Some Steffensen-type dynamic inequalities on time scales

Abstract

We consider some new Steffensen-type dynamic inequalities on an arbitrary time scale by utilizing the diamond-α dynamic integrals, which are characterized as a combination of the delta and nabla integrals. These inequalities expand some known dynamic inequalities on time scales, bind together and broaden some integral inequalities and their discrete analogs.

1 Introduction

The renowned integral Steffensen inequality [28] is written as

$$ \int _{b-\lambda }^{b}\phi (t)\,dt\leq \int _{a}^{b}\phi (t)\psi (t)\,dt \leq \int _{a}^{a+\lambda }\phi (t)\,dt, $$

(1.1)

where u is nonincreasing, $\lambda =\int _{a}^{b}\psi (t)\,dt$ and $0\leq \psi (t)\leq 1$ on $[a,b]$. It is simple to notice that inequalities (1.1) are reversed if u is nondecreasing.

Also we have

$$ \sum_{k=n-\lambda _{2}+1}^{n} \phi (k) \leq \sum_{k=1}^{n}\phi (k)\psi (k) \leq \sum _{k=1}^{\lambda _{1}} \phi (k) $$

(1.2)

such that $0 \leq \psi (k) \leq 1$, $\lambda _{1}, \lambda _{2}\in \{1, \dots ,n\}$ with $\lambda _{2}\leq \sum_{k=1}^{n}\psi (k)\leq \lambda _{1}$. Inequality (1.2) is known as discrete Steffensen’s inequality [15].

Stefan Hilger started the hypotheses of time scales in his PhD thesis [16] so as to bring together discrete and continuous analysis (see [17]). From that point onward, this theory has gotten a ton of consideration. The book due to Bohner and Peterson [9] regarding the matter of time scales briefs and sorts out a lot of time scales calculus.

Over the previous decade, a reasonable number of dynamic inequalities on time scales has been proven by many analysts who were propelled by certain applications (see [1,2,3,4, 9,10,11,12,13,14, 18, 29]). A few researchers created different outcomes concerning fractional calculus on time scales to deliver related dynamic inequalities (see [5,6,7, 24]).

Anderson, in [8], extended Steffensen’s inequality to times scale with nabla integrals as follows:

$$ \int _{b-\lambda }^{b}\phi (t)\nabla t\leq \int _{a}^{b}\phi (t)\psi (t) \nabla t\leq \int _{a}^{a+\lambda }\phi (t)\nabla t, $$

(1.3)

where u is of one sign and nonincreasing, $0 \leq \psi (t) \leq 1$ for every $t\in [a,b]_{\mathbb{T}}$, $\lambda =\int _{a}^{b}\psi (t)\nabla t$, and $b-\lambda , a+\lambda \in [a,b]_{\mathbb{T}}$.

By employing diamond-α integrals, Ozkan and Yildirim [21] gave a generalization of inequality (1.3) of the form:

If the following

$$\begin{aligned}& \int _{l}^{b}w(t)\diamondsuit _{\alpha }t\leq \int _{a}^{b}\phi (t) \diamondsuit _{\alpha }t \leq \int _{a}^{\eta }w(t)\diamondsuit _{\alpha }t \quad \mbox{if } u\geq 0, t\in [a,b]_{\mathbb{T}}, \\& \int _{l}^{b}w(t)\diamondsuit _{\alpha }t\geq \int _{a}^{b}\phi (t) \diamondsuit _{\alpha }t \geq \int _{a}^{\eta }w(t)\diamondsuit _{\alpha }t \quad \mbox{if } u\leq 0, t\in [a,b]_{\mathbb{T}}, \end{aligned}$$

hold, then

$$ \int _{l}^{b}u(t)w(t)\diamondsuit _{\alpha }t \leq \int _{a}^{b}u(t)v(t) \diamondsuit _{\alpha }t \leq \int _{a}^{\eta }u(\tau )w(t) \diamondsuit _{\alpha }t, $$

(1.4)

where $0 \leq \psi (t) \leq w(t)$ for all $t\in [a,b]_{\mathbb{T}}$ with $l, \eta \in [a,b]_{\mathbb{T}}$.

Also in [21], the authors have given the following interesting result:

$$\begin{aligned}& \int _{b-\lambda }^{b}\phi (t)w(t)\diamondsuit _{\alpha }t+ \int _{a} ^{b} \bigl\vert \bigl[\phi (t)-\phi (b- \lambda )\bigr]z(t) \bigr\vert \diamondsuit _{\alpha }t \\& \quad \leq \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t \\& \quad \leq \int _{a}^{a+\lambda }\phi (t)w(t)\diamondsuit _{\alpha }t- \int _{a}^{b} \bigl\vert \bigl[\phi (t)-\phi (a+ \lambda ) \bigr]z(t) \bigr\vert \diamondsuit _{\alpha }t, \end{aligned}$$

with u is nonincreasing, $0 \leq z(t)\leq \psi (t)\leq w(t)-z(t)$ for every $t\in [a,b]_{\mathbb{T}}$, $\int _{b-\lambda }^{b}w(t) \diamondsuit _{\alpha }t=\int _{a}^{b}\psi (t)\diamondsuit _{\alpha }t = \int _{a}^{a+\lambda }w(t)\diamondsuit _{\alpha }t$, and $b-\lambda , a+\lambda \in [a,b]_{\mathbb{T}}$.

