1. Introduction
The calculus of time scales was initially developed by Stefan Hilger (see [1]). A time scale is an arbitrary nonempty closed subset of the real numbers. The three commonly known examples of calculus on time scales are differential calculus, difference calculus, and quantum
calculus, i.e., when \(\mathbb{T}=\mathbb{R}\), \(\mathbb{T}=\mathbb{N}\) and \(\mathbb{T}=q^{\mathbb{N}_{0}}=\{q^{t}:t\in\mathbb{N}_{0}\}\) where \(q>1\). The time scales calculus is divided into delta calculus, nabla calculus and diamond–alpha calculus. During the last two decades, many researchers have established several dynamic inequalities (see [2,3,4,5,6,7,8,9,10]). The fundamental work on dynamic inequalities is done by Ravi Agarwal, George Anastassiou, Martin Bohner, Allan Peterson, Donal O’Regan, Samir Saker and many other researchers.
There have been recent developments and refinements of the theory and applications of dynamic inequalities on time scales. From the theoretical perspective, the work provides a coalition and amplification of conventional differential, difference and quantum equations. Moreover, it is a key mechanism in many mathematical, computational, biological, economical and numerical applications.
In this research article, it is accepted that all considerable integrals exist and are finite and \(\mathbb{T}\) denotes as usual the time scale, \(a,b\in\mathbb{T}\) with \(a< b\) and an interval \([a,b]_{\mathbb{T}}\) means the intersection of a real interval with the given time scale.
2. Preliminaries
We need here basic concepts of delta calculus. The results of delta calculus are adopted from monographs [
6,
11].
For \(t\in \mathbb{T}\), the forward jump operator \(\sigma:\mathbb{T}
\rightarrow \mathbb{T}\) is defined by
\begin{equation*}\sigma(t):=\inf\{s\in\mathbb{T}:s>t\}.\end{equation*}
The mapping \(\mu:\mathbb{T}\rightarrow \mathbb{R}^{+}_{0}=[0,+\infty)\) such
that \(\mu(t):=\sigma(t)-t\) is called the forward graininess function.
The backward jump operator \(\rho:\mathbb{T} \rightarrow\mathbb{T}\) is
defined by
\begin{equation*}\rho(t) :=\sup \{s\in\mathbb{T}:st\), we
say that \(t\) is right–scattered, while if \(\rho(t)< t\), we say that \(t\) is
left–scattered. Also, if \(t\inf\mathbb{T}\) and \(\rho(t)=t\), then \(t\) is
called left–dense. If \(\mathbb{T}\) has a left–scattered maximum \(M\), then
\(\mathbb{T}^{k}=\mathbb{T}-\{M\}\), otherwise \(\mathbb{T}^{k}=\mathbb{T}\).
For a function \(f:\mathbb{T}\rightarrow
\mathbb{R}\), the delta derivative \(f^{\Delta}\) is defined as follows; Let \(t\in\mathbb{T}^{k}\). If there exists \(f^{\Delta}(t)\in\mathbb{R}\) such
that for all \(\epsilon>0\), there is a neighborhood \(U\) of \(t\), such that
\begin{equation*}|f(\sigma(t))-f(s)-f^{\Delta}(t)(\sigma(t)-s)|\leq\epsilon
|\sigma(t)-s|,\end{equation*}
for all \(s\in U\), then \(f\) is said to be delta differentiable at \(t\), and \(f^{\Delta}(t)\) is called the delta derivative of \(f\) at \(t\).
A function \(f : \mathbb{T}\rightarrow\mathbb{R}\) is said to be right-dense
continuous (rd-continuous), if it is continuous at each right-dense
point and there exists a finite left-sided limit at every left-dense point. The set of all rd-continuous functions is denoted by \(C_{rd}(\mathbb{T},\mathbb{R})\).
The next definition is given in [6,11].
Definition 1.
A function \(F : \mathbb{T}\rightarrow \mathbb{R}\) is called a delta antiderivative of \(f : \mathbb{T}\rightarrow \mathbb{R}\), provided that \(F^{\Delta}(t)=f(t)\) holds for all \(t\in \mathbb{T}^{k}\). Then the delta
integral of \(f\) is defined by
\begin{equation*}\int^{b}_{a} f(t)\Delta t=F(b)-F(a).\end{equation*}
The following results of nabla calculus are taken from [6,11,12].
