1 Introduction

The integral inequality of Hermite–Hadamard type (or briefly HH-type) is a very interesting topic of mathematical analysis, this challenging topic has been developing very rapidly in the last three decades; see e.g. [1–5].

In the literature, there are two well-known types of HH-type inequalities which were obtained by Sarikaya et al. in [6] and [7], respectively. Their results are, respectively,

$$ \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr)\leq \frac{\Gamma (\varepsilon +1)}{2(\kappa _{2}-\kappa _{1})^{\varepsilon }} \bigl[{}^{RL}{I}_{\kappa _{1}+}^{\varepsilon }\Upsilon (\kappa _{2})+ {}^{RL}{I}_{\kappa _{2}-}^{\varepsilon }\Upsilon ( \kappa _{1}) \bigr]\leq \frac{\Upsilon (\kappa _{1})+\Upsilon (\kappa _{2})}{2} $$
(1.1)

and

$$ \begin{aligned}[b] \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr)&\leq \frac{2^{\varepsilon -1}\Gamma (\varepsilon +1)}{(\kappa _{2}-\kappa _{1})^{\varepsilon }} \bigl[{}^{RL}{I}_{ (\frac{\kappa _{1}+\kappa _{2}}{2} )+}^{\varepsilon }\Upsilon (\kappa _{2}) +{}^{RL}{I}_{ (\frac{\kappa _{1}+\kappa _{2}}{2} )-}^{\varepsilon } \Upsilon ( \kappa _{1}) \bigr]\\&\leq \frac{\Upsilon (\kappa _{1})+\Upsilon (\kappa _{2})}{2}. \end{aligned} $$
(1.2)

These results are valid for any \(\varepsilon >0\) and any \(L^{1}\) convex function \(\Upsilon :[\kappa _{1},\kappa _{2}]\to \mathbb{R}\). The difference between (1.1) and (1.2) is that in (1.1) is that we are considering fractional integration from the two respective ends of the interval \([\kappa _{1},\kappa _{2}]\) instead of from the center of the interval \([\kappa _{1},\kappa _{2}]\) as used in (1.2).

In the last ten years, the study of inequalities on time scales has received a lot of attention in the literature and has become important in both the fields of pure and of applied mathematics; see e.g. [8–13].

In 2016, Atici and Yildiz [10] obtained the discrete Hermite–Hadamard inequalities corresponding to (1.1) on the time scale \(\top _{[\kappa _{1},\kappa _{2}]}:= \{ h; h= \frac{\kappa _{2}-\mathsf{c} }{\kappa _{2}-\kappa _{1}} \text{ for } \mathsf{c}\in [\kappa _{1},\kappa _{2} ]_{\mathbb{Z}} \} \), their results are as follows.

Theorem 1.1

Let \(\Upsilon :\mathbb{Z}\to \mathbb{R}\) be a convex function on \([\kappa _{1},\kappa _{2}]_{\mathbb{Z}}\) and \(\kappa _{1},\kappa _{2}\in \mathbb{Z}\) with \(\kappa _{1}<\kappa _{2}\). If \(\kappa _{1}+\kappa _{2}\) is an even number, then, for \(\mathsf{c}\in \top _{[\kappa _{1},\kappa _{2}]}\setminus \{0,1\}\), we have

$$\begin{aligned} \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr)\leq \frac{1}{2(\kappa _{2}-\kappa _{1})} \biggl[ \int _{\kappa _{1}}^{ \kappa _{2}}\Upsilon (\mathsf{c})\Delta \mathsf{c}+ \int _{\kappa _{1}}^{ \kappa _{2}}\Upsilon (\mathsf{c})\nabla \mathsf{c} \biggr]\leq \frac{\Upsilon (\kappa _{1})+\Upsilon (\kappa _{2})}{2}. \end{aligned}$$
(1.3)

Theorem 1.2

Let \(\Upsilon :\mathbb{Z}\to \mathbb{R}\) be a convex function on \([\kappa _{1},\kappa _{2}]_{\mathbb{Z}}\) and \(\kappa _{1},\kappa _{2}\in \mathbb{Z}\) with \(\kappa _{1}<\kappa _{2}\). If \(\kappa _{1}+\kappa _{2}\) is an even number, then, for \(\varepsilon >0\) and \(\mathsf{c}\in \top _{[\kappa _{1},\kappa _{2}]}\setminus \{0,1\}\), we have