The following inequality is a special case of the above inequality: if we put $z(t)=M$ and $w(t)=1$, so

$$\begin{aligned}& \int _{b-\lambda }^{b}\phi (t)\diamondsuit _{\alpha }t+M \int _{a}^{b} \bigl\vert \phi (t)-\phi (b-\lambda ) \bigr\vert \diamondsuit _{\alpha }t \\& \quad \leq \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t \\& \quad \leq \int _{a}^{a+\lambda }\phi (t)\diamondsuit _{\alpha }t-M \int _{a} ^{b} \bigl\vert \phi (t)-\phi (a+\lambda ) \bigr\vert \diamondsuit _{\alpha }t, \end{aligned}$$

$a, b\in \mathbb{T_{\kappa }^{\kappa }}$ with $a< b$, $\lambda =\int _{a}^{b}\psi (t)\diamondsuit _{\alpha }t$, and $0\leq M\leq \psi (t) \leq 1-M$ for all $t\in [a,b]_{\mathbb{T}}$.

Since its establishment, Steffensen’s inequality has played crucial roles in numerous fields of mathematics, particularly in mathematical analysis. In the past several decades, numerous speculations and refinements of Steffensen’s inequality have been given by different authors. A few researchers have focused on Steffensen’s inequality related to local and conformable fractional integrals (see [22, 25, 26, 30]). For a comprehensive review, we refer the interested reader to the books [19, 20] and the references cited in them.

This article is about to extend some Steffensen-type inequalities given in [23] to a general time scale, and build up some new generalizations of the diamond-α dynamic Steffensen inequality on time scales. As special cases of our outcomes, we recapture the integral inequalities presented in the above mentioned paper. Our outcomes additionally give several original discrete Steffensen’s inequalities.

We get the unique Steffensen inequalities by utilizing the diamond-α integrals on time scales. For $\alpha = 1$, the diamond-α integral moves toward becoming delta integral and for $\alpha = 0$ it moves toward becoming nabla integral. An excellent review about the diamond-α calculus can be viewed in the paper [27].

2 Basics of time scales

For our convenience, $\mathbb{R}$ is the set of real numbers, $\mathbb{Z}$ is the set of integers, and a time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the set of real numbers $\mathbb{R}$. If $\mathbb{T}$ has a left-scattered maximum $t_{1}$, then $\mathbb{T}^{\kappa } = \mathbb{T}-\{t_{1}\}$, otherwise $\mathbb{T} ^{\kappa } = \mathbb{T}$. If $\mathbb{T}$ has a right-scattered minimum $t_{2}$, then $\mathbb{T}^{\kappa } = \mathbb{T}-\{t_{2}\}$, otherwise $\mathbb{T}_{\kappa } = \mathbb{T}$. Finally, we have $\mathbb{T}^{ \kappa }_{\kappa }=\mathbb{T}^{\kappa }\cap \mathbb{T}_{\kappa }$. The interval $[a,b]_{\mathbb{T}}=\{t\in \mathbb{T}:a\leq t\leq b\}$.

Assume the function $\phi : \mathbb{T} \rightarrow \mathbb{R}$, $t\in \mathbb{T}^{\kappa }$, then $\phi ^{\Delta }(t)\in \mathbb{R}$, $\phi ^{\nabla }(t)\in \mathbb{R}$ are said to be the delta derivative and nabla derivative of ϕ at t, respectively, if for any $\varepsilon > 0$ there exist a neighborhood U and a neighborhood V of t such that, for all $s\in U$ and $s\in V$ simultaneously, we have

$$ \bigl\vert \bigl[\phi \bigl(\sigma (t)\bigr)-\phi (s)\bigr]-\phi ^{\Delta }(t)\bigl[\sigma (t)-s\bigr] \bigr\vert \leq \varepsilon \bigl\vert \sigma (t)-s \bigr\vert $$

and

$$ \bigl\vert \bigl[\phi \bigl(\rho (t)\bigr)-\phi (s)\bigr]-\phi ^{\nabla }(t)\bigl[\rho (t)-s\bigr] \bigr\vert \leq \varepsilon \bigl\vert \rho (t)-s \bigr\vert . $$

Moreover, ϕ is said to be delta differentiable on $\mathbb{T} ^{\kappa }$ if it is delta differentiable at every $t\in \mathbb{T} ^{\kappa }$ and is said to be nabla differentiable on $\mathbb{T}_{ \kappa }$ if it is nabla differentiable at each $t\in \mathbb{T}_{ \kappa }$.