If \(\mathbb{T}\) has a right–scattered minimum \(m\), then
\(\mathbb{T}_{k}=\mathbb{T}-\{m\}\), otherwise \(\mathbb{T}_{k}=\mathbb{T}\). A function \(f:\mathbb{T}_{k}\rightarrow\mathbb{R}\) is called nabla
differentiable at \(t\in \mathbb{T}_{k}\), with nabla derivative \(f^{\nabla}(t)\), if there exists \(f^{\nabla}(t)\in
\mathbb{R}\) such that given any \(\epsilon>0\), there is a neighborhood \(V\) of \(t\), such that
\begin{equation*}|f(\rho(t))-f(s)-f^{\nabla}(t)(\rho(t)-s)|\leq
\epsilon|\rho(t)-s|,\end{equation*}
for all \(s\in V\).
A function \(f:\mathbb{T} \rightarrow\mathbb{R}\) is said to be left-dense continuous (ld-continuous), provided it is continuous at all left-dense points in \(\mathbb{T}\)
and its right-sided limits exist (finite) at all right-dense points in
\(\mathbb{T}\). The set of all ld-continuous functions is denoted by \(C_{ld}(\mathbb{T},\mathbb{R})\).
The next definition is given in [6,11,12].
Definition 2.
A function \(G : \mathbb{T}\rightarrow \mathbb{R}\) is called a nabla antiderivative of \(g : \mathbb{T}\rightarrow \mathbb{R}\), provided that \(G^{\nabla}(t)=g(t)\) holds for all \(t\in \mathbb{T}_{k}\). Then the nabla
integral of \(g\) is defined by
\begin{equation*}\int^{b}_{a} g(t)\nabla t=G(b)-G(a).\end{equation*}
The following definition is taken from [3,5].
Definition 3.
For \(\alpha\geq 1\), the time scale \(\Delta\)-Riemann-Liouville type
fractional integral for a function \(f\in C_{rd}\) is defined by
\begin{equation}
\mathcal{I}^{\alpha}_{a}f(t)=\int^{t}_{a}h_{\alpha-1}(t,\sigma({\tau}))f(\tau)\Delta \tau,
\end{equation}
(1)
which is an integral on \([a,t)_{\mathbb{T}}\), see [
13]
and \(h_{\alpha}:\mathbb{T}\times \mathbb{T}\rightarrow \mathbb{R}\), \(\alpha\geq0\) are the coordinate wise rd–continuous functions, such that \(h_{0}(t,s)=1\),
\begin{equation}
h_{\alpha+1}(t,s)=\int^{t}_{s}h_{\alpha}(\tau,s)\Delta \tau, ~\forall s,t\in\mathbb{T}.
\end{equation}
(2)
Notice that
\begin{equation*}
\mathcal{I}^{1}_{a}f(t)=\int^{t}_{a}f(\tau)\Delta \tau,
\end{equation*}
which is absolutely continuous in \(t\in [a,b]_{\mathbb{T}}\), see [
13].
The following definition is taken from [4,5].
Definition 4.
For \(\alpha\geq 1\), the time scale \(\nabla\)-Riemann-Liouville type
fractional integral for a function \(f\in C_{ld}\) is defined by
\begin{equation}
\mathcal{J}^{\alpha}_{a}f(t)=\int^{t}_{a}\hat{h}_{\alpha -1}(t,\rho({\tau}))f(\tau)\nabla \tau,
\end{equation}
(3)
which is an integral on \((a,t]_{\mathbb{T}}\), see [
13] and
\(\hat{h}_{\alpha}:\mathbb{T}\times \mathbb{T}\rightarrow \mathbb{R}\), \(\alpha\geq0\) are the coordinate wise ld–continuous functions, such that \(\hat{h}_{0}(t,s)=1\),
\begin{equation}
\hat{h}_{\alpha+1}(t,s)=\int^{t}_{s}\hat{h}_{\alpha}(\tau,s)\nabla \tau, ~\forall s,t\in\mathbb{T}.
\end{equation}
(4)
Notice that
\begin{equation*}
\mathcal{J}^{1}_{a}f(t)=\int^{t}_{a}f(\tau)\nabla \tau,
\end{equation*}
which is absolutely continuous in \(t\in [a,b]_{\mathbb{T}}\), see [
13].
3. Dynamic Young’s inequality
In order to present our main results, first we give a straightforward proof for an extension of dynamic Young’s inequalities by using the time scale \(\Delta\)-Riemann-Liouville type fractional integral.
Theorem 5.
