# Diversity of several estimates transformed on time scales

## Abstract

In this research article, we prove several generalizations of reverse Callebaut, Rogers–Hölder, and Cauchy–Schwarz inequalities via reverses of Young inequalities on time scales. Discrete, continuous, and quantum versions of the results are unified and extended on time scales.

## 1 Introduction

The calculus of time scales was accomplished by Stefan Hilger . A time scale is an arbitrary nonempty closed subset of the real numbers. Let $$\mathbb{T}$$ be a time scale, $$\xi ,\omega \in \mathbb{T}$$ with $$\xi <\omega$$, and an interval $$[\xi ,\omega ]_{\mathbb{T}}$$ means the intersection of the real interval with the given time scale. The major aim of the calculus of time scales is to establish results in general, comprehensive, unified, and extended forms. This hybrid theory is also widely applied in dynamic inequalities, see [2, 812]. The basic ideas about time scale calculus are given in the monographs [3, 4].

We state here the different versions of reverses of Callebaut, Rogers–Hölder, and Cauchy–Schwarz inequalities, see .

Let $$x_{k}>0$$, $$y_{k}>0$$, and $$w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$ with $$\sum^{\eta}_{k=1}w_{k}=1$$. If there exist constants m, $$M>0$$ such that $$0< m\leq \frac{x_{k}}{y_{k}}\leq M<\infty$$ for any $$k\in \{1,2,\ldots ,\eta \}$$, then

\begin{aligned} \sum^{\eta}_{k=1}w_{k}x^{2(1-v)}_{k}y^{2v}_{k} \sum^{ \eta}_{k=1}w_{k}x^{2v}_{k}y^{2(1-v)}_{k} &\leq \sum^{\eta}_{k=1}w_{k}x^{2}_{k} \sum^{\eta}_{k=1}w_{k}y^{2}_{k} \\ & \leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr)\sum ^{ \eta}_{k=1}w_{k}x^{2(1-v)}_{k}y^{2v}_{k} \sum^{\eta}_{k=1}w_{k}x^{2v}_{k}y^{2(1-v)}_{k}, \end{aligned}
(1.1)

for any $$v\in [0, 1]$$ and, in particular,

$$\Biggl(\sum^{\eta}_{k=1}w_{k}x_{k}y_{k} \Biggr)^{2} \leq \sum^{\eta}_{k=1}w_{k}x^{2}_{k} \sum^{\eta}_{k=1}w_{k}y^{2}_{k} \leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \Biggl(\sum ^{\eta}_{k=1}w_{k}x_{k}y_{k} \Biggr)^{2}.$$
(1.2)

Let $$\frac{1}{p}+\frac{1}{q}=1$$ with $$p>1$$. If there exist constants m, M, n, N such that $$0< m\leq x_{k}\leq M<\infty$$ and $$0< n\leq y_{k}\leq N<\infty$$ for any $$k\in \{1,2,\ldots ,\eta \}$$, then we have the following reverse of Rogers–Hölder discrete inequality:

$$\Biggl(\sum^{\eta}_{k=1}w_{k}x^{p} \Biggr)^{\frac{1}{p}} \Biggl(\sum^{\eta}_{k=1}w_{k}y^{q} \Biggr)^{\frac{1}{q}} \leq S \biggl( \biggl(\frac{M}{m} \biggr)^{p} \biggl(\frac{N}{n} \biggr)^{q} \biggr)\sum ^{\eta}_{k=1}w_{k}x_{k}y_{k},$$
(1.3)

and, in particular, the reverse of Cauchy–Bunyakovsky–Schwarz inequality

$$\Biggl(\sum^{\eta}_{k=1}w_{k}x^{2} \Biggr)^{\frac{1}{2}} \Biggl(\sum^{\eta}_{k=1}w_{k}y^{2} \Biggr)^{\frac{1}{2}} \leq S \biggl( \biggl(\frac{MN}{mn} \biggr)^{2} \biggr)\sum^{ \eta}_{k=1}w_{k}x_{k}y_{k}.$$
(1.4)

## 2 Preliminaries

First, we present a short introduction to the diamond-α derivative as given in [1, 13].

Let $$\mathbb{T}$$ be a time scale and $$f(\tau )$$ be differentiable on $$\mathbb{T}$$ in the Δ and sense. For $$\tau \in \mathbb{T}$$, the diamond-α dynamic derivative $$f^{\diamond _{\alpha}}(\tau )$$ is defined by

$$f^{\diamond _{\alpha}}(\tau )=\alpha f^{\Delta}(\tau )+(1-\alpha )f^{ \nabla}(\tau ),\quad 0 \leq \alpha \leq 1.$$

Thus f is diamond-α differentiable if and only if f is Δ and differentiable.

The diamond-α derivative reduces to the standard Δ-derivative for $$\alpha =1$$, or the standard -derivative for $$\alpha =0$$. It represents a weighted dynamic derivative for $$\alpha \in (0,1)$$.

The following definition is given in .