There is the following formula of delta integration by parts on time scales:

$$ \int _{a}^{b}\psi ^{\Delta }(t)\phi (t)\Delta t= \psi (b)\phi (b)-\psi (a) \phi (a)- \int _{a}^{b}\psi ^{\sigma }(t)\phi (t)\Delta t, $$

(2.1)

the nabla integration by parts on time scales is given by

$$ \int _{a}^{b}\psi ^{\nabla }(t)\phi (t)\nabla t=\psi (b)\phi (b)-\psi (a) \phi (a)- \int _{a}^{b}\psi ^{\rho }(t)\phi (t)\nabla t. $$

(2.2)

We will use the following relations between calculus on time scales $\mathbb{T}$ and either differential calculus on $\mathbb{R}$ or difference calculus on $\mathbb{Z}$. Note that:

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

  $$ \begin{aligned} &\sigma (t)=\rho (t)=t, \qquad \mu (t)= \nu (t)=0, \qquad \phi ^{\Delta }(t)= \phi ^{\nabla }(t)=\phi '(t), \\ & \int _{a}^{b}\phi (t)\Delta t= \int _{a}^{b}\phi (t)\nabla t= \int _{a} ^{b}\phi (t)\,dt. \end{aligned} $$

  (2.3)

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

  $$ \begin{aligned} &\sigma (t)=t+1, \qquad \rho (t)=t-1, \qquad \mu (t)=\nu (t)=1, \\ &\phi ^{\Delta }(t)=\Delta \phi (t), \qquad \phi ^{\nabla }(t)=\nabla \phi (t), \\ & \int _{a}^{b}\phi (t)\Delta t=\sum _{t=a}^{b-1}\phi (t),\qquad \int _{a} ^{b}\phi (t)\nabla t=\sum _{t=a+1}^{b}\phi (t), \end{aligned} $$

  (2.4)

  where the forward and backward difference operators are denoted by Δ and ∇, respectively.

We dedicate the rest of this section to the diamond-α calculus on time scales, and we recommend the paper [27] for further knowledge.

For any $t\in \mathbb{T}$, the diamond-α dynamic derivative of u at t is defined by

$$ u^{\diamondsuit _{\alpha }}(t) = \alpha u^{\Delta }(t)+(1-\alpha )u ^{\nabla }(t), \quad 0\leq \alpha \leq 1, $$

(2.5)

and denoted by $u^{\diamondsuit _{\alpha }}(t)$, where $\mathbb{T}$ is a time scale, and u is a function that is delta and nabla differentiable on $\mathbb{T}$.

Now, it is time to discuss our main results.

3 Main results

Lemma 3.1

Assume that

• (B1) k is a positive $\diamondsuit _{\alpha }$-integrable function on $[a, b]_{\mathbb{T}}$.

• (B2) ϕ, ψ, $h: [a, b]_{\mathbb{T}}\rightarrow \mathbb{R}$ are $\diamondsuit _{\alpha }$-integrable functions on $[a, b]_{ \mathbb{T}}$.

• (B3) $[c, d]_{\mathbb{T}}\subseteq [a, b]_{\mathbb{T}}$ with $\int _{c}^{d}h(t)k(t) \diamondsuit _{\alpha }t =\int _{a}^{b}\psi (t)k(t) \diamondsuit _{\alpha }t$.

• (B4) $z\in [a, b]_{\mathbb{T}}$.

Then

$$\begin{aligned} \int _{c}^{d}\phi (t)h(t)\diamondsuit _{\alpha }t- \int _{a}^{b}\phi (t) \psi (t)\diamondsuit _{\alpha }t =& \int _{c}^{a} \biggl( \frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t \\ &{} + \int _{c}^{d} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (z)}{k(z)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\diamondsuit _{\alpha }t \\ &{} + \int _{d}^{b} \biggl(\frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\diamondsuit _{\alpha }t. \end{aligned}$$

(3.1)

Proof

By straightforward calculations, we get

$$\begin{aligned}& \int _{c}^{d}\phi (t)h(t)\diamondsuit _{\alpha }t- \int _{a}^{b}\phi (t) \psi (t)\diamondsuit _{\alpha }t \\& \quad = \int _{c}^{d}k(t) \bigl[h(t)-\psi (t) \bigr] \frac{\phi (t)}{k(t)} \diamondsuit _{\alpha }t- \biggl[ \int _{a}^{c}\frac{\phi (t)}{k(t)} \psi (t)k(t) \diamondsuit _{\alpha }t+ \int _{d}^{b} \frac{\phi (t)}{k(t)}\psi (t)k(t) \diamondsuit _{\alpha }t \biggr] \\& \quad = \int _{a}^{c} \biggl(\frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\diamondsuit _{\alpha }t + \int _{c}^{d} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (z)}{k(z)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\diamondsuit _{\alpha }t \\& \qquad {} + \int _{d}^{b} \biggl(\frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\diamondsuit _{\alpha }t \\& \qquad {} +\frac{\phi (z)}{k(z)} \biggl[ \int _{c}^{d}k(t)h(t) \diamondsuit _{\alpha }t- \int _{a}^{c}\psi (t)k(t)\diamondsuit _{\alpha }t \\& \qquad {}- \int _{c}^{d}\psi (t)k(t)\diamondsuit _{\alpha }t- \int _{d}^{b}\psi (t)k(t) \diamondsuit _{\alpha }t \biggr]. \end{aligned}$$

