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\begin{document}

\title{Complex left Caputo fractional inequalities}
\author{George A. Anastassiou \\
%EndAName
Department of Mathematical Sciences\\
University of Memphis\\
Memphis, TN 38152, U.S.A.\\
ganastss@memphis.edu}
\date{}
\maketitle

\begin{abstract}
Here we present several complex left Caputo type fractional inequalities of
well known kinds, such as of Ostrowski, Poincare,\ Sobolev, Opial and
Hilbert-Pachpatte.
\end{abstract}

\noindent \textbf{2010 Mathematics Subject Classification : }26D10, 26D15,
30A10.

\noindent \textbf{Keywords and phrases:} Complex inequalities, fractional
inequalities, Caputo fractional derivative.

\section{Introduction}

We are motivated by the following result for functions of complex variable:
Complex Ostrowski type inequality

\begin{theorem}
\label{ta}(see \cite{3}) Let $f$ be holomorphic in $G$, an open domain and
suppose $\gamma \subset G$ is a smooth path from $z\left( a\right) =u$ to $%
z\left( b\right) =w$. If $v=z\left( x\right) $ with $x\in \left( a,b\right) $%
, then $\gamma _{u,w}=\gamma _{u,v}\cup \gamma _{v,w}$,
\begin{equation*}
\left\vert f\left( v\right) \left( w-u\right) -\int_{\gamma }f\left(
z\right) dz\right\vert \leq
\end{equation*}%
\begin{equation*}
\left\Vert f^{\prime }\right\Vert _{\gamma _{u,v};\infty }\int_{\gamma
_{u,v}}\left\vert z-u\right\vert \left\vert dz\right\vert +\left\Vert
f^{\prime }\right\Vert _{\gamma _{v,w};\infty }\int_{\gamma
_{v,w}}\left\vert z-w\right\vert \left\vert dz\right\vert \leq
\end{equation*}%
\begin{equation*}
\left[ \int_{\gamma _{u,v}}\left\vert z-u\right\vert \left\vert
dz\right\vert +\int_{\gamma _{v,w}}\left\vert z-w\right\vert \left\vert
dz\right\vert \right] \left\Vert f^{\prime }\right\Vert _{\gamma
_{u,w};\infty },
\end{equation*}%
and
\begin{equation*}
\left\vert f\left( v\right) \left( w-u\right) -\int_{\gamma }f\left(
z\right) dz\right\vert \leq
\end{equation*}%
\begin{equation*}
\underset{z\in \gamma _{u,v}}{\max }\left\vert z-u\right\vert \left\Vert
f^{\prime }\right\Vert _{\gamma _{u,v};1}+\underset{z\in \gamma _{v,w}}{\max
}\left\vert z-w\right\vert \left\Vert f^{\prime }\right\Vert _{\gamma
_{v,w};1}\leq
\end{equation*}%
\begin{equation*}
\max \left\{ \underset{z\in \gamma _{u,v}}{\max }\left\vert z-u\right\vert ,%
\underset{z\in \gamma _{v,w}}{\max }\left\vert z-w\right\vert \right\}
\left\Vert f^{\prime }\right\Vert _{\gamma _{u,w};1}.
\end{equation*}%
If $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then
\begin{equation*}
\left\vert f\left( v\right) \left( w-u\right) -\int_{\gamma }f\left(
z\right) dz\right\vert \leq
\end{equation*}%
\begin{equation*}
\left( \int_{\gamma _{u,v}}\left\vert z-u\right\vert ^{q}\left\vert
dz\right\vert \right) ^{\frac{1}{q}}\left\Vert f^{\prime }\right\Vert
_{\gamma _{u,v};p}+\left( \int_{\gamma _{v,w}}\left\vert z-w\right\vert
^{q}\left\vert dz\right\vert \right) ^{\frac{1}{q}}\left\Vert f^{\prime
}\right\Vert _{\gamma _{v,w};p}\leq
\end{equation*}%
\begin{equation*}
\left( \int_{\gamma _{u,v}}\left\vert z-u\right\vert ^{q}\left\vert
dz\right\vert +\int_{\gamma _{v,w}}\left\vert z-w\right\vert ^{q}\left\vert
dz\right\vert \right) ^{\frac{1}{q}}\left\Vert f^{\prime }\right\Vert
_{\gamma _{u,w};p}.
\end{equation*}%
Above $\left\vert \cdot \right\vert $ is the complex absolute value.
\end{theorem}

We are also motivated by the next complex Opial type inequality:

\begin{theorem}
\label{tb}(see \cite{2}) Let $f:D\subseteq \mathbb{C}\rightarrow \mathbb{C}$
be an analytic function on the domain $D$ and let $x,y,w\in D$. Suppose $%
\gamma $ is a smooth path parametrized by $z\left( t\right) $, $t\in \left[
a,b\right] $ with $z\left( a\right) =x$, $z\left( c\right) =y$, and $z\left(
b\right) =w$, where $c\in \left[ a,b\right] $ is floating. Assume that $%
f^{\left( k\right) }\left( x\right) =0$, $k=0,1,...,n$, $n\in \mathbb{Z}_{+}$%
, and $p,q>1:\frac{1}{p}+\frac{1}{q}=1$. Then