Let \(w,f,g\in C_{rd}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\Delta\)-integrable functions and \(h_{\alpha-1}(.,.),\) \(h_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):
\begin{align}
\notag&\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)||g(x)|^{p-1}\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||f(x)||g(x)|^{q-1}\right)\leq
\frac{1}{p}\mathcal{I}^{\alpha}_{a}
\left(|w(x)||f(x)|^{p}\right)\mathcal{I}^{\beta}_{a}\left(
|w(x)||g(x)|^{q}\right)\\ &+\frac{1}{q}\mathcal{I}^{\alpha}_{a}\left(
|w(x)||g(x)|^{p}\right)\mathcal{I}^{\beta}_{a}\left(
|w(x)||f(x)|^{q}\right),\label{3.1}
\end{align}
(5)
\begin{align}
\notag&\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)g(x)|\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||f(x)|^{p-1}|g(x)|^{q-1}\right)\leq
\frac{1}{p}\mathcal{I}^{\alpha}_{a}
\left(|w(x)||f(x)|^{p}\right)\mathcal{I}^{\beta}_{a}\left(
|w(x)||g(x)|^{q}\right)\\ &+\frac{1}{q}\mathcal{I}^{\alpha}_{a}\left(
|w(x)||g(x)|^{q}\right)\mathcal{I}^{\beta}_{a}\left(
|w(x)||f(x)|^{p}\right)\label{3.2}
\end{align}
(6)
and
\begin{align}
\label{3.3}
\notag&\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)g(x)|\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||f(x)g(x)|\right)\leq
\frac{1}{p}\mathcal{I}^{\alpha}_{a}\left(
|w(x)||f(x)|^{p}\right)\mathcal{I}^{\beta}_{a}\left(
|w(x)||g(x)|^{p}\right)\\ &+\frac{1}{q}\mathcal{I}^{\alpha}_{a}\left(
|w(x)||g(x)|^{q}\right)\mathcal{I}^{\beta}_{a}\left(
|w(x)||f(x)|^{q}\right).
\end{align}
(7)
Proof.
For the proof of inequality (5), we set \(\psi=\frac{|f(y)|}{|g(y)|}\) and \(\omega=\frac{|f(z)|}{|g(z)|}\), \(|g(y)|,|g(z)|\neq 0\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young’s inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\), we obtain
\begin{equation}
\label{3.4}
\frac{|f(y)f(z)|}{|g(y)g(z)|}\leq\frac{1}{p}\frac{|f(y)|^{p}}{|g(y)|^{p}}+
\frac{1}{q}\frac{|f(z)|^{q}}{|g(z)|^{q}}.
\end{equation}
(8)
Multiplying inequality (8) by \(h_{\alpha-1}(x,\sigma(y))h_{\beta-1}(x,\sigma(z))|w(y)w(z)|\), \(y,z\in[a,x)_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\) on both sides and double integrating over \(y\) and \(z\), respectively, from \(a\) to \(x\), we get
\begin{align}
\label{3.5}
\notag&\int^{x}_{a}h_{\alpha-1}(x,\sigma(y))|w(y)f(y)||g(y)|^{p-1}\Delta y\int^{x}_{a}h_{\beta-1}(x,\sigma(z))|w(z)f(z)||g(z)|^{q-1}\Delta z\\ \notag&\leq\frac{1}{p}\left(\int^{x}_{a}h_{\alpha-1}(x,\sigma(y))|w(y)||f(y)|^{p}\Delta y\right)
\left(\int^{x}_{a}h_{\beta-1}(x,\sigma(z))|w(z)||g(z)|^{q}\Delta z\right)\\
&\;\;\;+\frac{1}{q}\left(\int^{x}_{a}h_{\alpha-1}(x,\sigma(y))|w(y)||g(y)|^{p}\Delta y\right)\left(\int^{x}_{a}h_{\beta-1}(x,\sigma(z))|w(z)||f(z)|^{q}\Delta z\right).
\end{align}
(9)
Inequality (5) follows from inequality (9).
For the proof of inequality (6), we set \(\psi=\frac{|f(y)|}{|f(z)|}\) and \(\omega=\frac{|g(y)|}{|g(z)|}\), \(|f(z)|,|g(z)|\neq 0\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young’s inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (5), we obtain the desired result.
Now, for the proof of inequality (7), we set \(\psi=|f(y)g(z)|\) and \(\omega=|f(z)g(y)|\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young’s inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (5), we obtain the desired result.
This completes the proof of Theorem 5.
Next, we give a straightforward proof for an extension of dynamic Young’s inequalities by using the time scale \(\nabla\)-Riemann-Liouville type fractional integral.
Theorem 6.