Let $$\xi ,\tau \in \mathbb{T}$$ and $$h:\mathbb{T} \rightarrow \mathbb{R}$$. Then the diamond-α integral from ξ to τ of h is defined by

$$\int ^{\tau}_{\xi}h(\lambda )\diamond _{\alpha} \lambda =\alpha \int ^{ \tau}_{\xi}h(\lambda )\Delta \lambda +(1-\alpha ) \int ^{\tau}_{\xi}h( \lambda )\nabla \lambda ,\quad 0 \leq \alpha \leq 1,$$

provided that there exist delta and nabla integrals of h on $$\mathbb{T}$$.

The following well-known Young inequality holds:

For $$\Phi , \Psi >0$$ and $$v\in [0, 1]$$, we have

$$\Phi ^{1-v}\Psi ^{v}\leq (1-v)\Phi +v\Psi .$$
(2.1)

The following inequalities are given in .

For any $$\Phi , \Psi \in [m,M]\subset (0,\infty )$$ and $$v\in [0,1]$$, we have

$$(1-v)\Phi +v\Psi \leq S \biggl(\frac{M}{m} \biggr)\Phi ^{1-v}\Psi ^{v},$$
(2.2)

where Specht ratio [6, 14] is defined by

$$S(h)=\frac{h^{\frac{1}{h-1}}}{e\log h^{\frac{1}{h-1}}},$$

with $$h>0$$, $$h\neq 1$$.

Let $$v\in [0,1]$$ and $$\Phi ,\Psi >0$$. Then

$$(1-v)\Phi +v\Psi \leq S(L)\Phi ^{1-v}\Psi ^{v},$$
(2.3)

where $$0< L^{-1}\leq \frac{\Phi}{\Psi}\leq L<\infty$$ and $$L>1$$.

Let $$v\in [0,1]$$ and $$\Phi ,\Psi >0$$. Then

$$(1-v)\Phi +v\Psi \leq \max \bigl\{ S(l),S(L) \bigr\} \Phi ^{1-v}\Psi ^{v},$$
(2.4)

where $$0< l^{-1}\leq \frac{\Phi}{\Psi}\leq L<\infty$$ and L, $$l>0$$, with $$Ll>1$$.

In this paper, it is assumed that all considered integrals exist and are finite.

## 3 Main results

In the following, we give an extension of reverse Callebaut inequality on time scales. Throughout this section, we assume that neither $$s\equiv 0$$ nor $$t\equiv 0$$.

### Theorem 3.1

Let $$z,s,t\in C ([\xi , \omega ]_{\mathbb{T}}, \mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions. Assume further that $$0< m\leq \frac{|s(\lambda )|}{|t(\lambda )|}\leq M<\infty$$ on the set $$[\xi , \omega ]_{\mathbb{T}}$$. Let $$v\in [0, 1]$$. Then the following inequalities hold true:

\begin{aligned} & \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2(1-v)} \bigl\vert t( \lambda ) \bigr\vert ^{2v} \diamond _{\alpha}\lambda \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2v} \bigl\vert t( \lambda ) \bigr\vert ^{2(1-v)} \diamond _{\alpha} \lambda \\ &\quad\leq \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2} \diamond _{ \alpha} \lambda \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2} \diamond _{\alpha} \lambda \\ &\quad\leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2(1-v)} \bigl\vert t(\lambda ) \bigr\vert ^{2v} \diamond _{ \alpha}\lambda \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2v} \bigl\vert t( \lambda ) \bigr\vert ^{2(1-v)} \diamond _{\alpha}\lambda . \end{aligned}
(3.1)

### Proof

For $$\lambda ,\zeta \in [\xi , \omega ]_{\mathbb{T}}$$, we observe that

$$m^{2}\leq \frac{ \vert s(\lambda ) \vert ^{2}}{ \vert t(\lambda ) \vert ^{2}}, \frac{ \vert s(\zeta ) \vert ^{2}}{ \vert t(\zeta ) \vert ^{2}}\leq M^{2}.$$
(3.2)

Let $$\Phi (\lambda )=\frac{|s(\lambda )|^{2}}{|t(\lambda )|^{2}}$$ and $$\Psi (\zeta )=\frac{|s(\zeta )|^{2}}{|t(\zeta )|^{2}}$$, $$\lambda ,\zeta \in [\xi , \omega ]_{\mathbb{T}}$$. Then using the inequalities (2.1) and (2.2), we have

\begin{aligned} \biggl(\frac{ \vert s(\lambda ) \vert ^{2}}{ \vert t(\lambda ) \vert ^{2}} \biggr)^{1-v} \biggl(\frac{ \vert s(\zeta ) \vert ^{2}}{ \vert t(\zeta ) \vert ^{2}} \biggr)^{v} &\leq (1-v)\frac{ \vert s(\lambda ) \vert ^{2}}{ \vert t(\lambda ) \vert ^{2}}+v \frac{ \vert s(\zeta ) \vert ^{2}}{ \vert t(\zeta ) \vert ^{2}} \\ &\leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \biggl( \frac{ \vert s(\lambda ) \vert ^{2}}{ \vert t(\lambda ) \vert ^{2}} \biggr)^{1-v} \biggl( \frac{ \vert s(\zeta ) \vert ^{2}}{ \vert t(\zeta ) \vert ^{2}} \biggr)^{v}. \end{aligned}
(3.3)

