d_{2}\). Assume that \(\wp , \wp ^{\rho }, \wp ^{\nabla }\in \mathbb{C}_{\mathcal{LD}} ([d_{1}, d_{2}]_{\mathrm{T_{0}}}, \mathrm{R} )\) and \(p>1\). Then

$$\begin{aligned} \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s})+ \wp ^{\rho }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\nabla }(\mathbf{s}) \bigr\vert \nabla \mathbf{s} \biggr)^{\frac{1}{p}} \leq& \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp (\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\nabla }(\mathbf{s}) \bigr\vert \nabla \mathbf{s} \biggr)^{\frac{1}{p}} \\ & {}+ \biggl( \int _{d_{1}}^{d_{2}} \bigl\vert \wp ^{\rho }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\nabla }(\mathbf{s}) \bigr\vert \nabla \mathbf{s} \biggr)^{ \frac{1}{p}}. \end{aligned}$$
(4.7)

By making use of Theorems 4.3 and 4.4, we can deduce the following inequalities, respectively.

Corollary 4.3

Let \(0, \mu \in \mathrm{T_{0}}\). Assume that \(\wp , \wp ^{\rho }, \wp ^{\nabla }\in \mathbb{C}_{\mathcal{LD}} ([0, \mu ]_{\mathrm{T_{0}}}, \mathrm{R} )\). For a nabla differentiable and increasing function \(\wp :[0, \mu ]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp (0)= \wp ^{\rho }(0)=0\) and \(p>1\), then

$$\begin{aligned} \biggl( \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s})+ \wp ^{\rho }( \mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\nabla }(\mathbf{s}) \bigr\vert \nabla \mathbf{s} \biggr)^{\frac{1}{p}}\leq \biggl\vert \wp \biggl( \frac{\mu }{2} \biggr) \biggr\vert ^{\frac{1}{p}} \bigl( \bigl\vert \wp (\mu ) \bigr\vert + \bigl\vert \wp ^{\rho }(\mu ) \bigr\vert \bigr). \end{aligned}$$
(4.8)

Corollary 4.4

Let \(0, \mu \in \mathrm{T_{0}}\). Assume that \(\wp , \wp ^{\rho }, \wp ^{\nabla }\in \mathbb{C}_{\mathcal{LD}} ([0, \mu ]_{\mathrm{T_{0}}}, \mathrm{R} )\). For a nabla differentiable and increasing function \(\wp :[0, \mu ]_{\mathrm{T_{0}}}\to \mathrm{R}\) with \(\wp (0)= \wp ^{\rho }(0)=0\) and \(p\geq 1\), then

$$\begin{aligned} \int _{0}^{\mu } \bigl\vert \wp (\mathbf{s})+\wp ^{\rho }(\mathbf{s}) \bigr\vert ^{p} \bigl\vert \wp ^{\nabla }(\mathbf{s}) \bigr\vert \nabla \mathbf{s}\leq 2^{p-1} \biggl\vert \wp \biggl(\frac{\mu }{2} \biggr) \biggr\vert \bigl( \bigl\vert \wp (\mu ) \bigr\vert ^{p}+ \bigl\vert \wp ^{\rho }(\mu ) \bigr\vert ^{p} \bigr). \end{aligned}$$
(4.9)

Conclusion

In this article, by making use of the well-known dynamic inequalities, a dynamic version of integration by parts and chain rule formulas, we obtained some useful dynamic H-H and Opial inequalities on time scales. The derived inequalities generalize some well-known dynamic inequalities in the literature. For this purpose, the reader can see corollaries and remarks after each theorem of the main results.

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References

  1. 1.

    Hadamard, J.: Étude sur les propriétés des fonctions entières en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 58, 171–215 (1893)

    MATH  Google Scholar 

  2. 2.

    Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013)

    Article  Google Scholar 

  3. 3.

    Srivastava, H.M., Zhang, Z.-H., Wu, Y.-D.: Some further refinements and extensions of the Hermite–Hadamard and Jensen inequalities in several variables. Math. Comput. Model. 54, 2709–2717 (2011)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Han, J., Mohammed, P.O., Zeng, H.: Generalized fractional integral inequalities of Hermite–Hadamard-type for a convex function. Open Math. 18, 794–806 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Kashuri, A., Meftah, B., Mohammed, P.O.: Some weighted Simpson type inequalities for differentiable s-convex functions and their applications. J. Frac. Calc. Nonlinear Sys. 1, 75–94 (2021)

    Article  Google Scholar 

  6. 6.

    Mohammed, P.O., Abdeljawad, T., Zeng, S., Kashuri, A.: Fractional Hermite–Hadamard integral inequalities for a new class of convex functions. Symmetry 12, 1485 (2020)

    Article  Google Scholar 

  7. 7.