1)
\begin{equation*}
\left\vert \int_{a}^{b}f\left( z\left( t\right) \right) f^{\left( n+1\right)
}\left( z\left( t\right) \right) z^{\prime }\left( t\right) dt\right\vert
\leq \int_{a}^{b}\left\vert f\left( z\left( t\right) \right) \right\vert
\left\vert f^{\left( n+1\right) }\left( z\left( t\right) \right) \right\vert
\left\vert z^{\prime }\left( t\right) \right\vert dt\leq
\end{equation*}%
\begin{equation*}
\frac{1}{2^{\frac{1}{q}}n!}\left[ \int_{a}^{b}\left( \int_{a}^{c}\left\vert
z\left( c\right) -z\left( t\right) \right\vert ^{pn}\left\vert z^{\prime
}\left( t\right) \right\vert dt\right) \left\vert z^{\prime }\left( c\right)
\right\vert dc\right] ^{\frac{1}{p}}\cdot
\end{equation*}%
\begin{equation*}
\left( \int_{a}^{b}\left\vert f^{\left( n+1\right) }\left( z\left( t\right)
\right) \right\vert ^{q}\left\vert z^{\prime }\left( t\right) \right\vert
dt\right) ^{\frac{2}{q}},
\end{equation*}%
equivalently it holds

2)
\begin{equation*}
\left\vert \int_{\gamma _{x,w}}f\left( z\right) f^{\left( n+1\right) }\left(
z\right) dz\right\vert \leq \int_{\gamma _{x,w}}\left\vert f\left( z\right)
\right\vert \left\vert f^{\left( n+1\right) }\left( z\right) \right\vert
\left\vert dz\right\vert \leq
\end{equation*}%
\begin{equation*}
\frac{1}{2^{\frac{1}{q}}n!}\left[ \int_{a}^{b}\left( \int_{\gamma
_{x,y}}\left\vert z\left( c\right) -z\right\vert ^{pn}\left\vert
dz\right\vert \right) \left\vert z^{\prime }\left( c\right) \right\vert dc%
\right] ^{\frac{1}{p}}\left( \int_{\gamma _{x,w}}\left\vert f^{\left(
n+1\right) }\left( z\right) \right\vert ^{q}\left\vert dz\right\vert \right)
^{\frac{2}{q}}.
\end{equation*}
\end{theorem}

Here we utilize on $\mathbb{C}$ the results of \cite{1} which are for
general Banach space valued functions.

Mainly we give different cases of the left fractional $\mathbb{C}$-Ostrowski
type inequality and we continue with the left fractional: $\mathbb{C}$%
-Poincar\'{e} like and Sobolev like inequalities.

We present an Opial type left $\mathbb{C}$-fractional inequality, and we
finish with the Hilbert-Pachpatte left $\mathbb{C}$-fractional inequalities.

\section{Background}

In this section all integrals are of Bochner type.

We need

\begin{definition}
\label{d1}(see \cite{4}) A definition of the Hausdorff measure $h_{\alpha }$
goes as follows: if $\left( T,d\right) $ is a metric space, $A\subseteq T$
and $\delta >0$, let $\Lambda \left( A,\delta \right) $ be the set of all
arbitrary collections $\left( C\right) _{i}$ of subsets of $T$, such that $%
A\subseteq \cup _{i}C_{i}$ and $diam\left( C_{i}\right) \leq \delta $ ($%
diam= $diameter) for every $i$. Now, for every $\alpha >0$ define
\begin{equation}
h_{\alpha }^{\delta }\left( A\right) :=\inf \left\{ \sum \left(
diamC_{i}\right) ^{\alpha }|\left( C_{i}\right) _{i}\in \Lambda \left(
A,\delta \right) \right\} .  \tag{1}  \label{1.}
\end{equation}%
Then there exists $\underset{\delta \rightarrow 0}{\lim }h_{\alpha }^{\delta
}\left( A\right) =\underset{\delta >0}{\sup }h_{\alpha }^{\delta }\left(
A\right) $, and $h_{\alpha }\left( A\right) :=\underset{\delta \rightarrow 0}%
{\lim }h_{\alpha }^{\delta }\left( A\right) $ gives an outer measure on the
power set $\mathcal{P}\left( T\right) $, which is countably additive on the $%
\sigma $-field of all Borel subsets of $T$. If $T=\mathbb{R}^{n}$, then the
Hausdorff measure $h_{n}$, restricted to the $\sigma $-field of the Borel
subsets of $\mathbb{R}^{n}$, equals the Lebesgue measure on $\mathbb{R}^{n}$
up to a constant multiple. In particular, $h_{1}\left( C\right) =\mu \left(
C\right) $ for every Borel set $C\subseteq \mathbb{R}$, where $\mu $ is the
Lebesgue measure.
\end{definition}

\begin{definition}
\label{d2}(\cite{1}) Let $\left[ a,b\right] \subset \mathbb{R}$, $X$ be a
Banach space, $\nu >0$; $n:=\left\lceil \nu \right\rceil \in \mathbb{N}$, $%
\left\lceil \cdot \right\rceil $ is the ceiling of the number, $f:\left[ a,b%
\right] \rightarrow X$. We assume that $f^{\left( n\right) }\in L_{1}\left( %
\left[ a,b\right] ,X\right) $. We call the Caputo-Bochner left fractional
derivative of order $\nu $:
\begin{equation}
\left( D_{\ast a}^{\nu }f\right) \left( x\right) :=\frac{1}{\Gamma \left(
n-\nu \right) }\int_{a}^{x}\left( x-t\right) ^{n-\nu -1}f^{\left( n\right)
}\left( t\right) dt,\text{ \ }\forall \text{ }x\in \left[ a,b\right] .
\tag{2}  \label{2.}
\end{equation}%
If $\nu \in \mathbb{N}$, we set $D_{\ast a}^{\nu }f:=f^{\left( \nu \right) }$
the ordinary $X$-valued derivative, defined similarly to the numerical one,
and also set $D_{\ast a}^{0}f:=f.$
\end{definition}