Let \(w,f,g\in C_{ld}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\nabla\)-integrable functions and \(\hat{h}_{\alpha-1}(.,.),\) \(\hat{h}_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):
\begin{align}
\notag&\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)||g(x)|^{p-1}\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||f(x)||g(x)|^{q-1}\right) \leq
\frac{1}{p}\mathcal{J}^{\alpha}_{a}
\left(|w(x)||f(x)|^{p}\right)\mathcal{J}^{\beta}_{a}\left(
|w(x)||g(x)|^{q}\right)\\&+\frac{1}{q}\mathcal{J}^{\alpha}_{a}\left(
|w(x)||g(x)|^{p}\right)\mathcal{J}^{\beta}_{a}\left(
|w(x)||f(x)|^{q}\right),
\end{align}
(10)
\begin{align}
\notag&\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)g(x)|\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||f(x)|^{p-1}|g(x)|^{q-1}\right) \leq
\frac{1}{p}\mathcal{J}^{\alpha}_{a}
\left(|w(x)||f(x)|^{p}\right)\mathcal{J}^{\beta}_{a}\left(
|w(x)||g(x)|^{q}\right)\\&+\frac{1}{q}\mathcal{J}^{\alpha}_{a}\left(
|w(x)||g(x)|^{q}\right)\mathcal{J}^{\beta}_{a}\left(
|w(x)||f(x)|^{p}\right)
\end{align}
(11)
and
\begin{align}
\notag&\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)g(x)|\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||f(x)g(x)|\right) \leq
\frac{1}{p}\mathcal{J}^{\alpha}_{a}\left(
|w(x)||f(x)|^{p}\right)\mathcal{J}^{\beta}_{a}\left(
|w(x)||g(x)|^{p}\right)\\&+\frac{1}{q}\mathcal{J}^{\alpha}_{a}\left(
|w(x)||g(x)|^{q}\right)\mathcal{J}^{\beta}_{a}\left(
|w(x)||f(x)|^{q}\right).
\end{align}
(12)
Proof.
Similar to the proof of Theorem 5.
Remark 1.
Let \(\alpha=\beta=1\), \(\mathbb{T}=\mathbb{Z}\), \(a=1\), \(x=b=n+1\), \(w\equiv1\), \(f(k)=x_{k}\in[0,+\infty)\) and \(g(k)=y_{k}\in[0,+\infty)\) for \(k= 1,2,\ldots,n\). Then inequalities (5), (6) and (7) become
\begin{equation}
\sum\limits^{n}_{k=1}x_{k}y^{p-1}_{k}\sum\limits^{n}_{k=1}x_{k}y^{q-1}_{k}
\leq\frac{1}{p}\sum\limits^{n}_{k=1}x^{p}_{k}\sum\limits^{n}_{k=1}y^{q}_{k}+\frac{1}{q}
\sum\limits^{n}_{k=1}x^{q}_{k}\sum\limits^{n}_{k=1}y^{p}_{k},
\end{equation}
(13)
\begin{equation}
\sum\limits^{n}_{k=1}x_{k}y_{k}\sum\limits^{n}_{k=1}x^{p-1}_{k}y^{q-1}_{k}
\leq\sum\limits^{n}_{k=1}x^{p}_{k}\sum\limits^{n}_{k=1}y^{q}_{k}
\end{equation}
(14)
and
\begin{equation}
\left(\sum\limits^{n}_{k=1}x_{k}y_{k}\right)^{2}
\leq\frac{1}{p}\sum\limits^{n}_{k=1}x^{p}_{k}\sum\limits^{n}_{k=1}y^{p}_{k}+\frac{1}{q}
\sum\limits^{n}_{k=1}x^{q}_{k}\sum\limits^{n}_{k=1}y^{q}_{k}.
\end{equation}
(15)
We give an extension of more dynamic Young’s inequalities by using the time scale \(\Delta\)-Riemann-Liouville type fractional integral.
Theorem 7.
Let \(w,f,g\in C_{rd}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\Delta\)-integrable functions and \(h_{\alpha-1}(.,.),\) \(h_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):
\begin{align}
\label{3.12}
\notag&\mathcal{I}^{\alpha}_{a}\left(
|w(x)||f(x)g(x)|\right)
\mathcal{I}^{\beta}_{a}
\left(|w(x)||f(x)|^{\frac{2}{q}}|g(x)|^{\frac{2}{p}}\right) \leq\frac{1}{p}\mathcal{I}^{\alpha}_{a}\left
(|w(x)||f(x)|^{p}\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||g(x)|^{2}\right)\\&+
\frac{1}{q}\mathcal{I}^{\alpha}_{a}\left(|w(x)||g(x)|^{q}\right)\mathcal{I}^{\beta}_{a}
\left(|w(x)||f(x)|^{2}\right),
\end{align}
(16)
\begin{align}
\label{3.13}
\notag&\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|^{\frac{2}{p}}|g(x)|^{\frac{2}{q}}\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||f(x)|^{p-1}|g(x)|^{q-1}\right)
\leq\frac{1}{p}\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|^{2}\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||g(x)|^{q}\right)\\&
+\frac{1}{q}\mathcal{I}^{\alpha}_{a}\left(|w(x)||g(x)|^{2}\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||f(x)|^{p}\right)
\end{align}
(17)
and
\begin{align}
\label{3.14}
\notag&\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|^{\frac{2}{p}}|g(x)|\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||f(x)|^{\frac{2}{q}}|g(x)|\right)
\leq\frac{1}{p}\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|^{2}\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||g(x)|^{p}\right)
\\&+\frac{1}{q}\mathcal{I}^{\alpha}_{a}\left(|w(x)||g(x)|^{q}\right)
\mathcal{I}^{\beta}_{a}\left(|w(x)||f(x)|^{2}\right).