Multiplying by $$|t(\lambda )|^{2}|t(\zeta )|^{2}$$, $$\lambda ,\zeta \in [\xi ,\omega ]_{\mathbb{T}}$$, (3.3) takes the form

\begin{aligned} &\bigl\vert s(\lambda ) \bigr\vert ^{2(1-v)} \bigl\vert t(\lambda ) \bigr\vert ^{2v} \bigl\vert s(\zeta ) \bigr\vert ^{2v} \bigl\vert t( \zeta ) \bigr\vert ^{2(1-v)} \\ &\quad \leq (1-v) \bigl\vert s(\lambda ) \bigr\vert ^{2} \bigl\vert t(\zeta ) \bigr\vert ^{2}+v \bigl\vert t(\lambda ) \bigr\vert ^{2} \bigl\vert s( \zeta ) \bigr\vert ^{2} \\ &\quad\leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \bigl\vert s(\lambda ) \bigr\vert ^{2(1-v)} \bigl\vert t( \lambda ) \bigr\vert ^{2v} \bigl\vert s(\zeta ) \bigr\vert ^{2v} \bigl\vert t(\zeta ) \bigr\vert ^{2(1-v)}. \end{aligned}
(3.4)

Multiplying by $$|z(\lambda )|$$ and integrating (3.4) with respect to λ from ξ to ω, we obtain

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2(1-v)} \bigl\vert t( \lambda ) \bigr\vert ^{2v} \diamond _{\alpha}\lambda \biggr) \bigl\vert s(\zeta ) \bigr\vert ^{2v} \bigl\vert t( \zeta ) \bigr\vert ^{2(1-v)} \\ &\quad\leq (1-v) \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2} \diamond _{\alpha} \lambda \biggr) \bigl\vert t(\zeta ) \bigr\vert ^{2} +v \biggl( \int ^{ \omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2}\diamond _{\alpha} \lambda \biggr) \bigl\vert s(\zeta ) \bigr\vert ^{2} \\ &\quad\leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \biggl( \int ^{ \omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2(1-v)} \bigl\vert t(\lambda ) \bigr\vert ^{2v} \diamond _{\alpha}\lambda \biggr) \bigl\vert s(\zeta ) \bigr\vert ^{2v} \bigl\vert t(\zeta ) \bigr\vert ^{2(1-v)}. \end{aligned}
(3.5)

Again, multiplying by $$|z(\zeta )|$$ and integrating (3.5) with respect to ζ from ξ to ω, we obtain the desired inequality (3.1). □

The following reverse of Callebaut inequality holds:

### Corollary 3.1

Let $$z,s,t\in C ([\xi ,\omega ]_{\mathbb{T}}, \mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions. Assume further that $$0< m\leq \frac{|s(\lambda )|}{|t(\lambda )|}\leq M<\infty$$ on the set $$[\xi ,\omega ]_{\mathbb{T}}$$. Then the following inequalities hold true:

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert \bigl\vert t( \lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr)^{2} \\ &\quad\leq \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2} \diamond _{ \alpha} \lambda \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2} \diamond _{\alpha} \lambda \\ &\quad\leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \biggl( \int ^{ \omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert \diamond _{\alpha} \lambda \biggr)^{2}. \end{aligned}
(3.6)

### Proof

Take $$v=\frac{1}{2}$$ in Theorem 3.1, and the result follows. □

The following another reverse of Callebaut inequality holds:

### Corollary 3.2

Let $$z,s,t\in C ([\xi ,\omega ]_{\mathbb{T}},\mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions. Assume further that $$0< m\leq \frac{|s(\lambda )|}{|t(\lambda )|}\leq M<\infty$$ on the set $$[\xi ,\omega ]_{\mathbb{T}}$$. Let $$v\in [0, 1]$$. Then the following inequalities hold true:

\begin{aligned} & \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{1+v} \bigl\vert t( \lambda ) \bigr\vert ^{1-v} \diamond _{\alpha}\lambda \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{1-v} \bigl\vert t( \lambda ) \bigr\vert ^{1+v} \diamond _{\alpha} \lambda \\ &\quad\leq \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2} \diamond _{ \alpha} \lambda \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2} \diamond _{\alpha} \lambda \\ &\quad\leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{1+v} \bigl\vert t(\lambda ) \bigr\vert ^{1-v} \diamond _{ \alpha}\lambda \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{1-v} \bigl\vert t( \lambda ) \bigr\vert ^{1+v} \diamond _{\alpha}\lambda . \end{aligned}
(3.7)