By \cite{1} $\left( D_{\ast a}^{\nu }f\right) \left( x\right) $ exists
almost everywhere in $x\in \left[ a,b\right] $ and $D_{\ast a}^{\nu }f\in
L_{1}\left( \left[ a,b\right] ,X\right) $.

If $\left\Vert f^{\left( n\right) }\right\Vert _{L_{\infty }\left( \left[ a,b%
\right] ,X\right) }<\infty $, then by \cite{1} $D_{\ast a}^{\nu }f\in
C\left( \left[ a,b\right] ,X\right) .$

We need the left-fractional Taylor's formula:

\begin{theorem}
\label{t3}(\cite{1}) Let $n\in \mathbb{N}$ and $f\in C^{n-1}\left( \left[ a,b%
\right] ,X\right) ,$ where $\left[ a,b\right] \subset \mathbb{R}$ and $X$ is
a Banach space, and let $\nu \geq 0:n=\left\lceil \nu \right\rceil $. Set
\begin{equation}
F_{x}\left( t\right) :=\sum_{i=0}^{n-1}\frac{\left( x-t\right) ^{i}}{i!}%
f^{\left( i\right) }\left( t\right) \text{, \ \ }\forall \text{ }t\in \left[
a,x\right] \text{,}  \tag{3}  \label{3.}
\end{equation}%
where $x\in \left[ a,b\right] .$

Assume that $f^{\left( n\right) }$ exists outside a $\mu $-null Borel set $%
B_{x}\subseteq \left[ a,x\right] $, such that
\begin{equation}
h_{1}\left( F_{x}\left( B_{x}\right) \right) =0\text{, }\forall \text{ }x\in %
\left[ a,b\right] .  \tag{4}  \label{4.}
\end{equation}%
We also assume that $f^{\left( n\right) }\in L_{1}\left( \left[ a,b\right]
,X\right) $. Then
\begin{equation}
f\left( x\right) =\sum_{i=0}^{n-1}\frac{\left( x-a\right) ^{i}}{i!}f^{\left(
i\right) }\left( a\right) +\frac{1}{\Gamma \left( \nu \right) }%
\int_{a}^{x}\left( x-z\right) ^{\nu -1}\left( D_{\ast a}^{\nu }f\right)
\left( z\right) dz,  \tag{5}  \label{5.}
\end{equation}%
$\forall $ $x\in \left[ a,b\right] .$
\end{theorem}

Next we mention an Ostrowski type inequality at left fractional level for
Banach valued functions.

\begin{theorem}
\label{t4}(\cite{1}) Let $\nu \geq 0$, $n=\left\lceil \nu \right\rceil $.
Here all as in Theorem \ref{t3}. Assume that $f^{\left( i\right) }\left(
a\right) =0$, $i=1,...,n-1$, and that $D_{\ast a}^{\nu }f\in L_{\infty
}\left( \left[ a,b\right] ,X\right) $. Then
\begin{equation}
\left\Vert \frac{1}{b-a}\int_{a}^{b}f\left( x\right) dx-f\left( a\right)
\right\Vert \leq \frac{\left\Vert D_{\ast a}^{\nu }f\right\Vert _{L_{\infty
}\left( \left[ a,b\right] ,X\right) }}{\Gamma \left( \nu +2\right) }\left(
b-a\right) ^{\nu }.  \tag{6}  \label{6.}
\end{equation}
\end{theorem}

We mention an Ostrowski type $L_{p}$ fractional inequality:

\begin{theorem}
\label{t5}(\cite{1}) Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu >\frac{1%
}{q}$, $n=\left\lceil \nu \right\rceil $. Here all as in Theorem \ref{t3}.
Assume that $f^{\left( k\right) }\left( a\right) =0$, $k=1,...,n-1,$ and $%
D_{\ast a}^{\nu }f\in L_{q}\left( \left[ a,b\right] ,X\right) $, where $X$
is a Banach space. Then
\begin{equation}
\left\Vert \frac{1}{b-a}\int_{a}^{b}f\left( x\right) dx-f\left( a\right)
\right\Vert \leq \frac{\left\Vert D_{\ast a}^{\nu }f\right\Vert
_{L_{q}\left( \left[ a,b\right] ,X\right) }}{\Gamma \left( \nu \right)
\left( p\left( \nu -1\right) +1\right) ^{\frac{1}{p}}\left( \nu +\frac{1}{p}%
\right) }\left( b-a\right) ^{\nu -\frac{1}{q}}.  \tag{7}  \label{7.}
\end{equation}
\end{theorem}