\end{align}
(18)
Proof.
For the proof of inequality (16), we set \(\psi=|f(y)||g(z)|^{\frac{2}{p}}\) and \(\omega=|f(z)|^{\frac{2}{q}}|g(y)|\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young’s inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\), we obtain
\begin{equation}
\label{3.15}
|f(y)g(y)||f(z)|^{\frac{2}{q}}|g(z)|^{\frac{2}{p}}
\leq\frac{1}{p}|f(y)|^{p}|g(z)|^{2}+
\frac{1}{q}|f(z)|^{2}|g(y)|^{q}.
\end{equation}
(19)
Multiplying (19) by \(h_{\alpha-1}(x,\sigma(y))h_{\beta-1}(x,\sigma(z))|w(y)w(z)|\), \(y,z\in[a,x)_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\) on both sides and double integrating over \(y\) and \(z\), respectively, from \(a\) to \(x\), we get
\begin{align}
\label{3.16}
\notag\int^{x}_{a}h_{\alpha-1}(x,\sigma(y))|w(y)||f(y)g(y)|\Delta y\int^{x}_{a}h_{\beta-1}(x,\sigma(z))|w(z)||f(z)|^{\frac{2}{q}}|g(z)|^{\frac{2}{p}}
\Delta z\\\notag \leq\frac{1}{p}\left(\int^{x}_{a}h_{\alpha-1}(x,\sigma(y))|w(y)||f(y)|^{p}\Delta y\right)
\left(\int^{x}_{a}h_{\beta-1}(x,\sigma(z))|w(z)||g(z)|^{2}\Delta z\right)\\+\frac{1}{q}\left(\int^{x}_{a}h_{\alpha-1}(x,\sigma(y))|w(y)||g(y)|^{q}\Delta y\right)\left(\int^{x}_{a}h_{\beta-1}(x,\sigma(z))|w(z)||f(z)|^{2}\Delta z\right).
\end{align}
(20)
Inequality (16) follows from inequality (20).
For the proof of inequality (17), we set \(\psi=\frac{|f(y)|^{\frac{2}{p}}}{|f(z)|}\) and \(\omega=\frac{|g(y)|^{\frac{2}{q}}}{|g(z)|}\), \(|f(z)|,|g(z)|\) \(\neq0\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young’s inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (16), we obtain the desired result.
Now, for the proof of inequality (18), we set \(\psi=|f(y)|^{\frac{2}{p}}|g(z)|\) and \(\omega=|f(z)|^{\frac{2}{q}}|g(y)|\), \(y,z\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), in the classical Young’s inequality \(\psi\omega\leq\frac{\psi^{p}}{p}+\frac{\omega^{q}}{q}\), \(\psi,\omega\geq0\) and following the same steps used in the proof of inequality (16), we obtain the desired result.
This completes the proof of Theorem 7.
Next, we give an extension of more dynamic Young’s inequalities by using the time scale \(\nabla\)-Riemann-Liouville type fractional integral.
Theorem 8.