(3.2)

Consider

$$ \int _{c}^{d}k(t)h(t) \diamondsuit _{\alpha }t= \int _{a}^{b}k(t)\psi (t) \diamondsuit _{\alpha }t, $$

therefore

$$\begin{aligned}& \frac{\phi (z)}{k(z)} \biggl[ \int _{c}^{d}k(t)h(t)\diamondsuit _{\alpha }t- \int _{a}^{c}\psi (t)k(t)\diamondsuit _{\alpha }t \\& \quad {}- \int _{c}^{d}\psi (t)k(t) \diamondsuit _{\alpha }t- \int _{d}^{b}\psi (t)k(t)\diamondsuit _{\alpha }t \biggr]=0. \end{aligned}$$

(3.3)

Our desired result follows directly from (3.2) and (3.3). □

Corollary 3.2

Setting $\alpha =1$ in Lemma 3.1, we get the delta form of inequality (3.1) by

$$\begin{aligned} \int _{c}^{d}\phi (t)h(t)\Delta t- \int _{a}^{b}\phi (t)\psi (t)\Delta t =& \int _{c}^{a} \biggl(\frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr) \psi (t)k(t)\Delta t \\ &{} + \int _{c}^{d} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (z)}{k(z)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\Delta t \\ &{} + \int _{d}^{b} \biggl(\frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\Delta t. \end{aligned}$$

(3.4)

Corollary 3.3

Letting $\alpha =0$ in Lemma 3.1, we obtain the nabla version of (3.1) as follows:

$$\begin{aligned} \int _{c}^{d}\phi (t)h(t)\nabla t- \int _{a}^{b}\phi (t)\psi (t)\nabla t =& \int _{c}^{a} \biggl(\frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr) \psi (t)k(t)\nabla t \\ &{} + \int _{c}^{d} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (z)}{k(z)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\nabla t \\ &{} + \int _{d}^{b} \biggl(\frac{\phi (z)}{k(z)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\nabla t. \end{aligned}$$

(3.5)

Corollary 3.4

If $\mathbb{T}=\mathbb{R}$ in Corollary 3.2, then, with the help of relation (2.3), we recapture [23, Lemma 2.1].

Corollary 3.5

If $\mathbb{T}=\mathbb{Z}$ in Corollary 3.2, then, with the help of relation (2.4), inequality (3.4) becomes

$$\begin{aligned} \sum_{t=c}^{d-1}\phi (t)h(t)-\sum _{a}^{b}\phi (t)\psi (t)\,dt =&\sum _{t=c}^{a-1} \biggl(\frac{\phi (z)}{k(z)}-\frac{\phi (t)}{k(t)} \biggr) \psi (t)k(t) \\ &{} +\sum_{t=c}^{d-1} \biggl( \frac{\phi (t)}{k(t)}-\frac{\phi (z)}{k(z)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr] \\ &{} +\sum_{t=d}^{b-1} \biggl( \frac{\phi (z)}{k(z)}-\frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t). \end{aligned}$$

Theorem 3.6

Let (B1)–(B3) of Lemma 3.1,

• (B5) $\phi /k$ is nonincreasing, and

• (B6) $0\leq \psi (t)\leq h(t)$ $\forall t\in [a,b]_{\mathbb{T}}$

be satisfied, then the following inequalities hold:

$$\begin{aligned} (\mathrm{i})&\quad \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t\leq \int _{c}^{d}\phi (t)\psi (t)\diamondsuit _{\alpha }t+ \int _{a}^{c} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t, \end{aligned}$$

(3.6)

$$\begin{aligned} (\mathrm{ii})&\quad \int _{c}^{d}\phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{a}^{c} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t\leq \int _{a}^{b}\phi (t)\psi (t) \diamondsuit _{\alpha }t, \end{aligned}$$

(3.7)

$$\begin{aligned} (\mathrm{iii})&\quad \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t \leq \int _{c}^{d}\phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{c}^{d} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\diamondsuit _{\alpha }t \\ &\quad \hphantom{\int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t \leq{}}{}+ \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\diamondsuit _{\alpha }t \\ &\quad \hphantom{\int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t} \leq \int _{c}^{d}\phi (t)\psi (t)\diamondsuit _{\alpha }t + \int _{a} ^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t, \end{aligned}$$

(3.8)

$$\begin{aligned} (\mathrm{iv})&\quad \int _{c}^{d}\phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{a}^{c} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t \\ &\qquad \leq \int _{c}^{d}\phi (t)\psi (t)\diamondsuit _{\alpha }t + \int _{c} ^{d} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)k(t) \bigl[h(t)- \psi (t) \bigr]\diamondsuit _{\alpha }t \\ &\quad \qquad {} - \int _{a}^{c} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\diamondsuit _{\alpha }t \\ &\qquad \leq \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t. \end{aligned}$$

(3.9)

If $\phi /k$ is nondecreasing, then inequalities (3.6), (3.7), (3.8), and (3.9) should be switched.