It follows

\begin{corollary}
\label{c6}(\cite{1}) (to Theorem \ref{t5}, case of $p=q=2$). Let $\nu >\frac{%
1}{2}$, $n=\left\lceil \nu \right\rceil $. Here all as in Theorem \ref{t3}.
Assume that $f^{\left( k\right) }\left( a\right) =0$, $k=1,...,n-1$, and $%
D_{\ast a}^{\nu }f\in L_{2}\left( \left[ a,b\right] ,X\right) $. Then
\begin{equation}
\left\Vert \frac{1}{b-a}\int_{a}^{b}f\left( x\right) dx-f\left( a\right)
\right\Vert \leq \frac{\left\Vert D_{\ast a}^{\nu }f\right\Vert
_{L_{2}\left( \left[ a,b\right] ,X\right) }}{\Gamma \left( \nu \right)
\left( \sqrt{2\nu -1}\right) \left( \nu +\frac{1}{2}\right) }\left(
b-a\right) ^{\nu -\frac{1}{2}}.  \tag{8}  \label{8.}
\end{equation}
\end{corollary}

Next comes the $L_{1}$ case of fractional Ostrowski inequality:

\begin{theorem}
\label{t7}(\cite{1}) Let $\nu \geq 1$, $n=\left\lceil \nu \right\rceil $,
and all as in Theorem \ref{t3}. Assume that $f^{\left( k\right) }\left(
a\right) =0$, $k=1,...,n-1$, and $D_{\ast a}^{\nu }f\in L_{1}\left( \left[
a,b\right] ,X\right) $. Then
\begin{equation}
\left\Vert \frac{1}{b-a}\int_{a}^{b}f\left( x\right) dx-f\left( a\right)
\right\Vert \leq \frac{\left\Vert D_{\ast a}^{\nu }f\right\Vert
_{L_{1}\left( \left[ a,b\right] ,X\right) }}{\Gamma \left( \nu +1\right) }%
\left( b-a\right) ^{\nu -1}.  \tag{9}  \label{9.}
\end{equation}
\end{theorem}

We continue with a Poincar\'{e} like fractional inequality:

\begin{theorem}
\label{t8}(\cite{1}) Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu >\frac{1%
}{q}$, $n=\left\lceil \nu \right\rceil $. Here all as in Theorem \ref{t3}.
Assume that $f^{\left( k\right) }\left( a\right) =0$, $k=0,1,...,n-1$, and $%
D_{\ast a}^{\nu }f\in L_{q}\left( \left[ a,b\right] ,X\right) $, where $X$
is a Banach space. Then
\begin{equation}
\left\Vert f\right\Vert _{L_{q}\left( \left[ a,b\right] ,X\right) }\leq
\frac{\left( b-a\right) ^{\nu }}{\Gamma \left( \nu \right) \left( p\left(
\nu -1\right) +1\right) ^{\frac{1}{p}}\left( q\nu \right) ^{\frac{1}{q}}}%
\left\Vert D_{\ast a}^{\nu }f\right\Vert _{L_{q}\left( \left[ a,b\right]
,X\right) }.  \tag{10}  \label{10.}
\end{equation}
\end{theorem}

Next comes a Sobolev like fractional inequality.

\begin{theorem}
\label{t9}(\cite{1}) All as in the last Theorem \ref{t8}. Let $r>0$. Then
\begin{equation}
\left\Vert f\right\Vert _{L_{r}\left( \left[ a,b\right] ,X\right) }\leq
\frac{\left( b-a\right) ^{\nu -\frac{1}{q}+\frac{1}{r}}}{\Gamma \left( \nu
\right) \left( p\left( \nu -1\right) +1\right) ^{\frac{1}{p}}\left( r\left(
\nu -\frac{1}{q}\right) +1\right) ^{\frac{1}{r}}}\left\Vert D_{\ast a}^{\nu
}f\right\Vert _{L_{q}\left( \left[ a,b\right] ,X\right) }.  \tag{11}
\label{11.}
\end{equation}
\end{theorem}

We mention the following Opial type fractional inequality:

\begin{theorem}
\label{t10}(\cite{1}) Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu >\frac{%
1}{q}$, $n:=\left\lceil \nu \right\rceil $. Let $\left[ a,b\right] \subset
\mathbb{R}$, $X$ a Banach space, and $f\in C^{n-1}\left( \left[ a,b\right]
,X\right) $. Set
\begin{equation}
F_{x}\left( t\right) :=\sum_{i=0}^{n-1}\frac{\left( x-t\right) ^{i}}{i!}%
f^{\left( i\right) }\left( t\right) \text{, \ }\forall \text{ }t\in \left[
a,x\right] ,\text{ where }x\in \left[ a,b\right] .  \tag{12}  \label{12.}
\end{equation}%
Assume that $f^{\left( n\right) }$ exists outside a $\mu $-null Borel set $%
B_{x}\subseteq \left[ a,x\right] $, such that
\begin{equation}
h_{1}\left( F_{x}\left( B_{x}\right) \right) =0\text{, \ }\forall \text{ }%
x\in \left[ a,b\right] .  \tag{13}  \label{13.}
\end{equation}%
We also assume that $f^{\left( n\right) }\in L_{\infty }\left( \left[ a,b%
\right] ,X\right) $. Assume also that $f^{\left( k\right) }\left( a\right) =0
$, $k=0,1,...,n-1$. Then
\begin{equation*}
\int_{a}^{x}\left\Vert f\left( w\right) \right\Vert \left\Vert \left(
D_{\ast a}^{\nu }f\right) \left( w\right) \right\Vert dw\leq
\end{equation*}%
\begin{equation}
\frac{\left( x-a\right) ^{\nu -1+\frac{2}{p}}}{2^{\frac{1}{q}}\Gamma \left(
\nu \right) \left( \left( p\left( \nu -1\right) +1\right) \left( p\left( \nu
-1\right) +2\right) \right) ^{\frac{1}{p}}}\left( \int_{a}^{x}\left\Vert
\left( D_{\ast a}^{\nu }f\right) \left( z\right) \right\Vert ^{q}dz\right) ^{%
\frac{2}{q}},  \tag{14}  \label{14.}
\end{equation}%
$\forall $ $x\in \left[ a,b\right] .$
\end{theorem}