$$\begin{aligned} \begin{aligned}[b] \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr) &\leq \frac{\Gamma (\varepsilon )}{2\Omega (\kappa _{2}-\kappa _{1})} \bigl[ \bigl(\Delta _{\kappa _{2}-1}^{-\varepsilon }\Upsilon \bigr) (a-\varepsilon ) + \bigl( {}_{\kappa _{1}+1}{\nabla } ^{-\varepsilon }\Upsilon \bigr) ( \kappa _{2}) \bigr]\\&\leq \frac{\Upsilon (\kappa _{1})+\Upsilon (\kappa _{2})}{2}, \end{aligned} \end{aligned}$$
(1.4)

where

$$\begin{aligned} \Omega = \int _{\top _{[\kappa _{1},\kappa _{2}]}} \bigl((\kappa _{2}- \kappa _{1})\mathsf{c}+\varepsilon -1 \bigr)^{ \overline{\varepsilon -1}}\hat{\Delta } \mathsf{c}. \end{aligned}$$

In the literature, the inequalities of HH-type are often connected with further integral inequalities which are called trapezoidal type (where the ends a, b of the interval are used) or midpoint type (where the midpoint \(\frac{\kappa _{1}+\kappa _{2}}{2}\) of the interval is used). Many inequalities of such a type have been obtained by many researchers; see e.g. [14–23].

In the present study, we obtain the new discrete inequalities of HH-type corresponding to (1.2) on the new time scales \(\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}\), where

$$\begin{aligned} \top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}&:= \biggl\{ h; h= \frac{2(\kappa _{2}-\mathsf{c} )}{\kappa _{2}-\kappa _{1}} \text{ such that } \mathsf{c}\in \biggl[ \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} \biggr]_{\mathbb{Z}} \biggr\} , \end{aligned}$$
(1.5)

where \([\kappa _{1},\kappa _{2}]_{\mathbb{Z}}=[\kappa _{1},\kappa _{2}] \cap \mathbb{Z}\). We can observe that \(\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}\) is a finite subset of the interval \([0,1]\).

2 Discrete inequality of HH-type

At first, we need to recall the following preliminary definitions and theorems of discrete time scales.

Definition 2.1

([10])

Let \(z_{1}\), \(z_{2}\) be two elements of a time scale ⊤ with \({{z_{1}< z_{2}}}\). A function \(\Upsilon :\top \to \mathbb{R}\) is said to be convex on ⊤, if

$$\begin{aligned} {{\Upsilon \bigl(\mathsf{c} z_{1}+(1-\mathsf{c})z_{2}\bigr) \leq \mathsf{c} \Upsilon (z_{1})+(1-\mathsf{c})\Upsilon (z_{2})}} \end{aligned}$$

holds for each \(\mathsf{c}\in \top _{[z_{1},z_{2}]}\).

In the following theorem, we recall the time scale substitution rule.

Theorem 2.1

([24])

Let \({w}:\mathbb{Z}\to \mathbb{R}\) be strictly increasing and \(\hat{\top }:= {w}(\top )\) be a time scale. If \(\Upsilon :\mathbb{Z}\to \mathbb{R}\) is an rd-continuous function and w is differentiable with rd-continuous derivative, then for \(\kappa _{1},\kappa _{2}\in \top _{[\kappa _{1},\kappa _{2}]}\) we have

$$\begin{aligned} \int _{\kappa _{1}}^{\kappa _{2}}\Upsilon (\mathsf{c}) {w}^{\Delta }( \mathsf{c})\Delta \mathsf{c} = \int _{{w}(\kappa _{1})}^{{w}(\kappa _{2})} \bigl(\Upsilon \circ {w}^{-1} \bigr) (s)\hat{\Delta }s, \end{aligned}$$

or

$$\begin{aligned} \int _{\kappa _{1}}^{\kappa _{2}}\Upsilon (\mathsf{c}) {w}^{\nabla }( \mathsf{c})\nabla \mathsf{c} = \int _{{w}(\kappa _{1})}^{{w}(\kappa _{2})} \bigl(\Upsilon \circ {w}^{-1} \bigr) (s)\hat{\nabla }s. \end{aligned}$$

Remark 2.1

Note that

$$\begin{aligned} \int _{\kappa _{1}}^{\kappa _{2}}\Upsilon (\mathsf{c})\Delta \mathsf{c} &=\sum_{\mathsf{c}=\kappa _{1}}^{\kappa _{2}-1} \Upsilon (\mathsf{c}) \quad \text{and}\quad \int _{\kappa _{1}}^{ \kappa _{2}}\Upsilon (\mathsf{c})\nabla \mathsf{c} =\sum_{ \mathsf{c}=\kappa _{1}+1}^{\kappa _{2}}\Upsilon (\mathsf{c}). \end{aligned}$$