### Proof

Replace v by $$\frac{1}{2}(1-v)$$ in Theorem 3.1, and the result follows. □

The following another reverse of Callebaut inequality holds:

### Corollary 3.3

Let $$z,s,t\in C ([\xi ,\omega ]_{\mathbb{T}},\mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions. Assume further that $$0< m\leq \frac{|s(\lambda )|}{|t(\lambda )|}\leq M<\infty$$ on the set $$[\xi ,\omega ]_{\mathbb{T}}$$. Let $$\nu \in [0, 2]$$. Then the following inequalities hold true:

\begin{aligned} & \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2-\nu} \bigl\vert t( \lambda ) \bigr\vert ^{\nu} \diamond _{\alpha}\lambda \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{\nu} \bigl\vert t( \lambda ) \bigr\vert ^{2-\nu} \diamond _{\alpha} \lambda \\ &\quad\leq \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2} \diamond _{ \alpha} \lambda \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2} \diamond _{\alpha} \lambda \\ &\quad\leq S \biggl( \biggl(\frac{M}{m} \biggr)^{2} \biggr) \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2-\nu} \bigl\vert t(\lambda ) \bigr\vert ^{\nu} \diamond _{ \alpha}\lambda \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{\nu} \bigl\vert t( \lambda ) \bigr\vert ^{2-\nu} \diamond _{\alpha}\lambda . \end{aligned}
(3.8)

### Proof

Take $$v=\frac{1}{2}\nu$$ in Theorem 3.1, and the result follows. □

In the following, we give an extension of reverse Rogers–Hölder inequality on time scales.

### Theorem 3.2

Let $$z,s,t\in C ([\xi , \omega ]_{\mathbb{T}}, \mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions satisfying $$\int ^{\omega}_{\xi}|z(\lambda )| \diamond _{\alpha}\lambda =1$$. Assume further that $$0< m\leq |s(\lambda )|\leq M<\infty$$ and $$0< n\leq |t(\lambda )|\leq N<\infty$$ on the set $$[\xi ,\omega ]_{\mathbb{T}}$$. Let $$\frac{1}{p}+\frac{1}{q}=1$$ with $$p>1$$. Then the following inequality holds true:

\begin{aligned} &\biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{p}\diamond _{ \alpha} \lambda \biggr)^{\frac{1}{p}} \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{q}\diamond _{\alpha}\lambda \biggr)^{ \frac{1}{q}} \\ &\quad \leq S \biggl( \biggl(\frac{M}{m} \biggr)^{p} \biggl( \frac{N}{n} \biggr)^{q} \biggr) \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda )t(\lambda ) \bigr\vert \diamond _{\alpha}\lambda . \end{aligned}
(3.9)

### Proof

Using the given conditions, for $$\lambda \in [\xi ,\omega ]_{\mathbb{T}}$$, we have

$$m^{p}\leq \bigl\vert s(\lambda ) \bigr\vert ^{p}\leq M^{p}\quad \text{and}\quad n^{q}\leq \bigl\vert t(\lambda ) \bigr\vert ^{q}\leq N^{q},$$

which imply that

$$\biggl(\frac{m}{M} \biggr)^{p}\leq \frac{ \vert s(\lambda ) \vert ^{p}}{\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert s(\lambda ) \vert ^{p}\diamond _{\alpha}\lambda} \leq \biggl(\frac{M}{m} \biggr)^{p}$$
(3.10)

and

$$\biggl(\frac{n}{N} \biggr)^{q}\leq \frac{ \vert t(\lambda ) \vert ^{q}}{\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert t(\lambda ) \vert ^{q}\diamond _{\alpha}\lambda} \leq \biggl(\frac{N}{n} \biggr)^{q}.$$
(3.11)

Therefore,

\begin{aligned} \biggl[ \biggl(\frac{M}{m} \biggr)^{p} \biggl(\frac{N}{n} \biggr)^{q} \biggr]^{-1} &\leq \biggl( \frac{ \vert z(\lambda ) \vert \vert s(\lambda ) \vert ^{p}}{\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert s(\lambda ) \vert ^{p}\diamond _{\alpha}\lambda} \biggr) \biggl( \frac{\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert t(\lambda ) \vert ^{q}\diamond _{\alpha}\lambda}{ \vert z(\lambda ) \vert \vert t(\lambda ) \vert ^{q}} \biggr) \\ &\leq \biggl(\frac{M}{m} \biggr)^{p} \biggl(\frac{N}{n} \biggr)^{q}. \end{aligned}
(3.12)