We finish this section with a Hilbert-Pachpatte left fractional inequality:

\begin{theorem}
\label{t11}(\cite{1}) Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu _{1}>%
\frac{1}{q}$, $\nu _{2}>\frac{1}{p}$, $n_{i}:=\left\lceil \nu
_{i}\right\rceil $, $i=1,2$. Here $\left[ a_{i},b_{i}\right] \subset \mathbb{%
R}$, $i=1,2;$ $X$ is a Banach space. Let $f_{i}\in C^{n_{i}-1}\left( \left[
a_{i},b_{i}\right] ,X\right) $, $i=1,2$. Set
\begin{equation}
F_{x_{i}}\left( t_{i}\right) :=\sum_{j_{i}=0}^{n_{i}-1}\frac{\left(
x_{i}-t_{i}\right) ^{j_{i}}}{j_{i}!}f_{i}^{\left( j_{i}\right) }\left(
t_{i}\right) \text{,}  \tag{15}  \label{15.}
\end{equation}%
$\forall $ $t_{i}\in \left[ a_{i},x_{i}\right] $, where $x_{i}\in \left[
a_{i},b_{i}\right] $; $i=1,2$. Assume that $f_{i}^{\left( n_{i}\right) }$
exists outside a $\mu $-null Borel set $B_{x_{i}}\subseteq \left[ a_{i},x_{i}%
\right] $, such that
\begin{equation}
h_{1}\left( F_{x_{i}}\left( B_{x_{i}}\right) \right) =0\text{, \ }\forall
\text{ }x_{i}\in \left[ a_{i},b_{i}\right] ;\text{ }i=1,2.  \tag{16}
\label{16.}
\end{equation}%
We also assume that $f_{i}^{\left( n_{i}\right) }\in L_{1}\left( \left[
a_{i},b_{i}\right] ,X\right) $, and
\begin{equation}
f_{i}^{\left( k_{i}\right) }\left( a_{i}\right) =0\text{, \ }%
k_{i}=0,1,...,n_{i}-1;\text{ }i=1,2,  \tag{17}  \label{17.}
\end{equation}%
and
\begin{equation*}
\left( D_{\ast a_{1}}^{\nu _{1}}f_{1}\right) \in L_{q}\left( \left[
a_{1},b_{1}\right] ,X\right) ,\text{ \ }\left( D_{\ast a_{2}}^{\nu
_{2}}f_{2}\right) \in L_{p}\left( \left[ a_{2},b_{2}\right] ,X\right) .
\end{equation*}%
Then
\begin{equation*}
\int_{a_{1}}^{b_{1}}\int_{a_{2}}^{b_{2}}\frac{\left\Vert f_{1}\left(
x_{1}\right) \right\Vert \left\Vert f_{2}\left( x_{2}\right) \right\Vert
dx_{1}dx_{2}}{\left( \frac{\left( x_{1}-a_{1}\right) ^{p\left( \nu
_{1}-1\right) +1}}{p\left( p\left( \nu _{1}-1\right) +1\right) }+\frac{%
\left( x_{2}-a_{2}\right) ^{q\left( \nu _{2}-1\right) +1}}{q\left( q\left(
\nu _{2}-1\right) +1\right) }\right) }\leq
\end{equation*}%
\begin{equation}
\frac{\left( b_{1}-a_{1}\right) \left( b_{2}-a_{2}\right) }{\Gamma \left(
\nu _{1}\right) \Gamma \left( \nu _{2}\right) }\left\Vert D_{\ast
a_{1}}^{\nu _{1}}f_{1}\right\Vert _{L_{q}\left( \left[ a_{1},b_{1}\right]
,X\right) }\left\Vert D_{\ast a_{2}}^{\nu _{2}}f_{2}\right\Vert
_{L_{p}\left( \left[ a_{2},b_{2}\right] ,X\right) }.  \tag{18}  \label{18.}
\end{equation}
\end{theorem}

\section{Main Results}

We need a special case of Definition \ref{d2} over $\mathbb{C}.$

\begin{definition}
\label{d12} Let $\left[ a,b\right] \subset \mathbb{R}$, $\nu >0$; $%
n:=\left\lceil \nu \right\rceil \in \mathbb{N}$, $\left\lceil \cdot
\right\rceil $ is the ceiling of the number and $f\in C^{n}\left( \left[ a,b%
\right] ,\mathbb{C}\right) $. We call Caputo-Complex left fractional
derivative of order $\nu $:
\begin{equation}
\left( D_{\ast a}^{\nu }f\right) \left( x\right) :=\frac{1}{\Gamma \left(
n-\nu \right) }\int_{a}^{x}\left( x-t\right) ^{n-\nu -1}f^{\left( n\right)
}\left( t\right) dt,\text{ \ }\forall \text{ }x\in \left[ a,b\right] ,
\tag{19}  \label{19.}
\end{equation}%
where the derivatives $f^{\prime },...f^{\left( n\right) }$ are defined as
the numerical derivative.