Theorem 2.2

(Dual time scale substitution rule [24])

Let ⊤ be a time scale and \(\hat{\top }:= \{s\in \mathbb{R}; -s\in \top \}\). Let \({w}:\mathbb{Z}\to \mathbb{R}\) be strictly increasing and \(\hat{\top }:= {w}(\top )\) be a time scale. If \(\Upsilon :\mathbb{Z}\to \mathbb{R}\) is a continuous function and w is differentiable with rd-continuous derivative, then for \(\kappa _{1},\kappa _{2}\in \top _{[\kappa _{1},\kappa _{2}]}\) we have

$$\begin{aligned} \int _{\kappa _{1}}^{\kappa _{2}}\Upsilon (\mathsf{c}) \bigl(-{w}^{ \Delta } \bigr) (\mathsf{c})\Delta \mathsf{c} = \int _{{w}(\kappa _{2})}^{{w}( \kappa _{1})} \bigl(\Upsilon \circ {w}^{-1} \bigr) (s)\hat{\nabla }s, \end{aligned}$$

or

$$\begin{aligned} \int _{\kappa _{1}}^{\kappa _{2}}\Upsilon (\mathsf{c}) \bigl(-{w}^{ \nabla } \bigr) (\mathsf{c})\nabla \mathsf{c} = \int _{{w}(\kappa _{2})}^{{w}( \kappa _{1})} \bigl(\Upsilon \circ {w}^{-1} \bigr) (s)\hat{\Delta }s. \end{aligned}$$

The first result starts from the following main theorem.

Theorem 2.3

Let \(\Upsilon :\mathbb{Z}\to \mathbb{R}\) be a convex function on \([\kappa _{1},\kappa _{2}]_{\mathbb{Z}}\) and \(\kappa _{1},\kappa _{2}\in \mathbb{Z}\) with \(\kappa _{1}<\kappa _{2}\). If \(\kappa _{1}+\kappa _{2}\) is an even number, then we have

$$\begin{aligned} \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr)\leq \frac{2}{\kappa _{2}-\kappa _{1}} \biggl[ \int _{ \frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}}\Upsilon (\mathsf{c}) \Delta \mathsf{c}+ \int _{\frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}} \Upsilon (\mathsf{c})\nabla \mathsf{c} \biggr]\leq \Upsilon (\kappa _{1})+ \Upsilon (\kappa _{2}). \end{aligned}$$
(2.1)

Proof

Let \(\mathsf{c}\in \top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}\setminus \{0,1\}\) be fixed. Then we can see that

$$\begin{aligned} z_{1}=\frac{2-\mathsf{c} }{2}\kappa _{1}+\frac{\mathsf{c}}{2}\kappa _{2} ,\qquad z_{2}=\frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2} \kappa _{2} \end{aligned}$$

are in \([\kappa _{1},\kappa _{2}]_{\mathbb{Z}}\) and \(z_{1}+z_{2}=\kappa _{1}+\kappa _{2}\) is even. Since \(\frac{1}{2}\in \top _{[z_{1},z_{2}]}\) (or \(\top _{[z_{2},z_{1}]}\)) and ϒ is convex on \([z_{1},z_{2}]_{\mathbb{Z}}\) (or \([z_{2},z_{1}]_{\mathbb{Z}}\)), we can deduce

$$\begin{aligned} \begin{aligned} \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr)&=\Upsilon \biggl(\frac{z_{1}+z_{2}}{2} \biggr)\leq \frac{\Upsilon (z_{1})+\Upsilon (z_{2})}{2} \\&=\frac{1}{2} \biggl[ \Upsilon \biggl( \frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2}\kappa _{2} \biggr) +\Upsilon \biggl( \frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2}\kappa _{2} \biggr) \biggr].\end{aligned} \end{aligned}$$

Integrating both sides over \(\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}\) we get

$$\begin{aligned} & \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}}\Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \hat{\Delta }\mathsf{c} \\ &\quad \leq \frac{1}{2} \biggl[ \int _{\top _{ [ \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \Upsilon \biggl(\frac{2-\mathsf{c} }{2}\kappa _{1}+\frac{\mathsf{c}}{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\ &\quad \quad {}+ \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}}\Upsilon \biggl(\frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \biggr] =\frac{1}{2}[{h}_{1}+{h}_{2}], \end{aligned}$$
(2.2)

where Δ̂ is the derivative operator on the time scale \(\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}\).