Using the inequality (2.3) with $$v=\frac{1}{q}$$, $$L= (\frac{M}{m} )^{p} (\frac{N}{n} )^{q}$$, $$\Phi (\lambda )= \frac{|z(\lambda )||s(\lambda )|^{p}}{\int ^{\omega}_{\xi}|z(\lambda )||s(\lambda )|^{p}\diamond _{\alpha}\lambda}$$, and $$\Psi (\lambda )= \frac{|z(\lambda )||t(\lambda )|^{q}}{\int ^{\omega}_{\xi}|z(\lambda )||t(\lambda )|^{q}\diamond _{\alpha}\lambda}$$, we get

\begin{aligned} &\frac{1}{p} \frac{ \vert z(\lambda ) \vert \vert s(\lambda ) \vert ^{p}}{\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert s(\lambda ) \vert ^{p}\diamond _{\alpha}\lambda} +\frac{1}{q} \frac{ \vert z(\lambda ) \vert \vert t(\lambda ) \vert ^{q}}{\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert t(\lambda ) \vert ^{q}\diamond _{\alpha}\lambda} \\ &\quad \leq S(L) \frac{ \vert z(\lambda ) \vert \vert s(\lambda )t(\lambda ) \vert }{ (\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert s(\lambda ) \vert ^{p}\diamond _{\alpha}\lambda )^{\frac{1}{p}} (\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert t(\lambda ) \vert ^{q}\diamond _{\alpha}\lambda )^{\frac{1}{q}}}. \end{aligned}
(3.13)

Integrating (3.13) with respect to λ from ξ to ω, we obtain

$$1\leq S(L) \frac{\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert s(\lambda )t(\lambda ) \vert \diamond _{\alpha}\lambda}{ (\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert s(\lambda ) \vert ^{p}\diamond _{\alpha}\lambda )^{\frac{1}{p}} (\int ^{\omega}_{\xi} \vert z(\lambda ) \vert \vert t(\lambda ) \vert ^{q}\diamond _{\alpha}\lambda )^{\frac{1}{q}}}.$$
(3.14)

This completes the proof of Theorem 3.2. □

Next, we give an extension of reverse Cauchy–Schwarz inequality on time scales.

### Corollary 3.4

Let $$z,s,t\in C ([\xi ,\omega ]_{\mathbb{T}},\mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions satisfying $$\int ^{\omega}_{\xi}|z(\lambda )| \diamond _{\alpha}\lambda =1$$. Assume further that $$0< m\leq |s(\lambda )|\leq M<\infty$$ and $$0< n\leq |t(\lambda )|\leq N<\infty$$ on the set $$[\xi ,\omega ]_{\mathbb{T}}$$. Then the following inequality holds true:

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2}\diamond _{ \alpha} \lambda \biggr)^{\frac{1}{2}} \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2}\diamond _{\alpha}\lambda \biggr)^{ \frac{1}{2}} \\ &\quad \leq S \biggl( \biggl(\frac{MN}{mn} \biggr)^{2} \biggr) \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda )t(\lambda ) \bigr\vert \diamond _{\alpha}\lambda . \end{aligned}
(3.15)

### Proof

Take $$p=q=2$$ in Theorem 3.2, and the result follows. □

### Remark 3.1

We have the following:

1. (i)

Let $$\alpha =1$$, $$\mathbb{T}=\mathbb{Z}$$, $$\xi =1$$, $$\omega =\eta +1$$, $$s(k)=x_{k}>0$$, $$t(k)=y_{k}>0$$, and $$z(k)=w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$ with $$\sum^{\eta}_{k=1}w_{k}=1$$. Then inequality (3.1) reduces to inequality (1.1).

2. (ii)

Let $$\alpha =1$$, $$\mathbb{T}=\mathbb{Z}$$, $$\xi =1$$, $$\omega =\eta +1$$, $$s(k)=x_{k}>0$$, $$t(k)=y_{k}>0$$, and $$z(k)=w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$ with $$\sum^{\eta}_{k=1}w_{k}=1$$. Then inequality (3.6) reduces to inequality (1.2).

3. (iii)

Let $$\alpha =1$$, $$\mathbb{T}=\mathbb{Z}$$, $$\xi =1$$, $$\omega =\eta +1$$, $$s(k)=x_{k}>0$$, $$t(k)=y_{k}>0$$, and $$z(k)=w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$. Then inequality (3.9) reduces to inequality (1.3).

4. (iv)

Let $$\alpha =1$$, $$\mathbb{T}=\mathbb{Z}$$, $$\xi =1$$, $$\omega =\eta +1$$, $$s(k)=x_{k}>0$$, $$t(k)=y_{k}>0$$, and $$z(k)=w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$. Then inequality (3.15) reduces to inequality (1.4).

Finally, we give another extension of reverse Rogers–Hölder dynamic inequality.