Let \(w,f,g\in C_{ld}\left([a,b]_{\mathbb{T}},\mathbb{R}\right)\) be \(\nabla\)-integrable functions and \(\hat{h}_{\alpha-1}(.,.),\) \(\hat{h}_{\beta-1}(.,.)>0\). If \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then the following inequalities hold true for \(\alpha,\beta\geq1\):
\begin{align}
\notag&\mathcal{J}^{\alpha}_{a}\left(
|w(x)||f(x)g(x)|\right)
\mathcal{J}^{\beta}_{a}
\left(|w(x)||f(x)|^{\frac{2}{q}}|g(x)|^{\frac{2}{p}}\right) \leq\frac{1}{p}\mathcal{J}^{\alpha}_{a}\left
(|w(x)||f(x)|^{p}\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||g(x)|^{2}\right)\\&+
\frac{1}{q}\mathcal{J}^{\alpha}_{a}\left(|w(x)||g(x)|^{q}\right)\mathcal{J}^{\beta}_{a}
\left(|w(x)||f(x)|^{2}\right),
\end{align}
(21)
\begin{align}
\notag&\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|^{\frac{2}{p}}|g(x)|^{\frac{2}{q}}\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||f(x)|^{p-1}|g(x)|^{q-1}\right)
\leq\frac{1}{p}\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|^{2}\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||g(x)|^{q}\right)\\&+
\frac{1}{q}\mathcal{J}^{\alpha}_{a}\left(|w(x)||g(x)|^{2}\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||f(x)|^{p}\right)
\end{align}
(22)
and
\begin{align}
\notag&\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|^{\frac{2}{p}}|g(x)|\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||f(x)|^{\frac{2}{q}}|g(x)|\right)
\leq\frac{1}{p}\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|^{2}\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||g(x)|^{p}\right)
\\&+\frac{1}{q}\mathcal{J}^{\alpha}_{a}\left(|w(x)||g(x)|^{q}\right)
\mathcal{J}^{\beta}_{a}\left(|w(x)||f(x)|^{2}\right).
\end{align}
(23)
Proof.
Similar to the proof of Theorem 7.
Remark 2.
Let \(\alpha=\beta=1\), \(\mathbb{T}=\mathbb{Z}\), \(a=1\), \(x=b=n+1\), \(w\equiv1\), \(f(k)=x_{k}\in[0,+\infty)\) and \(g(k)=y_{k}\in[0,+\infty)\) for \(k= 1,2,\ldots,n\). Then inequalities (16), (17) and (18) become
\begin{equation}
\sum\limits^{n}_{k=1}x_{k}y_{k}\sum\limits^{n}_{k=1}x^{\frac{2}{q}}_{k}y^{\frac{2}{p}}_{k}
\leq\frac{1}{p}\sum\limits^{n}_{k=1}x^{p}_{k}\sum\limits^{n}_{k=1}y^{2}_{k}+\frac{1}{q}
\sum\limits^{n}_{k=1}x^{2}_{k}\sum\limits^{n}_{k=1}y^{q}_{k},
\end{equation}
(24)
\begin{equation}
\sum\limits^{n}_{k=1}x^{\frac{2}{p}}_{k}y^{\frac{2}{q}}_{k}
\sum\limits^{n}_{k=1}x^{p-1}_{k}y^{q-1}_{k}
\leq\frac{1}{p}\sum\limits^{n}_{k=1}x^{2}_{k}\sum\limits^{n}_{k=1}y^{q}_{k}+
\frac{1}{q}\sum\limits^{n}_{k=1}x^{p}_{k}\sum\limits^{n}_{k=1}y^{2}_{k}
\end{equation}
(25)
and
\begin{equation}
\sum\limits^{n}_{k=1}x^{\frac{2}{p}}_{k}y_{k}\sum\limits^{n}_{k=1}x^{\frac{2}{q}}_{k}y_{k}
\leq\sum\limits^{n}_{k=1}x^{2}_{k}
\sum\limits^{n}_{k=1}\left(\frac{y^{p}_{k}}{p}+
\frac{y^{q}_{k}}{q}\right).
\end{equation}
(26)
4. Dynamic Qi’s inequality
In this section, we give an extension of dynamic Qi’s inequalities by using the time scale \(\Delta\)-Riemann-Liouville type fractional integral.
Theorem 9.
Let \(w,f,g,h\in C_{rd}\left([a,b]_{\mathbb{T}},\mathbb{R}-\{0\}\right)\) be \(\Delta\)-integrable functions with \(0< m\leq\frac{|f(y)|}{|g(y)|}\leq M0\) and \(\alpha\geq1\).
- (i) If \(p>0\), \(q>0\), \(r< 0\), then
\begin{equation}
\label{4.1}
\left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}}
\left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}}
\left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \geq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}.
\end{equation}
(27)
- (ii) If \(p< 0\), \(q 0\), then
\begin{equation}
\left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}}
\left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}}
\left\{\mathcal{I}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \leq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}.
\end{equation}
(28)
Proof.
Case (i). The given condition \(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0\) can be rearranged to yield
\(\frac{1}{\left(-\frac{p}{r}\right)}+\frac{1}{\left(-\frac{q}{r}\right)}=1\).
Applying dynamic Rogers–Hölder’s inequality [2] for \(-\frac{p}{r}>1\) and \(-\frac{q}{r}>1\), we get
\begin{align}
\label{4.3}
\int^{x}_{a}|w(y)||f(y)|^{-\frac{r}{p}}|g(y)|^{-\frac{r}{q}}\Delta y \leq\left(\int^{x}_{a}|w(y)||f(y)|\Delta y\right)^{-\frac{r}{p}}\left(\int^{x}_{a}|w(y)||g(y)|\Delta y\right)^{-\frac{r}{q}}.