Making use of Theorem 2.1 with \({w}(\mathsf{c}):= \frac{2(\mathsf{c}-\kappa _{1})}{\kappa _{2}-\kappa _{1}}\) we get

$$\begin{aligned} \begin{aligned}[b] {h}_{1}&= \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}}\Upsilon \biggl(\frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\&= \int _{1}^{2}\Upsilon \biggl(\frac{2-\mathsf{c} }{2} \kappa _{1}+ \frac{\mathsf{c}}{2}\kappa _{2} \biggr)\hat{\Delta }\mathsf{c} = \frac{2}{\kappa _{2}-\kappa _{1}} \int _{ \frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}}\Upsilon (\mathsf{c}) \Delta \mathsf{c},\end{aligned} \end{aligned}$$
(2.3)

where we used that \({w}^{\Delta }(\mathsf{c})=\frac{2}{\kappa _{2}-\kappa _{1}}\) and \({{{w} ( [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]_{\mathbb{Z}} ) =\top _{ [\kappa _{1}, \frac{\kappa _{1}+\kappa _{2}}{2} ]}}}\) is also a time scale.

On the other hand, for \({w}(\mathsf{c}):= \frac{2(\kappa _{2}-\mathsf{c} )}{\kappa _{2}-\kappa _{1}}\), we have

$$\begin{aligned} {h}_{2}&= \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}}\Upsilon \biggl(\frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} = \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \bigl(\Upsilon \circ {w}^{-1} \bigr) ( \mathsf{c}) \hat{\Delta }\mathsf{c} \\ &= \int _{0}^{1} \bigl(\Upsilon \circ {w}^{-1} \bigr) (\mathsf{c}) \hat{\Delta }\mathsf{c} = \int _{-1}^{0} \bigl(\Upsilon \circ {w}^{-1} \bigr)^{\ast }(s)\hat{\nabla }s \\ &= \int _{-1= (f^{-1}\circ {w} ) ( \frac{\kappa _{1}+\kappa _{2}}{2} )}^{0= (f^{-1}\circ {w} )(\kappa _{2})} \Upsilon \bigl( \bigl(f^{-1} \circ {w} \bigr)^{-1} \bigr) (s)\hat{\nabla }s, \end{aligned}$$

where \(f(s)=-s\) and we used that

$$\begin{aligned} \bigl(\Upsilon \circ {w}^{-1} \bigr)^{\ast }(s)= \bigl(\Upsilon \circ {w}^{-1} \bigr) (-s) =\Upsilon \bigl({w}^{-1}(-s) \bigr) = \Upsilon \bigl( \bigl({w}^{-1}\circ f \bigr) (s) \bigr)= \Upsilon \bigl( \bigl(f^{-1}\circ {w} \bigr)^{-1} \bigr) (s). \end{aligned}$$

Since

$$\begin{aligned} \bigl(f^{-1}\circ {w} \bigr) (s)=f^{-1} \bigl({w}(s) \bigr)=f^{-1} \biggl(\frac{2(\kappa _{2}-s)}{\kappa _{2}-\kappa _{1}} \biggr) = \frac{2(s-\kappa _{2})}{\kappa _{2}-\kappa _{1}} \end{aligned}$$

we have \({{ (f^{-1}\circ {w} )^{\Delta }(s)= \frac{2}{\kappa _{2}-\kappa _{1}}}}>0\) and hence \(f^{-1}\circ {w}\) is strictly increasing. Thus, by making use of Theorem 2.2, we get

$$\begin{aligned} {h}_{2}= \int _{-1}^{0}\Upsilon \bigl(f^{-1}\circ {w} \bigr)^{-1}( \mathsf{c})\hat{\nabla }\mathsf{c} =\frac{2}{\kappa _{2}-\kappa _{1}} \int _{\frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}}\Upsilon ( \mathsf{c})\nabla \mathsf{c}. \end{aligned}$$
(2.4)

Thus, the one half of the inequality in (2.1) follows from (2.2)–(2.4).

To prove the other half of the inequality in (2.1), we use the convexity of ϒ and the following inequalities:

$$\begin{aligned}& \Upsilon \biggl(\frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2} \kappa _{2} \biggr)\leq \frac{2-\mathsf{c} }{2}\Upsilon (\kappa _{1})+\frac{\mathsf{c}}{2} \Upsilon (\kappa _{2}); \\& \Upsilon \biggl(\frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2} \kappa _{2} \biggr)\leq \frac{\mathsf{c}}{2}\Upsilon (\kappa _{1})+\frac{2-\mathsf{c} }{2} \Upsilon (\kappa _{2}). \end{aligned}$$