If $\nu \in \mathbb{N}$, we set $D_{\ast a}^{\nu }f:=f^{\left( \nu \right) }$
the ordinary $\mathbb{C}$-valued derivative and also set $D_{\ast a}^{0}f:=f.
$
\end{definition}

Notice here (by \cite{1}) that $D_{\ast a}^{\nu }f\in C\left( \left[ a,b%
\right] ,\mathbb{C}\right) .$

We make

\begin{remark}
\label{r13}Suppose $\gamma $ is a smooth path parametrized by $z\left(
t\right) $, $t\in \left[ a,b\right] $ (i.e. there exists $z^{\prime }\left(
t\right) $ and is continuous) and from now on $f$ is a complex function
which is continuous on $\gamma $.

Put $z\left( a\right) =u$ and $z\left( b\right) =w$ with $u,w\in \mathbb{C}$%
. We define the integral of $f$ on $\gamma _{u,w}=\gamma $ as
\begin{equation}
\int_{\gamma }f\left( z\right) dz=\int_{\gamma _{u,w}}f\left( z\right)
dz:=\int_{a}^{b}f\left( z\left( t\right) \right) z^{\prime }\left( t\right)
dt=\int_{a}^{b}h\left( t\right) dt,  \tag{20}  \label{20.}
\end{equation}%
where $h\left( t\right) :=f\left( z\left( t\right) \right) z^{\prime }\left(
t\right) $, $t\in \left[ a,b\right] .$

We notice that the actual choice of parametrization of $\gamma $ does not
matter.

This definition immediately extends to paths that are piecewise smooth.
Suppose $\gamma $ is parametrized by $z\left( t\right) $, $t\in \left[ a,b%
\right] $, which is differentiable on the intervals $\left[ a,c\right] $ and
$\left[ c,b\right] $, then assuming that $f$ is continuous on $\gamma $ we
define
\begin{equation*}
\int_{\gamma _{u,w}}f\left( z\right) dz:=\int_{\gamma _{u,v}}f\left(
z\right) dz+\int_{\gamma _{v,w}}f\left( z\right) dz,
\end{equation*}%
where $v:=z\left( c\right) $. This can be extended for a finite number of
intervals.

We also define the integral with respect to arc-length
\begin{equation*}
\int_{\gamma _{u,w}}f\left( z\right) \left\vert dz\right\vert
:=\int_{a}^{b}f\left( z\left( t\right) \right) \left\vert z^{\prime }\left(
t\right) \right\vert dt
\end{equation*}%
and the length of the curve $\gamma $ is then
\begin{equation*}
l\left( \gamma \right) =\int_{\gamma _{u,w}}\left\vert dz\right\vert
:=\int_{a}^{b}\left\vert z^{\prime }\left( t\right) \right\vert dt.
\end{equation*}

We mention also the triangle inequality for the complex integral, namely
\begin{equation}
\left\vert \int_{\gamma }f\left( z\right) dz\right\vert \leq \int_{\gamma
}\left\vert f\left( z\right) \right\vert \left\vert dz\right\vert \leq
\left\Vert f\right\Vert _{\gamma ,\infty }l\left( \gamma \right) ,  \tag{21}
\label{21.}
\end{equation}%
where $\left\Vert f\right\Vert _{\gamma ,\infty }:=\underset{z\in \gamma }{%
\sup }\left\vert f\left( z\right) \right\vert $.
\end{remark}

We give the following left-fractional $\mathbb{C}$-Taylor's formula:

\begin{theorem}
\label{t14}Let $h\in C^{n}\left( \left[ a,b\right] ,\mathbb{C}\right) $, $%
n=\left\lceil \nu \right\rceil $, $\nu \geq 0$. Then
\begin{equation}
h\left( t\right) =\sum\limits_{i=0}^{n-1}\frac{\left( t-a\right) ^{i}}{i!}%
h^{\left( i\right) }\left( a\right) +\frac{1}{\Gamma \left( \nu \right) }%
\int_{a}^{t}\left( t-\lambda \right) ^{\nu -1}\left( D_{\ast a}^{\nu
}h\right) \left( \lambda \right) d\lambda ,  \tag{22}  \label{22.}
\end{equation}%
$\forall $ $t\in \left[ a,b\right] ,$

in particular it holds,
\begin{equation*}
f\left( z\left( t\right) \right) z^{\prime }\left( t\right)
=\sum\limits_{i=0}^{n-1}\frac{\left( t-a\right) ^{i}}{i!}\left( f\left(
z\left( a\right) \right) z^{\prime }\left( a\right) \right) ^{\left(
i\right) }+
\end{equation*}%
\begin{equation}
\frac{1}{\Gamma \left( \nu \right) }\int_{a}^{t}\left( t-\lambda \right)
^{\nu -1}\left( D_{\ast a}^{\nu }f\left( z\left( \cdot \right) \right)
z^{\prime }\left( \cdot \right) \right) \left( \lambda \right) d\lambda ,
\tag{23}  \label{23.}
\end{equation}%
$\forall $ $t\in \left[ a,b\right] .$
\end{theorem}