### Theorem 3.3

Let $$z,u_{1},u_{2},s,t\in C ([\xi , \omega ]_{\mathbb{T}}, \mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions. Assume further that $$0< m\leq |s(\lambda )|\leq M<\infty$$ and $$0< n\leq |t(\lambda )|\leq N<\infty$$ on the set $$[\xi , \omega ]_{\mathbb{T}}$$. Let $$\frac{1}{p}+\frac{1}{q}=1$$ with $$p>1$$. Then the following inequalities hold true:

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{1}( \lambda )s(\lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \biggl( \int ^{ \omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{2}(\lambda )t(\lambda ) \bigr\vert \diamond _{ \alpha}\lambda \biggr) \\ &\quad\leq \frac{1}{p} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{1}( \lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{p}\diamond _{\alpha}\lambda \biggr) \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{2}(\lambda ) \bigr\vert \diamond _{\alpha} \lambda \biggr) \\ &\quad\quad {}+\frac{1}{q} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{1}( \lambda ) \bigr\vert \diamond _{\alpha} \lambda \biggr) \biggl( \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{2}(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{q}\diamond _{\alpha} \lambda \biggr) \\ &\quad\leq \max \biggl\{ S \biggl(\frac{N^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{n^{q}} \biggr) \biggr\} \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert u_{1}(\lambda )s(\lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \\ &\quad\quad{}\times \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{2}(\lambda )t(\lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr). \end{aligned}
(3.16)

### Proof

For $$\lambda , \zeta \in [\xi , \omega ]_{\mathbb{T}}$$, it is clear that

$$\frac{m^{p}}{N^{q}}\leq \frac{ \vert s(\lambda ) \vert ^{p}}{ \vert t(\zeta ) \vert ^{q}} \leq \frac{M^{p}}{n^{q}}.$$
(3.17)

Let $$l=\frac{N^{q}}{m^{p}}$$, $$L=\frac{M^{p}}{n^{q}}$$, $$\Phi (\lambda )=|s(\lambda )|^{p}$$, $$\Psi (\zeta )=|t(\zeta )|^{q}$$, and $$v=\frac{1}{q}$$. Then using the inequalities (2.1) and (2.4), respectively, we have

$$\bigl\vert s(\lambda ) \bigr\vert \bigl\vert t(\zeta ) \bigr\vert \leq \frac{1}{p} \bigl\vert s(\lambda ) \bigr\vert ^{p}+\frac{1}{q} \bigl\vert t( \zeta ) \bigr\vert ^{q} \leq \max \biggl\{ S \biggl(\frac{N^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{n^{q}} \biggr) \biggr\} \bigl\vert s(\lambda ) \bigr\vert \bigl\vert t(\zeta ) \bigr\vert .$$
(3.18)

Multiplying by $$|z(\lambda )||u_{1}(\lambda )|$$ and integrating (3.18) with respect to λ from ξ to ω, we obtain

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{1}(\lambda )s( \lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \bigl\vert t(\zeta ) \bigr\vert \\ &\quad\leq \frac{1}{p} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{1}( \lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{p}\diamond _{\alpha}\lambda \biggr)+ \frac{1}{q} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert u_{1}(\lambda ) \bigr\vert \diamond _{\alpha} \lambda \biggr) \bigl\vert t(\zeta ) \bigr\vert ^{q} \\ &\quad\leq \max \biggl\{ S \biggl(\frac{N^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{n^{q}} \biggr) \biggr\} \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert u_{1}(\lambda )s(\lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \bigl\vert t(\zeta ) \bigr\vert . \end{aligned}
(3.19)

Multiplying by $$|z(\zeta )||u_{2}(\zeta )|$$ and integrating (3.19) with respect to ζ from ξ to ω, we obtain the desired inequality (3.16). □

Next, we give an extension of reverse Rogers–Hölder inequality on time scales.

### Corollary 3.5

Let $$z,s,t\in C ([\xi , \omega ]_{\mathbb{T}}, \mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions. Assume further that $$0< m\leq |s(\lambda )|\leq M<\infty$$ and $$0< n\leq |t(\lambda )|\leq N<\infty$$ on the set $$[\xi , \omega ]_{\mathbb{T}}$$. Let $$\frac{1}{p}+\frac{1}{q}=1$$ with $$p>1$$. Then the following inequalities hold true:

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda )t( \lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr)^{2} \\ &\quad\leq \frac{1}{p} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert \bigl\vert s( \lambda ) \bigr\vert ^{p}\diamond _{\alpha}\lambda \biggr) \biggl( \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \\ &\quad\quad {}+\frac{1}{q} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t( \lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \biggl( \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{q}\diamond _{\alpha} \lambda \biggr) \\ &\quad\leq \max \biggl\{ S \biggl(\frac{N^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{n^{q}} \biggr) \biggr\} \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert s(\lambda )t(\lambda ) \bigr\vert \diamond _{\alpha} \lambda \biggr)^{2}. \end{aligned}
(3.20)

### Proof

Put $$|u_{1}(\lambda )|=|t(\lambda )|$$ and $$|u_{2}(\lambda )|=|s(\lambda )|$$ on $$[\xi , \omega ]_{\mathbb{T}}$$ in Theorem 3.3, and then the result follows. □

Now, we give another extension of reverse Rogers–Hölder inequality on time scales.