Adding these we obtain

$$\begin{aligned} \Upsilon \biggl(\frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2} \kappa _{2} \biggr)+\Upsilon \biggl( \frac{\mathsf{c}}{2}\kappa _{1}+\frac{2-\mathsf{c} }{2}\kappa _{2} \biggr) &\leq \Upsilon ( \kappa _{1})+\Upsilon (\kappa _{2}). \end{aligned}$$

Integrating both sides over \({{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}}}\) we get

$$\begin{aligned} & \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}}\Upsilon \biggl(\frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} + \int _{ \top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \Upsilon \biggl(\frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\ &\quad \leq \Upsilon (\kappa _{1})+\Upsilon (\kappa _{2}). \end{aligned}$$

We can use the same method used above we get

$$\begin{aligned} \frac{2}{\kappa _{2}-\kappa _{1}} \biggl[ \int _{ \frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}}\Upsilon (\mathsf{c}) \Delta \mathsf{c}+ \int _{\frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}} \Upsilon (\mathsf{c})\nabla \mathsf{c} \biggr]\leq \Upsilon (\kappa _{1})+ \Upsilon (\kappa _{2}), \end{aligned}$$

and thus the result follows. □

3 Discrete fractional inequality of HH-type

The left nabla fractional sum of ϒ of order ε is defined by [25–27]

$$\begin{aligned} \bigl({}_{\kappa _{1}}{\nabla } ^{-\varepsilon }\Upsilon \bigr) ( \mathsf{c})&:= \frac{1}{\Gamma (\varepsilon )} \sum_{r= \kappa _{1}+1}^{\mathsf{c}} \bigl(\mathsf{c}-\rho (r)\bigr)^{ \overline{{\varepsilon -1}}}\Upsilon (r), \end{aligned}$$
(3.1)

and the right delta fractional sum of ϒ of order ε is defined by [25, 26]

$$\begin{aligned} \bigl(\Delta _{\kappa _{2}}^{-\varepsilon }\Upsilon \bigr) ( \mathsf{c})&:= \frac{1}{\Gamma (\varepsilon )} \sum_{r= \mathsf{c}+\varepsilon }^{\kappa _{2}} \bigl(r-\sigma (\mathsf{c})\bigr)^{( \varepsilon -1)}\Upsilon (r), \end{aligned}$$
(3.2)

for \(\varepsilon \in \mathbb{R}\setminus \{\cdots ,-2,-1,0\}\). For arbitrary \(\mathsf{c},\varepsilon \in \mathbb{R}\) and \(h>0\), the rising and falling factorial functions are, respectively, defined by [27]

$$\begin{aligned} \begin{gathered} \mathsf{c}^{\overline{\varepsilon }}= \frac{\Gamma (\mathsf{c}+\varepsilon )}{\Gamma (\mathsf{c} )}, \quad \mathsf{c}\in \mathbb{R}\setminus \{\ldots ,-2,-1,0\}, \\ \mathsf{c}^{(\varepsilon )}= \frac{\Gamma (\mathsf{c}+1 )}{\Gamma (\mathsf{c}+1-\varepsilon )}, \end{gathered} \end{aligned}$$
(3.3)

such that \(0^{\overline{\varepsilon }}=0\) and we use the convention that divisions at poles yield zero.

Remark 3.1

In view of the rising and falling factorial functions in (3.3), we have

$$\begin{aligned} (\kappa _{2}-r+\varepsilon -1 )^{(\varepsilon -1)}= ( \kappa _{2}-r+1 )^{\overline{\varepsilon -1}}. \end{aligned}$$

Theorem 3.1

Let \(\Upsilon :\mathbb{Z}\to \mathbb{R}\) be a convex function on \([\kappa _{1},\kappa _{2}]_{\mathbb{Z}}\) and \(\kappa _{1},\kappa _{2}\in \mathbb{Z}\) with \(\kappa _{1}<\kappa _{2}\). If \(\kappa _{1}+\kappa _{2}\) is an even number, then, for \(\varepsilon >0\), we have

$$\begin{aligned} \begin{aligned}[b] \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr) &\leq \frac{\Gamma (\varepsilon )}{\Omega (\kappa _{2}-\kappa _{1})} \biggl[ \bigl(\Delta _{\kappa _{2}-1}^{-\varepsilon }\Upsilon \bigr) \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}-\varepsilon \biggr) + \bigl( {}_{\frac{\kappa _{1}+\kappa _{2}}{2}}{\nabla }^{-\varepsilon } \Upsilon \bigr) (\kappa _{2}) \biggr]\\&\leq \Upsilon ( \kappa _{1})+ \Upsilon (\kappa _{2}), \end{aligned} \end{aligned}$$
(3.4)

where

$$\begin{aligned} \Omega = \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}} \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}( \mathsf{c}-1)+( \varepsilon -1) \biggr)^{\overline{\varepsilon -1}} \hat{\Delta }\mathsf{c}. \end{aligned}$$