\begin{proof}
By Theorem \ref{t3}.
\end{proof}

It follows a left fractional $\mathbb{C}$-Ostroswski type inequality

\begin{theorem}
\label{t15}Let $n\in \mathbb{N}$ and $h\in C^{n}\left( \left[ a,b\right] ,%
\mathbb{C}\right) $, where $\left[ a,b\right] \subset \mathbb{R}$, and let $%
\nu \geq 0:n=\left\lceil \nu \right\rceil $. Assume that $h^{\left( i\right)
}\left( a\right) =0$, $i=1,...,n-1$. Then
\begin{equation}
\left\vert \frac{1}{b-a}\int_{a}^{b}h\left( t\right) dt-f\left( a\right)
\right\vert \leq \frac{\left\Vert D_{\ast a}^{\nu }h\right\Vert _{\infty ,%
\left[ a,b\right] }}{\Gamma \left( \nu +2\right) }\left( b-a\right) ^{\nu },
\tag{24}  \label{24.}
\end{equation}%
in particular when $h\left( t\right) :=f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) $ and $\left( f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) \right) ^{\left( i\right) }|_{t=a}=0$, $i=1,...n-1
$, we get
\begin{equation*}
\left\vert \frac{1}{b-a}\int_{\gamma _{u,w}}f\left( z\right) dz-f\left(
u\right) z^{\prime }\left( a\right) \right\vert =\left\vert \frac{1}{b-a}%
\int_{a}^{b}f\left( z\left( t\right) \right) z^{\prime }\left( t\right)
dt-f\left( z\left( a\right) \right) z^{\prime }\left( a\right) \right\vert
\end{equation*}%
\begin{equation}
\leq \frac{\left\Vert D_{\ast a}^{\nu }f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) \right\Vert _{\infty ,\left[ a,b\right] }}{\Gamma
\left( \nu +2\right) }\left( b-a\right) ^{\nu }.  \tag{25}  \label{25.}
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t4}.
\end{proof}

The corresponding $\mathbb{C}$-Ostrowski type $L_{p}$ inequality follows:

\begin{theorem}
\label{t16} Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu >\frac{1}{q}$, $%
n=\left\lceil \nu \right\rceil $. Here $h\in C^{n}\left( \left[ a,b\right] ,%
\mathbb{C}\right) $. Assume that $h^{\left( i\right) }\left( a\right) =0$, $%
i=1,...,n-1.$ Then
\begin{equation}
\left\vert \frac{1}{b-a}\int_{a}^{b}h\left( t\right) dt-h\left( a\right)
\right\vert \leq \frac{\left\Vert D_{\ast a}^{\nu }h\right\Vert
_{L_{q}\left( \left[ a,b\right] ,\mathbb{C}\right) }}{\Gamma \left( \nu
\right) \left( p\left( \nu -1\right) +1\right) ^{\frac{1}{p}}\left( \nu +%
\frac{1}{p}\right) }\left( b-a\right) ^{\nu -\frac{1}{q}},  \tag{26}
\label{26.}
\end{equation}%
in particular when $h\left( t\right) :=f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) $ and $\left( f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) \right) ^{\left( i\right) }|_{t=a}=0$, $i=1,...n-1
$, we get:
\begin{equation*}
\left\vert \frac{1}{b-a}\int_{\gamma _{u,w}}f\left( z\right) dz-f\left(
u\right) z^{\prime }\left( a\right) \right\vert =\left\vert \frac{1}{b-a}%
\int_{a}^{b}f\left( z\left( t\right) \right) z^{\prime }\left( t\right)
dt-f\left( z\left( a\right) \right) z^{\prime }\left( a\right) \right\vert
\end{equation*}%
\begin{equation}
\leq \frac{\left\Vert D_{\ast a}^{\nu }\left( f\left( z\left( t\right)
\right) z^{\prime }\left( t\right) \right) \right\Vert _{L_{q}\left( \left[
a,b\right] ,\mathbb{C}\right) }}{\Gamma \left( \nu \right) \left( p\left(
\nu -1\right) +1\right) ^{\frac{1}{p}}\left( \nu +\frac{1}{p}\right) }\left(
b-a\right) ^{\nu -\frac{1}{q}}.  \tag{27}  \label{27.}
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t5}.
\end{proof}

It follows

\begin{corollary}
\label{c17} (to Theorem \ref{t16}, case of $p=q=2$). We have that
\begin{equation}
\left\vert \frac{1}{b-a}\int_{\gamma _{u,w}}f\left( z\right) dz-f\left(
u\right) z^{\prime }\left( a\right) \right\vert \leq \frac{\left\Vert
D_{\ast a}^{\nu }\left( f\left( z\left( t\right) \right) z^{\prime }\left(
t\right) \right) \right\Vert _{L_{2}\left( \left[ a,b\right] ,\mathbb{C}%
\right) }}{\Gamma \left( \nu \right) \sqrt{2\nu -1}\left( \nu +\frac{1}{2}%
\right) }\left( b-a\right) ^{\nu -\frac{1}{2}}.  \tag{28}  \label{28.}
\end{equation}
\end{corollary}