\end{align}
(29)
From (29), we have that
\begin{align}
\int^{x}_{a}|w(y)||f(y)|^{-\frac{r}{p}}|g(y)|^{-\frac{r}{q}}\Delta y \leq\left(\int^{x}_{a}|w(y)||f(y)|^{-\frac{r}{p}}|f(y)|^{-\frac{r}{q}}\Delta y\right)^{-\frac{r}{p}} \times\left(\int^{x}_{a}|w(y)||g(y)|^{-\frac{r}{p}}|g(y)|^{-\frac{r}{q}}
\Delta y\right)^{-\frac{r}{q}}.
\end{align}
(30)
From the given condition, we obtain
\begin{equation*}
|f(y)|^{-\frac{r}{p}}\leq \left(M|g(y)|\right)^{-\frac{r}{p}},~~~~~~|g(y)|^{-\frac{r}{q}}\leq m^{\frac{r}{q}}|f(y)|^{-\frac{r}{q}},
\end{equation*}
on the set \([a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\).
From the above inequality, it follows that
\begin{align}
\notag&\int^{x}_{a}|w(y)||f(y)|^{-\frac{r}{p}}|g(y)|^{-\frac{r}{q}}\Delta y\leq M^{\frac{r^{2}}{p^{2}}}m^{-\frac{r^{2}}{q^{2}}}
\left(\int^{x}_{a}|w(y)||g(y)|^{-\frac{r}{p}}|f(y)|^{-\frac{r}{q}}\Delta y\right)^{-\frac{r}{p}} \\& \times\left(\int^{x}_{a}|w(y)||g(y)|^{-\frac{r}{p}}|f(y)|^{-\frac{r}{q}}
\Delta y\right)^{-\frac{r}{q}}.
\end{align}
(31)
Therefore
\begin{align}
\label{4.6}
\int^{x}_{a}|w(y)||f(y)|^{-\frac{r}{p}}|g(y)|^{-\frac{r}{q}}\Delta y \leq M^{\frac{r^{2}}{p^{2}}}m^{-\frac{r^{2}}{q^{2}}}
\int^{x}_{a}|w(y)||g(y)|^{-\frac{r}{p}}|f(y)|^{-\frac{r}{q}}\Delta y.
\end{align}
(32)
Again, applying dynamic Rogers-Hölder’s inequality on the right-hand side of inequality (32), we obtain
\begin{align}
\label{4.7}
\int^{x}_{a}|w(y)||f(y)|^{-\frac{r}{p}}|g(y)|^{-\frac{r}{q}}\Delta y \leq M^{\frac{r^{2}}{p^{2}}}m^{-\frac{r^{2}}{q^{2}}}
\left(\int^{x}_{a}|w(y)||g(y)|\Delta y\right)^{-\frac{r}{p}}\left(\int^{x}_{a}|w(y)||f(y)|
\Delta y\right)^{-\frac{r}{q}}.
\end{align}
(33)
Using the condition that \(|f(y)|^{\frac{1}{p}}|g(y)|^{\frac{1}{q}}|h(y)|^{\frac{1}{r}}=c\), where \(c\) is a positive real number and \(y\in[a,x]_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), the inequality (33) becomes
\begin{align}
\label{4.8}
\int^{x}_{a}c^{-r}|w(y)||h(y)|\Delta y \leq \frac{M^{\frac{r^{2}}{p^{2}}}}{m^{\frac{r^{2}}{q^{2}}}}
\left(\int^{x}_{a}|w(y)||g(y)|\Delta y\right)^{-\frac{r}{p}}\left(\int^{x}_{a}|w(y)||f(y)|
\Delta y\right)^{-\frac{r}{q}}.
\end{align}
(34)
Taking power \(-\frac{1}{r}>0\) on both sides of inequality (34) and replacing \(|w(y)|\) by \(h_{\alpha-1}(x,\sigma(y))|w(y)|\), \(y\in[a,x)_{\mathbb{T}}\), \(\forall x\in[a,b]_{\mathbb{T}}\), we obtain the desired inequality (27).
The proof of Case (ii) is similar to that of Case (i). This completes the proof of Theorem 9.
Next, we give an extension of dynamic Qi’s inequalities by using the time scale \(\nabla\)-Riemann-Liouville type fractional integral.
Theorem 10.
Let \(w,f,g,h\in C_{ld}\left([a,b]_{\mathbb{T}},\mathbb{R}-\{0\}\right)\) be \(\nabla\)-integrable functions with \(0< m\leq\frac{|f(y)|}{|g(y)|}\leq M0\) and \(\alpha\geq1\).