Proof

Let \(\mathsf{c}\in \top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}\). Then we can see that

$$\begin{aligned} z_{1}=\frac{\mathsf{c}}{2}\kappa _{1}+\frac{2-\mathsf{c} }{2}\kappa _{2} ,\qquad z_{2}=\frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2} \kappa _{2} \end{aligned}$$

are in \([\kappa _{1},\kappa _{2}]_{\mathbb{Z}}\) and \(z_{1}+z_{2}=\kappa _{1}+\kappa _{2}\) is even. Since \(\frac{1}{2}\in [z_{1},z_{2}]_{\mathbb{Z}}\) (or \([z_{2},z_{1}]_{\mathbb{Z}}\)) and ϒ is convex on \([z_{1},z_{2}]_{\mathbb{Z}}\) (or \([z_{2},z_{1}]_{\mathbb{Z}}\)), we can deduce

$$\begin{aligned} \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr)&=\Upsilon \biggl(\frac{z_{1}+z_{2}}{2} \biggr) \leq \frac{\Upsilon (z_{1})+\Upsilon (z_{2})}{2} \\&=\frac{1}{2} \biggl[ \Upsilon \biggl( \frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2}\kappa _{2} \biggr) +\Upsilon \biggl( \frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2}\kappa _{2} \biggr) \biggr]. \end{aligned}$$

Multiplying both sides by \((\frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+(\varepsilon -1) )^{(\varepsilon -1)}\) and then integrating over \({{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}}}\) we get

$$\begin{aligned}& \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \int _{ \top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+(\varepsilon -1) \biggr)^{(\varepsilon -1)} \hat{\Delta }\mathsf{c} \\& \quad \leq \frac{1}{2} \biggl[ \int _{\top _{ [ \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \biggl( \frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)} \Upsilon \biggl(\frac{2-\mathsf{c} }{2} \kappa _{1}+\frac{\mathsf{c}}{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\& \qquad {}+ \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)} \Upsilon \biggl( \frac{\mathsf{c}}{2}\kappa _{1}+\frac{2-\mathsf{c} }{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \biggr] \\& \quad := \frac{1}{2}[{h}_{1}+{h}_{2}], \end{aligned}$$
(3.5)

where Δ̂ is the derivative operator on the time scale \(\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}\).

We assert that

$$\begin{aligned} {h}_{1}:= \frac{2\Gamma (\varepsilon )}{\kappa _{2}-\kappa _{1}} \bigl(\Delta _{ \kappa _{2}-1}^{-\varepsilon } \Upsilon \bigr) \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}-\varepsilon \biggr). \end{aligned}$$

To prove this, we define \({w}(\mathsf{c}):= \frac{2(\mathsf{c}-\kappa _{1})}{\kappa _{2}-\kappa _{1}}\), \(g( \mathsf{c}):= (\mathsf{c}- \frac{\kappa _{1}+\kappa _{2}}{2}+(\varepsilon -1) )^{ \overline{\varepsilon -1}}\) and \(F(\mathsf{c})=g(\mathsf{c})\Upsilon (\mathsf{c})\), then we have

$$\begin{aligned} F \bigl({w}^{-1}(\mathsf{c}) \bigr)&=(gf) \bigl({w}^{-1}( \mathsf{c}) \bigr)=g \bigl({w}^{-1}(\mathsf{c}) \bigr) \Upsilon \bigl({w}^{-1}( \mathsf{c}) \bigr) \\ &= \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)} \Upsilon \biggl( \frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2}\kappa _{2} \biggr). \end{aligned}$$

Then, by making use of Theorem 2.1 for the above findings, we get

$$\begin{aligned} {h}_{1}&= \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}} \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}( \mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)} \Upsilon \biggl(\frac{2-\mathsf{c} }{2}\kappa _{1}+\frac{\mathsf{c}}{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\ &= \int _{\frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}}F(\mathsf{c}){w}^{ \Delta }( \mathsf{c})\Delta \mathsf{c} = \frac{2}{\kappa _{2}-\kappa _{1}}\sum_{r= \frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}-1} \biggl(r- \frac{\kappa _{1}+\kappa _{2}}{2}+(\varepsilon -1) \biggr)^{( \varepsilon -1)}\Upsilon (r) \\ &=\frac{2\Gamma (\varepsilon )}{\kappa _{2}-\kappa _{1}} \bigl( \Delta _{\kappa _{2}-1}^{-\varepsilon }\Upsilon \bigr) \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}-\varepsilon \biggr). \end{aligned}$$

This completes the proof of our assertion.