Proof

(i) Since $\phi /k$ is nonincreasing, k is positive, and $0\leq \psi \leq h$, we have

$$ \int _{c}^{d} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\diamondsuit _{\alpha }t\geq 0 $$

(3.10)

and

$$ \int _{d}^{b} \biggl(\frac{\phi (d)}{k(d)}- \frac{\phi (t)}{k(t)} \biggr) \psi (t)k(t) \diamondsuit _{\alpha }t\geq 0. $$

(3.11)

From (3.1), (3.10), and (3.11) with $z=d$, we obtain

$$\begin{aligned}& \int _{c}^{d}\phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{a}^{b} \phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{a}^{c} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t \\& \quad = \int _{c}^{d} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\diamondsuit _{\alpha }t + \int _{d} ^{b} \biggl(\frac{\phi (d)}{k(d)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t\geq 0. \end{aligned}$$

This proves our claim.

(ii) Since $\phi /k$ is nonincreasing, k is positive, and $0\leq \psi \leq h$, we have

$$ \int _{c}^{d} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\diamondsuit _{\alpha }t\geq 0 $$

(3.12)

and

$$ \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (c)}{k(c)} \biggr) \psi (t)k(t) \diamondsuit _{\alpha }t\geq 0. $$

(3.13)

From (3.1) with $z=c$, (3.12), and (3.13), we have

$$\begin{aligned}& \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{c}^{d} \phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{d}^{b} \biggl(\frac{ \phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t \\& \quad = \int _{c}^{d} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\diamondsuit _{\alpha }t + \int _{a} ^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (c)}{k(c)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t\geq 0. \end{aligned}$$

This completes our proof.

The proof of (iii), (iv) is similar to (i), (ii) of Theorem 3.6, respectively. Details are omitted. □

Corollary 3.7

Substituting $\alpha =1$ and $\alpha =0$ in Theorem 3.6(i), (ii), (iii), (iv) simultaneously, we achieve the following delta and nabla versions of inequalities (3.6), (3.7), (3.8), and (3.9), respectively:

$$\begin{aligned} (\mathrm{v})&\quad \int _{a}^{b}\phi (t)\psi (t)\Delta t\leq \int _{c}^{d}\phi (t) \psi (t)\Delta t+ \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\Delta t, \end{aligned}$$

(3.14)

$$\begin{aligned} (\mathrm{vi})&\quad \int _{c}^{d}\phi (t)\psi (t)\Delta t- \int _{a}^{c} \biggl(\frac{ \phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\Delta t\leq \int _{a}^{b}\phi (t)\psi (t)\Delta t, \end{aligned}$$

(3.15)

$$\begin{aligned} (\mathrm{vii})&\quad \int _{a}^{b}\phi (t)\psi (t)\Delta t \leq \int _{c}^{d}\phi (t)\psi (t) \Delta t- \int _{c}^{d} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\Delta t \\ &\quad \hphantom{\int _{a}^{b}\phi (t)\psi (t)\Delta t \leq{}}{} + \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\Delta t \\ &\quad \hphantom{\int _{a}^{b}\phi (t)\psi (t)\Delta t}\leq \int _{c}^{d}\phi (t)\psi (t)\Delta t + \int _{a}^{c} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\Delta t, \end{aligned}$$

(3.16)

$$\begin{aligned} (\mathrm{viii})&\quad \int _{c}^{d}\phi (t)\psi (t)\Delta t- \int _{a}^{c} \biggl( \frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\Delta t \\ &\qquad \leq \int _{c}^{d}\phi (t)\psi (t)\Delta t + \int _{c}^{d} \biggl(\frac{ \phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\Delta t \\ &\qquad \quad {} - \int _{a}^{c} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\Delta t \\ &\qquad \leq \int _{a}^{b}\phi (t)\psi (t)\Delta t, \\ (\mathrm{ix})&\quad \int _{a}^{b}\phi (t)\psi (t)\nabla t\leq \int _{c}^{d} \phi (t)\psi (t)\nabla t+ \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{ \phi (d)}{k(d)} \biggr)\psi (t)k(t)\nabla t, \\ (\mathrm{x})&\quad \int _{c}^{d}\phi (t)\psi (t)\nabla t- \int _{a}^{c} \biggl(\frac{ \phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\nabla t\leq \int _{a}^{b}\phi (t)\psi (t)\nabla t, \\ (\mathrm{xi})&\quad \int _{a}^{b}\phi (t)\psi (t)\nabla t \leq \int _{c}^{d}\phi (t)\psi (t) \nabla t- \int _{c}^{d} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\nabla t \\ &\quad \hphantom{\int _{a}^{b}\phi (t)\psi (t)\nabla t \leq{}}{} + \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\nabla t \\ &\quad \hphantom{\int _{a}^{b}\phi (t)\psi (t)\nabla t}\leq \int _{c}^{d}\phi (t)\psi (t)\nabla t + \int _{a}^{c} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\nabla t, \\ (\mathrm{xii})&\quad \int _{c}^{d}\phi (t)\psi (t)\nabla t- \int _{a}^{c} \biggl( \frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\nabla t \\ &\qquad \leq \int _{c}^{d}\phi (t)\psi (t)\nabla t + \int _{c}^{d} \biggl(\frac{ \phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr]\nabla t \\ &\qquad \quad {} - \int _{a}^{c} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\nabla t \\ &\qquad \leq \int _{a}^{b}\phi (t)\psi (t)\nabla t. \end{aligned}$$

(3.17)

Corollary 3.8

If $\mathbb{T}=\mathbb{R}$ in Corollary 3.7(v), (vi), (vii), (viii), then with the help of relation (2.3), we recapture [23, Theorem 2.1, Theorem 2.2, Theorem 2.3, Theorem 2.4], respectively.