### Corollary 3.6

Let $$z,s,t\in C ([\xi , \omega ]_{\mathbb{T}}, \mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions. Assume further that $$0< m\leq |s(\lambda )|\leq M<\infty$$ and $$0< n\leq |t(\lambda )|\leq N<\infty$$ on the set $$[\xi , \omega ]_{\mathbb{T}}$$. Let $$\frac{1}{p}+\frac{1}{q}=1$$ with $$p>1$$. Then the following inequalities hold true:

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2} \diamond _{\alpha} \lambda \biggr) \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2}\diamond _{\alpha}\lambda \biggr) \\ &\quad\leq \frac{1}{p} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{p+1} \diamond _{\alpha} \lambda \biggr) \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \\ &\quad\quad {}+\frac{1}{q} \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert s( \lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr) \biggl( \int ^{\omega}_{ \xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{q+1}\diamond _{\alpha} \lambda \biggr) \\ &\quad\leq \max \biggl\{ S \biggl(\frac{N^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{n^{q}} \biggr) \biggr\} \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert s(\lambda ) \bigr\vert ^{2}\diamond _{\alpha}\lambda \biggr) \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert t(\lambda ) \bigr\vert ^{2}\diamond _{\alpha} \lambda \biggr). \end{aligned}
(3.21)

### Proof

Put $$|u_{1}(\lambda )|=|s(\lambda )|$$ and $$|u_{2}(\lambda )|=|t(\lambda )|$$ on $$[\xi , \omega ]_{\mathbb{T}}$$ in Theorem 3.3, and then the result follows. □

Next, we give another extension of reverse Rogers–Hölder inequality on time scales.

### Corollary 3.7

Let $$z,f_{1},f_{2}\in C ([\xi , \omega ]_{\mathbb{T}}, \mathbb{R} )$$ be $$\diamond _{\alpha}$$-integrable functions, with neither $$f_{1}\equiv 0$$ nor $$f_{2}\equiv 0$$. Assume further that $$0< m\leq \frac{|f_{1}(\lambda )|}{|f_{2}(\lambda )|}\leq M<\infty$$ on the set $$[\xi , \omega ]_{\mathbb{T}}$$. Let $$\frac{1}{p}+\frac{1}{q}=1$$ with $$p>1$$. Then the following inequalities hold true:

\begin{aligned} & \biggl( \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert f_{1}(\lambda )f_{2}( \lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr)^{2} \\ &\quad\leq \biggl[\frac{1}{p} \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert f_{1}( \lambda ) \bigr\vert ^{p} \bigl\vert f_{2}(\lambda ) \bigr\vert ^{2-p}\diamond _{\alpha} \lambda \\ &\quad\quad {}+\frac{1}{q} \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert f_{1}( \lambda ) \bigr\vert ^{q} \bigl\vert f_{2}(\lambda ) \bigr\vert ^{2-q}\diamond _{\alpha} \lambda \biggr] \int ^{\omega}_{\xi} \bigl\vert z(\lambda ) \bigr\vert \bigl\vert f_{2}(\lambda ) \bigr\vert ^{2} \diamond _{\alpha}\lambda \\ &\quad\leq \max \biggl\{ S \biggl(\frac{M^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{m^{q}} \biggr) \biggr\} \biggl( \int ^{\omega}_{\xi} \bigl\vert z( \lambda ) \bigr\vert \bigl\vert f_{1}(\lambda )f_{2}(\lambda ) \bigr\vert \diamond _{\alpha}\lambda \biggr)^{2}. \end{aligned}
(3.22)

### Proof

Put $$|s(\lambda )|=|t(\lambda )|= \frac{|f_{1}(\lambda )|}{|f_{2}(\lambda )|}$$, $$|u_{1}(\lambda )|=|u_{2}(\lambda )|=|f_{2}(\lambda )|^{2}$$ on $$[\xi , \omega ]_{\mathbb{T}}$$, $$M=N$$, and $$m=n$$ in Theorem 3.3, and then the result follows. □

### Remark 3.2

We have the following:

1. (i)

Let $$\alpha =1$$, $$\mathbb{T}=\mathbb{Z}$$, $$\xi =1$$, $$\omega =\eta +1$$, $$s(k)=x_{k}>0$$, $$t(k)=y_{k}>0$$, and $$z(k)=w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$ with $$\sum^{\eta}_{k=1}w_{k}=1$$. Then inequality (3.20) reduces to inequality 

\begin{aligned} \Biggl(\sum ^{\eta}_{k=1}w_{k}x_{k}y_{k} \Biggr)^{2} &\leq \frac{1}{p}\sum^{\eta}_{k=1}w_{k}y_{k}x^{p}_{k} \sum^{\eta}_{k=1}w_{k}x_{k}+ \frac{1}{q}\sum^{\eta}_{k=1}w_{k}y_{k} \sum^{\eta}_{k=1}w_{k}x_{k}y^{q}_{k} \\ &\leq \max \biggl\{ S \biggl(\frac{N^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{n^{q}} \biggr) \biggr\} \Biggl(\sum^{\eta}_{k=1}w_{k}x_{k}y_{k} \Biggr)^{2}. \end{aligned}
(3.23)
2. (ii)