On the other hand, we assert that

$$\begin{aligned} {h}_{2}:= \bigl( {}_{\frac{\kappa _{1}+\kappa _{2}}{2}}{\nabla }^{-\varepsilon } \Upsilon \bigr) (\kappa _{2}). \end{aligned}$$

Then, for \({w}(\mathsf{c}):= \frac{2(\kappa _{2}-\mathsf{c} )}{\kappa _{2}-\kappa _{1}}\), we have

$$\begin{aligned} {h}_{2}&= \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2}, \kappa _{2} ]}} \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}( \mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)} \Upsilon \biggl(\frac{\mathsf{c}}{2}\kappa _{1}+\frac{2-\mathsf{c} }{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\&= \int _{\top _{ [ \frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \bigl(F\circ {w}^{-1} \bigr) (\mathsf{c}) \hat{\Delta }\mathsf{c} \\ &= \int _{0}^{1} \bigl(F\circ {w}^{-1} \bigr) (\mathsf{c})\hat{\Delta } \mathsf{c} = \int _{-1}^{0} \bigl(F\circ {w}^{-1} \bigr)^{\ast }(s) \hat{\nabla }s \\ &= \int _{-1= (f^{-1}\circ {w} ) ( \frac{\kappa _{1}+\kappa _{2}}{2} )}^{0= (f^{-1}\circ {w} )(\kappa _{2})} F \bigl( \bigl(f^{-1}\circ {w} \bigr)^{-1} \bigr) (s)\hat{\nabla }s, \end{aligned}$$

where \(f(s)=-s\), \(g(\mathsf{c}):= (\kappa _{2}-\mathsf{c} +(\varepsilon -1) )^{\overline{\varepsilon -1}}\), \(F(\mathsf{c})=g(\mathsf{c})\Upsilon (\mathsf{c})\) and we used that

$$\begin{aligned} \bigl(F\circ {w}^{-1} \bigr)^{\ast }(s)=F \bigl({w}^{-1}(-s) \bigr) =F \bigl( \bigl({w}^{-1}\circ f \bigr) (s) \bigr)=F \bigl( \bigl(f^{-1} \circ {w} \bigr)^{-1} \bigr) (s). \end{aligned}$$

Since

$$\begin{aligned} \bigl(f^{-1}\circ {w} \bigr) (s)=f^{-1} \bigl({w}(s) \bigr)=f^{-1} \biggl(\frac{2(\kappa _{2}-s)}{\kappa _{2}-\kappa _{1}} \biggr) = \frac{2(s-\kappa _{2})}{\kappa _{2}-\kappa _{1}} \end{aligned}$$

we have \({{ (f^{-1}\circ {w} )^{\Delta }(s)= \frac{2}{\kappa _{2}-\kappa _{1}}>0}}\) and hence \(f^{-1}\circ {w}\) is strictly increasing. Therefore, by making use of Theorem 2.2 and Remark 3.1, we get

$$\begin{aligned} {h}_{2}&= \int _{-1}^{0}F \bigl(f^{-1}\circ {w} \bigr)^{-1}( \mathsf{c})\hat{\nabla }\mathsf{c} =\frac{2}{\kappa _{2}-\kappa _{1}} \int _{\frac{\kappa _{1}+\kappa _{2}}{2}}^{\kappa _{2}}F(\mathsf{c}) \nabla \mathsf{c} \\ &=\frac{2}{\kappa _{2}-\kappa _{1}}\sum_{r= \frac{\kappa _{1}+\kappa _{2}}{2}+1}^{\kappa _{2}} (\kappa _{2}-r+ \varepsilon -1 )^{(\varepsilon -1)}\Upsilon (r) \\ &=\frac{2}{\kappa _{2}-\kappa _{1}}\sum_{r= \frac{\kappa _{1}+\kappa _{2}}{2}+1}^{\kappa _{2}} \bigl( \kappa _{2}- \sigma (r) \bigr)^{\overline{\varepsilon -1}}\Upsilon (r) = \frac{2\Gamma (\varepsilon )}{\kappa _{2}-\kappa _{1}} \bigl( {}_{\frac{\kappa _{1}+\kappa _{2}}{2}}{\nabla }^{-\varepsilon } \Upsilon \bigr) (\kappa _{2}). \end{aligned}$$