Corollary 3.9

If $\mathbb{T}=\mathbb{Z}$ and applying (2.4), then inequalities (3.14), (3.15), (3.16), and (3.17), respectively, give

$$\begin{aligned}& \sum_{t=a}^{b-1}\phi (t)\psi (t)\leq \sum_{t=c}^{d-1}\phi (t)\psi (t)+ \sum_{t=a}^{c-1} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t), \end{aligned}$$

(3.18)

$$\begin{aligned}& \sum_{t=c}^{d-1}\phi (t)\psi (t)-\sum_{t=a}^{c-1} \biggl( \frac{\phi (c)}{k(c)}-\frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \leq \sum _{t=a}^{b-1}\phi (t)\psi (t), \end{aligned}$$

(3.19)

$$\begin{aligned}& \begin{aligned}[b] \sum_{t=a}^{b-1}\phi (t) \psi (t) &\leq \sum_{t=c}^{d-1}\phi (t)\psi (t)t- \sum_{t=c}^{d-1} \biggl( \frac{\phi (t)}{k(t)}-\frac{\phi (d)}{k(d)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr] \\ &\quad {} +\sum_{t=a}^{c-1} \biggl( \frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \\ &\leq \sum_{t=c}^{d-1}\phi (t)\psi (t)+\sum _{t=a}^{c-1} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t), \end{aligned} \end{aligned}$$

(3.20)

$$\begin{aligned}& \sum_{t=c}^{d-1}\phi (t)\psi (t)- \sum_{t=a}^{c-1} \biggl( \frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \\& \quad \leq \sum_{t=c}^{d-1}\phi (t)\psi (t)+\sum_{t=c}^{d-1} \biggl( \frac{ \phi (c)}{k(c)}-\frac{\phi (t)}{k(t)} \biggr)k(t) \bigl[h(t)-\psi (t) \bigr] \\& \qquad {} -\sum_{t=a}^{c-1} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \\& \quad \leq \sum_{t=a}^{b-1}\phi (t)\psi (t). \end{aligned}$$

(3.21)

Theorem 3.10

Let (B1)–(B3) and

• (B7) $\phi /k$ is nonincreasing in the Δ and ∇ sense,

be fulfilled.

• (i) If

  $$\begin{aligned}& \int _{c}^{\sigma (x)}k(t)\psi (t)\Delta t \leq \int _{c}^{\sigma (x)}k(t)h(t) \Delta t, \quad c\leq x \leq d, \\& \int _{c}^{\rho (x)}k(t)\psi (t)\nabla t \leq \int _{c}^{\rho (x)}k(t)h(t) \nabla t,\quad c\leq x \leq d, \\& \int _{\sigma (x)}^{b}k(t)\psi (t) \Delta t\geq 0, \quad d \leq x\leq b, \\& \int _{\rho (x)}^{b}k(t)\psi (t) \nabla t\geq 0, \quad d \leq x\leq b, \end{aligned}$$

  then

  $$ \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t\leq \int _{c}^{d} \phi (t)h(t)\diamondsuit _{\alpha }t+ \int _{a}^{c} \biggl( \frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t. $$

  (3.22)

• (ii) If

  $$\begin{aligned}& \int _{\sigma (x)}^{d}k(t)\psi (t) \Delta t \leq \int _{\sigma (x)} ^{d}k(t)h(t) \Delta t, \quad c\leq x \leq d, \\& \int _{\rho (x)}^{d}k(t)\psi (t) \nabla t \leq \int _{\rho (x)}^{d}k(t)h(t) \nabla t, \quad c\leq x \leq d, \\& \int _{a}^{\sigma (x)}k(t)\psi (t) \Delta t\geq 0, \quad a \leq x\leq c, \\& \int _{a}^{\rho (x)}k(t)\psi (t) \nabla t\geq 0, \quad a \leq x\leq c, \end{aligned}$$

  then

  $$ \int _{c}^{d}\phi (t)h(t)\diamondsuit _{\alpha }t- \int _{d}^{b} \biggl(\frac{ \phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t \leq \int _{a}^{b}\phi (t)\psi (t) \diamondsuit _{\alpha }t. $$

  (3.23)