    Mohammed, P.O., Aydi, H., Kashuri, A., Hamed, Y.S., Abualnaja, K.M.: Midpoint inequalities in fractional calculus defined using positive weighted symmetry function kernels. Symmetry 13, 550 (2021)

    Article  Google Scholar 

  8. 8.

    Khan, M.B., Mohammed, P.O., Noor, B., Hamed, Y.S.: New Hermite–Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities. Symmetry 13, 673 (2021)

    Article  Google Scholar 

  9. 9.

    Opial, Z.: Sur une inégalité. Ann. Pol. Math. 8, 29–32 (1960)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zhao, C.-J., Cheung, W.-S.: On some Opial-type inequalities. J. Inequal. Appl. 2011, 7 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Samraiz, M., Iqbal, S., Pečaric, J.: Generalized integral inequalities for fractional calculus. Cogent Math. Stat. 5, 1426205 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Başcı, Y., Baleanu, D.: New aspects of Opial-type integral inequalities. Adv. Differ. Equ. 2018, 452 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Sarikaya, M.Z., Bilisik, C.C., Mohammed, P.O.: Some generalizations of Opial type inequalities. Appl. Math. Inf. Sci. 14, 809–816 (2020)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Srivastava, H.M.: Some weighted Opial-type inequalities on time scale. Taiwan. J. Math. 14, 107–122 (2010)

    MathSciNet  Article  Google Scholar 

  15. 15.

    KH, F.M., El-Deeb, A.A., Abdeldaim, A., Khan, Z.A.: On some generalizations of dynamic Opial-type inequalities on time scales. Adv. Differ. Equ. 2019, 323 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Saker, S.H.: Opial’s type inequalities on time scales and some applications. Ann. Pol. Math. 104(3), 243–260 (2012)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Abdeljawad, T., Atici, F.M.: On the definitions of nabla fractional operators. Abstr. Appl. Anal. 2012, Article ID 406757 (2012)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Mohammed, P.O.: Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals of a convex function with respect to a monotone function. Math. Meth. Appl. Sci., 1–11 (2019). https://doi.org/10.1002/mma.5784

  19. 19.

    Mohammed, P.O., Sarikaya, M.Z., Baleanu, D.: On the generalized Hermite–Hadamard inequalities via the tempered fractional integrals. Symmetry 12, 595 (2020)

    Article  Google Scholar 

  20. 20.

    Fernandez, A., Mohammed, P.: Hermite–Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels. Math. Meth. Appl. Sci., 1–18 (2020). https://doi.org/10.1002/mma.6188

  21. 21.

    Mohammed, P.O., Abdeljawad, T.: Opial integral inequalities for generalized fractional operators with nonsingular kernel. J. Inequal. Appl. 2020, 148 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Tomovski, Z., Pečaric, J., Weighted, F.G.: Opial-type inequalities for fractional integral and differential operators involving generalized Mittag-Leffler functions. Eur. J. Pure Appl. Math. 10, 419–439 (2017)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Hilger, S.: Analysis on measure chains – a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)

    Book  Google Scholar 

  25. 25.

    Agarwal, R.P., O’Regan, D., Saker, S.: Dynamic Inequalities on Time Scales. Springer, London (2014)

    Book  Google Scholar 

  26. 26.

    Gu, J., Meng, F.W.: Some new nonlinear Volterra–Fredholm type dynamic integral inequalities on time scales. Appl. Math. Comput. 245, 235–242 (2014)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Pachpatte, D.P.: Explicit estimates on integral inequalities with time scales. J. Inequal. Pure Appl. Math. 17, Article ID 143 (2006)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Rehák, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 5, 495–507 (2005)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Mohammed, P.O.: Some integral inequalities of fractional quantum type. Malaya J. Mat. 4, 93–99 (2016)

    Google Scholar 

  30. 30.

    Srivastava, H.M., Tseng, K.-L., Tseng, S.-J., Lo, J.-C.: Some generalizations of Maroni’s inequality on time scales. Math. Inequal. Appl. 14, 469–480 (2011)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Atici, F.M., Yaldız, H.: Convex functions on discrete time domains. Can. Math. Bull. 59, 225–233 (2016)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This Research was supported by Taif University Researchers Supporting Project Number (TURSP-2020/217), Taif University, Taif, Saudi Arabia, and the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092).

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Correspondence to Pshtiwan Othman Mohammed or Cheon Seoung Ryoo.

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Mohammed, P.O., Ryoo, C.S., Kashuri, A. et al. Some Hermite–Hadamard and Opial dynamic inequalities on time scales. J Inequal Appl 2021, 89 (2021). https://doi.org/10.1186/s13660-021-02624-9

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MSC

  • 26D07
  • 26D10
  • 26D15
  • 26A33

Keywords

  • Time scales
  • H-H inequality
  • Steffensen inequalities
  • Opial inequality
  • Hölder’s inequality