This completes the second assertion and thus

$$\begin{aligned} \Upsilon \biggl(\frac{\kappa _{1}+\kappa _{2}}{2} \biggr)\leq \frac{\Gamma (\varepsilon )}{\Omega (\kappa _{2}-\kappa _{1})} \biggl[ \bigl(\Delta _{\kappa _{2}-1}^{-\varepsilon }\Upsilon \bigr) \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}-\varepsilon \biggr) + \bigl( {}_{\frac{\kappa _{1}+\kappa _{2}}{2}}{\nabla }^{-\varepsilon } \Upsilon \bigr) (\kappa _{2}) \biggr]. \end{aligned}$$
(3.6)

To prove the other half of the inequality in (2.1), we use the convexity of ϒ and the following inequalities:

$$\begin{aligned}& \Upsilon \biggl(\frac{2-\mathsf{c} }{2}\kappa _{1}+ \frac{\mathsf{c}}{2} \kappa _{2} \biggr)\leq \frac{2-\mathsf{c} }{2}\Upsilon (\kappa _{1})+\frac{\mathsf{c}}{2} \Upsilon (\kappa _{2}); \\& \Upsilon \biggl(\frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2} \kappa _{2} \biggr)\leq \frac{\mathsf{c}}{2}\Upsilon (\kappa _{1})+\frac{2-\mathsf{c} }{2} \Upsilon (\kappa _{2}). \end{aligned}$$

Adding these we obtain

$$\begin{aligned} \Upsilon \biggl(\frac{\mathsf{c}}{2}\kappa _{1}+ \frac{2-\mathsf{c} }{2} \kappa _{2} \biggr)+\Upsilon \biggl( \frac{2-\mathsf{c} }{2}\kappa _{1}+\frac{\mathsf{c}}{2}\kappa _{2} \biggr) &\leq \Upsilon ( \kappa _{1})+\Upsilon (\kappa _{2}). \end{aligned}$$

Multiplying both sides by \((\frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+(\varepsilon -1) )^{(\varepsilon -1)}\) and then integrating both sides over \(\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}\) we get

$$\begin{aligned}& \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)} \Upsilon \biggl( \frac{\mathsf{c}}{2}\kappa _{1}+\frac{2-\mathsf{c} }{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\& \qquad {}+ \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \biggl(\frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)} \Upsilon \biggl( \frac{2-\mathsf{c} }{2}\kappa _{1}+\frac{\mathsf{c}}{2}\kappa _{2} \biggr)\hat{\Delta } \mathsf{c} \\& \quad \leq \bigl[\Upsilon (\kappa _{1})+\Upsilon (\kappa _{2})\bigr] \int _{\top _{ [\frac{\kappa _{1}+\kappa _{2}}{2},\kappa _{2} ]}} \biggl( \frac{\kappa _{2}-\kappa _{1}}{2}(\mathsf{c}-1)+( \varepsilon -1) \biggr)^{(\varepsilon -1)}\hat{\Delta }\mathsf{c}. \end{aligned}$$

We can use the same method used above to get

$$\begin{aligned} \frac{\Gamma (\varepsilon )}{\Omega (\kappa _{2}-\kappa _{1})} \biggl[ \bigl(\Delta _{\kappa _{2}-1}^{-\varepsilon }\Upsilon \bigr) \biggl( \frac{\kappa _{1}+\kappa _{2}}{2}-\varepsilon \biggr) + \bigl( {}_{\frac{\kappa _{1}+\kappa _{2}}{2}}{ \nabla }^{-\varepsilon } \Upsilon \bigr) (\kappa _{2}) \biggr]\leq \Upsilon (\kappa _{1})+ \Upsilon (\kappa _{2}), \end{aligned}$$

and thus the result follows. □

Remark 3.2

In the literature of fractional integral inequalities there are three major HH-type inequalities, namely the endpoint, midpoint and end–midpoint HH-types inequalities; for more details we advise the reader to read the Discussion section of Ref. [28]. Fortunately, the endpoint version of HH-type inequality in the time scale notation has been established by Atıcı and Yaldız in [10]. Also, the inequality (3.4) obtained in Theorem 3.1 represents the midpoint version of HH-type inequality which has never been presented before.

4 Conclusion

The inequality of HH-type plays a crucial role in the theory and application of convex functions. During the last two decades, it has been used as an essential tool to obtain many results in approximation theory, integral inequalities, numerical analysis and optimization theory. In this study, we have considered new discrete time scales to obtain some inequalities of midpoint type for convex functions which have never presented before.