Let $$\alpha =1$$, $$\mathbb{T}=\mathbb{Z}$$, $$\xi =1$$, $$\omega =\eta +1$$, $$s(k)=x_{k}>0$$, $$t(k)=y_{k}>0$$, and $$z(k)=w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$ with $$\sum^{\eta}_{k=1}w_{k}=1$$. Then inequality (3.21) reduces to inequality 

\begin{aligned} \sum^{\eta}_{k=1}w_{k}x^{2}_{k} \sum^{ \eta}_{k=1}w_{k}y^{2}_{k} &\leq \frac{1}{p}\sum^{\eta}_{k=1}w_{k}x^{p+1}_{k} \sum^{\eta}_{k=1}w_{k}y_{k}+ \frac{1}{q}\sum^{\eta}_{k=1}w_{k}x_{k} \sum^{\eta}_{k=1}w_{k}y^{q+1}_{k} \\ &\leq \max \biggl\{ S \biggl(\frac{N^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{n^{q}} \biggr) \biggr\} \sum^{\eta}_{k=1}w_{k}x^{2}_{k} \sum^{\eta}_{k=1}w_{k}y^{2}_{k}. \end{aligned}
(3.24)
3. (iii)

Let $$\alpha =1$$, $$\mathbb{T}=\mathbb{Z}$$, $$\xi =1$$, $$\omega =\eta +1$$, $$f_{1}(k)=x_{k}>0$$, $$f_{2}(k)=y_{k}>0$$, and $$z(k)=w_{k}\geq 0$$ for any $$k\in \{1,2,\ldots ,\eta \}$$ with $$\sum^{\eta}_{k=1}w_{k}=1$$. Then inequality (3.22) reduces to inequality 

\begin{aligned} \Biggl(\sum ^{\eta}_{k=1}w_{k}x_{k}y_{k} \Biggr)^{2} &\leq \Biggl(\frac{1}{p}\sum ^{\eta}_{k=1}w_{k}x^{p}_{k}y^{2-p}_{k}+ \frac{1}{q}\sum^{\eta}_{k=1}w_{k}x^{q}_{k}y^{2-q}_{k} \Biggr) \sum^{\eta}_{k=1}w_{k}y^{2}_{k} \\ &\leq \max \biggl\{ S \biggl(\frac{M^{q}}{m^{p}} \biggr),S \biggl( \frac{M^{p}}{m^{q}} \biggr) \biggr\} \Biggl(\sum^{\eta}_{k=1}w_{k}x_{k}y_{k} \Biggr)^{2}. \end{aligned}
(3.25)

Not applicable.

## References

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

2. Akin, L.: On innovations of n-dimensional integral-type inequality on time scales. Adv. Differ. Equ. 2021, 148 (2021)

3. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser Boston, Boston (2001)

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

5. Dragomir, S.S.: Some results for isotonic functionals via an inequality due to Tominaga. Mem. Grad. Sci. Eng. Shimane Univ. Ser. B: Math. 50, 31–41 (2017)

6. Fujii, J.I., Izumino, S., Seo, Y.: Determinant for positive operators and Specht’s theorem. Sci. Math. 1, 307–310 (1998)

7. Hilger, S.: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD Thesis, Universität Würzburg (1988)

8. Sahir, M.J.S.: Consonancy of dynamic inequalities correlated on time scale calculus. Tamkang J. Math. 51(3), 233–243 (2020)

9. Sahir, M.J.S.: Homogeneity of classical and dynamic inequalities compatible on time scales. Int. J. Difference Equ. 15(1), 173–186 (2020)

10. Sahir, M.J.S.: Analogy of classical and dynamic inequalities merging on time scales. J. Math. Appl. 43, 139–152 (2020)

11. Sahir, M.J.S.: Integrity of variety of inequalities sketched on time scales. J. Abstr. Comput. Math. 6(2), 8–15 (2021)

12. Saker, S.H., Osman, M.M., Anderson, D.R.: On a new class of dynamic Hardy-type inequalities and some related generalizations. Aequ. Math. 96, 773–793 (2022)

13. Sheng, Q., Fadag, M., Henderson, J., Davis, J.M.: An exploration of combined dynamic derivatives on time scales and their applications. Nonlinear Anal., Real World Appl. 7(3), 395–413 (2006)

14. Specht, W.: Zur Theorie der elementaren Mittel. Math. Z. 74, 91–98 (1960)

## Acknowledgements

This research work was funded by Institutional Fund Projects under grant no. (IFPIP:1264-130-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

Not applicable.

## Author information

Authors

### Contributions

MJSS and DA carried out preparatory investigation. Formal analysis was done by MI and ASA. The manuscript was prepared by MJSS and DA. ASA was responsible for the main funding. MJSS and MI worked together to conclude the result. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Mustafa Inc.

## Ethics declarations

### Competing interests

The authors declare no competing interests. 