Proof

(i) Utilizing (3.4) and delta integration by parts formula on time scales, we get

$$\begin{aligned}& \int _{c}^{d} \phi (t)h(t) \Delta t+ \int _{a}^{b} \phi (t)\psi (t) \Delta t + \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \Delta t \\& \quad = \biggl[- \int _{c}^{d} \biggl( \int _{c}^{\sigma (x)}k(t) \bigl[h(t)-\psi (t) \bigr]\Delta t \biggr) \biggl(\frac{\phi (x)}{k(x)} \biggr)^{\Delta }\Delta x \biggr] \\& \qquad {} \times \biggl[- \int _{d}^{b} \biggl( \int _{\sigma (x)}^{b}\psi (t)k(t) \Delta t \biggr) \biggl( \frac{\phi (x)}{k(x)} \biggr)^{\Delta }\Delta x \biggr]\geq 0. \end{aligned}$$

In a similar manner, using (3.5) and nabla integration by parts formula on time scales, we have

$$\begin{aligned}& \int _{c}^{d} \phi (t)h(t) \nabla t+ \int _{a}^{b} \phi (t)\psi (t) \nabla t + \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \nabla t \\& \quad = \biggl[- \int _{c}^{d} \biggl( \int _{c}^{\rho (x)}k(t) \bigl[h(t)-\psi (t) \bigr]\nabla t \biggr) \biggl(\frac{\phi (x)}{k(x)} \biggr)^{\nabla }\nabla x \biggr] \\& \qquad {} \times \biggl[- \int _{d}^{b} \biggl( \int _{\rho (x)}^{b}\psi (t)k(t) \nabla t \biggr) \biggl( \frac{\phi (x)}{k(x)} \biggr)^{\nabla }\nabla x \biggr]\geq 0. \end{aligned}$$

Therefore

$$\begin{aligned}& \int _{c}^{d} \phi (t)h(t) \diamondsuit _{\alpha }t+ \int _{a}^{b} \phi (t)\psi (t) \diamondsuit _{\alpha }t + \int _{a}^{c} \biggl(\frac{ \phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t \\& \quad =\alpha \int _{c}^{d} \phi (t)h(t) \Delta t+(1-\alpha ) \int _{c}^{d} \phi (t)h(t) \nabla t \\& \qquad {}+ \alpha \int _{a}^{b} \phi (t)\psi (t) \Delta t +(1- \alpha ) \int _{a}^{b} \phi (t)\psi (t) \nabla t \\& \qquad {} +\alpha \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \Delta t \\& \qquad {}+(1-\alpha ) \int _{a} ^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) \nabla t\geq 0. \end{aligned}$$

Hence, (3.22) holds.

(ii) Using (3.4) and delta integration by parts, we have

$$\begin{aligned}& \int _{a}^{b} \phi (t)\psi (t) \Delta t- \int _{c}^{d} \phi (t)h(t) \Delta t + \int _{d}^{b} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \Delta t \\& \quad = \biggl[- \int _{a}^{c} \biggl( \int _{a}^{\sigma (x)}\psi (t)k(t)\Delta t \biggr) \biggl( \frac{\phi (x)}{k(x)} \biggr)^{\Delta }\Delta x \biggr] \\& \qquad {} \times \biggl[- \int _{c}^{d} \biggl( \int _{\sigma (x)}^{d}k(t) \bigl[h(t)-\psi (t) \bigr]\Delta t \biggr) \biggl(\frac{\phi (x)}{k(x)} \biggr)^{ \Delta }\Delta x \biggr]\geq 0. \end{aligned}$$

Now, (3.5) and nabla integration by parts yield

$$\begin{aligned}& \int _{a}^{b} \phi (t)\psi (t) \nabla t- \int _{c}^{d} \phi (t)h(t) \nabla t + \int _{d}^{b} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \nabla t \\& \quad = \biggl[- \int _{a}^{c} \biggl( \int _{a}^{\rho (x)}\psi (t)k(t)\nabla t \biggr) \biggl( \frac{\phi (x)}{k(x)} \biggr)^{\nabla }\nabla x \biggr] \\& \qquad {} \times \biggl[- \int _{c}^{d} \biggl( \int _{\rho (x)}^{d}k(t) \bigl[h(t)- \psi (t) \bigr]\nabla t \biggr) \biggl(\frac{\phi (x)}{k(x)} \biggr)^{\nabla } \nabla x \biggr]\geq 0, \end{aligned}$$

so that

$$\begin{aligned}& \int _{a}^{b} \phi (t)\psi (t)\diamondsuit _{\alpha }t- \int _{c}^{d} \phi (t)h(t)\diamondsuit _{\alpha }t + \int _{d}^{b} \biggl( \frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \diamondsuit _{\alpha }t \\& \quad =\alpha \int _{a}^{b} \phi (t)\psi (t)\Delta t+(1-\alpha ) \int _{a} ^{b} \phi (t)\psi (t)\nabla t \\& \qquad {}-\alpha \int _{c}^{d} \phi (t)h(t)\Delta t -(1-\alpha ) \int _{c}^{d} \phi (t)h(t)\nabla t \\& \qquad {} +\alpha \int _{d}^{b} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \Delta t \\& \qquad {}+(1-\alpha ) \int _{d} ^{b} \biggl(\frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \nabla t\geq 0, \end{aligned}$$

from which (3.23) is satisfied. □

Corollary 3.11

Setting $\alpha =1$ and $\alpha =0$ in Theorem 3.10(i), (ii) simultaneously, we obtain the delta and nabla versions of inequalities (3.22) and (3.23), respectively, as follows:

$$\begin{aligned} (\mathrm{i})&\quad \int _{a}^{b}\phi (t)\psi (t)\Delta t\leq \int _{c}^{d}\phi (t)h(t) \Delta t+ \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\Delta t, \end{aligned}$$