- (i) If \(p>0\), \(q>0\), \(r< 0\), then
\begin{equation}
\left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}}
\left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}}
\left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \geq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}.
\end{equation}
(35)
- (ii) If \(p< 0\), \(q0\), then
\begin{equation}
\left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||f(x)|\right)\right\}^{\frac{1}{q}}
\left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||g(x)|\right)\right\}^{\frac{1}{p}}
\left\{\mathcal{J}^{\alpha}_{a}\left(|w(x)||h(x)|\right)\right\}^{\frac{1}{r}} \leq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}.
\end{equation}
(36)
Proof.
Similar to the proof of Theorem 9.
Remark 3.
Let \(\alpha=1\), \(\mathbb{T}=\mathbb{R}\), \(x=b\), \(r=-1\), \(c=1\), \(w\equiv1\) and \(f(y),g(y)\in(0,+\infty)\), \(\forall y\in[a,b]\). Then inequality (27) reduces to
\begin{equation}
\label{4.11}
\int^{b}_{a}[f(y)]^{\frac{1}{p}}[g(y)]^{\frac{1}{q}}d y \leq \frac{M^{\frac{1}{p^{2}}}}{m^{\frac{1}{q^{2}}}}\left(\int^{b}_{a}f(y)d y\right)^{\frac{1}{q}}
\left(\int^{b}_{a}g(y)d y\right)^{\frac{1}{p}}.
\end{equation}
(37)
The inequality (37) can be found in [
14].
Remark 4.
Let \(\alpha=1\), \(x=b\), \(r=-1\), \(c=1\), \(w\equiv1\) and \(f(y),g(y)\in(0,+\infty)\), \(\forall y\in[a,b]_{\mathbb{T}}\). Then inequality (27) reduces to
\begin{equation}
\label{4.12}
\int^{b}_{a}[f(y)]^{\frac{1}{p}}[g(y)]^{\frac{1}{q}}\Delta y \leq \frac{M^{\frac{1}{p^{2}}}}{m^{\frac{1}{q^{2}}}}\left(\int^{b}_{a}f(y)\Delta y\right)^{\frac{1}{q}}
\left(\int^{b}_{a}g(y)\Delta y\right)^{\frac{1}{p}}.
\end{equation}
(38)
The inequality (38) may be found in [
10].
Corollary 1.
Let \(x_{k},y_{k},z_{k}\in(0,+\infty)\) with \(0 < m\leq\frac{x_{k}}{y_{k}}\leq M< \infty\) for \(k\in\{1,2,\ldots,n\}\) satisfying \(x^{\frac{1}{p}}_{k}y^{\frac{1}{q}}_{k}z^{\frac{1}{r}}_{k}=c\), where \(c\) is a positive real number. Assume further that \(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0\), \(p,q,r\in\mathbb{R}-\{0\}\).
- (i) If \(p>0\), \(q>0\), \(r< 0\), then
\begin{equation}
\label{4.13}
\left(\sum^{n}_{k=1}x_{k}\right)^{\frac{1}{q}}
\left(\sum^{n}_{k=1}y_{k}\right)^{\frac{1}{p}}
\left(\sum^{n}_{k=1}z_{k}\right)^{\frac{1}{r}} \geq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}.
\end{equation}
(39)
- (ii) If \(p< 0\), \(q 0\), then
\begin{equation}
\label{4.14}
\left(\sum^{n}_{k=1}x_{k}\right)^{\frac{1}{q}}
\left(\sum^{n}_{k=1}y_{k}\right)^{\frac{1}{p}}
\left(\sum^{n}_{k=1}z_{k}\right)^{\frac{1}{r}} \leq c\left(\frac{m^{\frac{1}{q^{2}}}}{M^{\frac{1}{p^{2}}}}\right)^{-r}.
\end{equation}
(40)
Proof.
Putting \(\alpha=1\), \(\mathbb{T}=\mathbb{Z}\), \(a=1\), \(x=b=n+1\) and \(w\equiv1\) in Theorem 9, we obtain the inequalities (39) and (40).
5. Conclusion
Young’s inequalities on fractional calculus by means of generalized fractional
integrals can be found in [
15]. Such inequalities on fractional calculus by Hadamard fractional
integral operator can be found in [
16]. Motivated by the work, we have obtained dynamic Young’s inequalities on fractional calculus of time scales, which has become a significant way in pure and applied mathematics. We have also developed several versions of dynamic Qi’s inequalities on fractional calculus of time scales.
Conflict of Interests
The author declares no conflict of interest.