(3.24)

$$\begin{aligned} (\mathrm{ii})&\quad \int _{c}^{d}\phi (t)h(t)\Delta t- \int _{d}^{b} \biggl( \frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\Delta t \leq \int _{a}^{b}\phi (t)\psi (t)\Delta t, \\ (\mathrm{iii})&\quad \int _{a}^{b}\phi (t)\psi (t)\nabla t\leq \int _{c}^{d}\phi (t)h(t) \nabla t+ \int _{a}^{c} \biggl(\frac{\phi (t)}{k(t)}- \frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t)\nabla t, \\ (\mathrm{iv})&\quad \int _{c}^{d}\phi (t)h(t)\nabla t- \int _{d}^{b} \biggl( \frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t)\nabla t \leq \int _{a}^{b}\phi (t)\psi (t)\nabla t. \end{aligned}$$

(3.25)

Corollary 3.12

If $\mathbb{T}=\mathbb{R}$ in Corollary 3.11, then, with the help of (2.3), (i), (ii) recover [23, Theorem 2.5, Theorem 2.6], respectively.

Corollary 3.13

If $\mathbb{T}=\mathbb{Z}$ in Corollary 3.11, then, with the help of relation (2.4), inequalities (3.24) and (3.25) turn into

$$ \sum_{t=a}^{b-1}\phi (t)\psi (t)\leq \sum _{t=c}^{d-1}\phi (t)h(t)+ \sum _{t=a}^{c-1} \biggl(\frac{\phi (t)}{k(t)}-\frac{\phi (d)}{k(d)} \biggr)\psi (t)k(t) $$

and

$$ \sum_{t=c}^{d-1}\phi (t)h(t)-\sum _{t=d}^{b-1} \biggl( \frac{\phi (c)}{k(c)}- \frac{\phi (t)}{k(t)} \biggr)\psi (t)k(t) \leq \sum_{t=a}^{b-1} \phi (t)\psi (t), $$

respectively.

The following theorem can be obtained by taking $c=a$ and $d=a+\lambda$ in Theorem 3.10.

Theorem 3.14

Let (B1)–(B3), (B7) hold.

• (i) If λ is defined by $\int _{a}^{a+\lambda }h(t)k(t) \diamondsuit _{\alpha }t=\int _{a}^{b}\psi (t)k(t) \diamondsuit _{\alpha }t$,

  $$\begin{aligned}& \int _{a}^{\sigma (x)}k(t)\psi (t)\Delta t \leq \int _{a}^{\sigma (x)}k(t)h(t) \Delta t, \quad a\leq x \leq a+\lambda , \\& \int _{a}^{\rho (x)}k(t)\psi (t)\nabla t \leq \int _{a}^{\rho (x)}k(t)h(t) \nabla t, \quad a\leq x \leq a+\lambda , \\& \int _{\sigma (x)}^{b}k(t)\psi (t) \Delta t\geq 0, \quad a+ \lambda \leq x\leq b, \end{aligned}$$

  and

  $$ \int _{\rho (x)}^{b}k(t)\psi (t) \nabla t\geq 0, \quad a+ \lambda \leq x\leq b, $$

  then

  $$ \int _{a}^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t\leq \int _{a}^{a+ \lambda }\phi (t)h(t)\diamondsuit _{\alpha }t. $$

• (ii) If λ is given by $\int _{b-\lambda }^{b}h(t)k(t) \diamondsuit _{\alpha }t=\int _{a}^{b}\psi (t)k(t) \diamondsuit _{\alpha }t$,

  $$\begin{aligned}& \int _{\sigma (x)}^{b}k(t)\psi (t) \Delta t \leq \int _{\sigma (x)}^{b}k(t)h(t) \Delta t, \quad b-\lambda \leq x \leq b, \\& \int _{\rho (x)}^{b}k(t)\psi (t) \nabla t \leq \int _{\rho (x)}^{b}k(t)h(t) \nabla t, \quad b-\lambda \leq x \leq b, \\& \int _{a}^{\sigma (x)}k(t)\psi (t) \Delta t\geq 0, \quad a \leq x\leq b-\lambda , \end{aligned}$$

  and

  $$ \int _{a}^{\rho (x)}k(t)\psi (t) \nabla t\geq 0, \quad a \leq x\leq b-\lambda , $$

  then

  $$ \int _{b-\lambda }^{b}\phi (t)h(t)\diamondsuit _{\alpha }t\leq \int _{a} ^{b}\phi (t)\psi (t)\diamondsuit _{\alpha }t. $$