We continue with an $L_{1}$ fractional $\mathbb{C}$-Ostrowski type
inequality:

\begin{theorem}
\label{t18} Let $\nu \geq 1$, $n=\left\lceil \nu \right\rceil $. Assume that
$h\in C^{n}\left( \left[ a,b\right] ,\mathbb{C}\right) $, where $h\left(
t\right) :=f\left( z\left( t\right) \right) z^{\prime }\left( t\right) $,
and such that $h^{\left( i\right) }\left( a\right) =0$, $i=1,...,n-1$. Then
\begin{equation}
\left\vert \frac{1}{b-a}\int_{\gamma _{u,w}}f\left( z\right) dz-f\left(
u\right) z^{\prime }\left( a\right) \right\vert \leq \frac{\left\Vert
D_{\ast a}^{\nu }\left( f\left( z\left( t\right) \right) z^{\prime }\left(
t\right) \right) \right\Vert _{L_{1}\left( \left[ a,b\right] ,\mathbb{C}%
\right) }}{\Gamma \left( \nu +1\right) }\left( b-a\right) ^{\nu -1}.
\tag{29}  \label{29.}
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t7}.
\end{proof}

It follows a Poincar\'{e} like $\mathbb{C}$-fractional inequality:

\begin{theorem}
\label{t19} Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu >\frac{1}{q}$, $%
n=\left\lceil \nu \right\rceil $. Let $h\in C^{n}\left( \left[ a,b\right] ,%
\mathbb{C}\right) $. Assume that $h^{\left( i\right) }\left( a\right) =0$, $%
i=1,...,n-1$. Then
\begin{equation}
\left\Vert h\right\Vert _{L_{q}\left( \left[ a,b\right] ,\mathbb{C}\right)
}\leq \frac{\left( b-a\right) ^{\nu }\left\Vert D_{\ast a}^{\nu
}h\right\Vert _{L_{q}\left( \left[ a,b\right] ,\mathbb{C}\right) }}{\Gamma
\left( \nu \right) \left( p\left( \nu -1\right) +1\right) ^{\frac{1}{p}%
}\left( q\nu \right) ^{\frac{1}{q}}},  \tag{30}  \label{30.}
\end{equation}%
in particular when $h\left( t\right) :=f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) $ and $\left( f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) \right) ^{\left( i\right) }|_{t=a}=0$, $i=1,...n-1
$, we get:
\begin{equation*}
\left\Vert f\left( z\left( t\right) \right) z^{\prime }\left( t\right)
\right\Vert _{L_{q}\left( \left[ a,b\right] ,\mathbb{C}\right) }\leq
\end{equation*}%
\begin{equation}
\frac{\left( b-a\right) ^{\nu }}{\Gamma \left( \nu \right) \left( p\left(
\nu -1\right) +1\right) ^{\frac{1}{p}}\left( q\nu \right) ^{\frac{1}{q}}}%
\left\Vert D_{\ast a}^{\nu }\left( f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) \right) \right\Vert _{L_{q}\left( \left[ a,b%
\right] ,\mathbb{C}\right) }.  \tag{31}  \label{31.}
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t8}.
\end{proof}

The corresponding Sobolev like inequality follows:

\begin{theorem}
\label{t20} All as in Theorem \ref{t19}. Let $r>0$. Then
\begin{equation*}
\left\Vert f\left( z\left( t\right) \right) z^{\prime }\left( t\right)
\right\Vert _{L_{r}\left( \left[ a,b\right] ,\mathbb{C}\right) }\leq
\end{equation*}%
\begin{equation}
\frac{\left( b-a\right) ^{\nu -\frac{1}{q}+\frac{1}{r}}}{\Gamma \left( \nu
\right) \left( p\left( \nu -1\right) +1\right) ^{\frac{1}{p}}\left( r\left(
\nu -\frac{1}{q}\right) +1\right) ^{\frac{1}{r}}}\left\Vert D_{\ast a}^{\nu
}\left( f\left( z\left( t\right) \right) z^{\prime }\left( t\right) \right)
\right\Vert _{L_{q}\left( \left[ a,b\right] ,\mathbb{C}\right) }.  \tag{32}
\label{32.}
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t9}.
\end{proof}

We continue with an Opial type $\mathbb{C}$-fractional inequality

\begin{theorem}
\label{t21} Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu >\frac{1}{q}$, $%
n:=\left\lceil \nu \right\rceil $, $h\in C^{n}\left( \left[ a,b\right] ,%
\mathbb{C}\right) $. Assume $h^{\left( k\right) }\left( a\right) =0$, $%
k=0,1,...,n-1$. Then
\begin{equation*}
\int_{a}^{x}\left\vert h\left( t\right) \right\vert \left\vert \left(
D_{\ast a}^{\nu }h\right) \left( t\right) \right\vert dt\leq
\end{equation*}%
\begin{equation}
\frac{\left( x-a\right) ^{\nu -1+\frac{2}{p}}}{2^{\frac{1}{q}}\Gamma \left(
\nu \right) \left( \left( p\left( \nu -1\right) +1\right) \left( p\left( \nu
-1\right) +2\right) \right) ^{\frac{1}{p}}}\left( \int_{a}^{x}\left\vert
\left( D_{\ast a}^{\nu }h\right) \left( t\right) \right\vert ^{q}dt\right) ^{%
\frac{2}{q}},  \tag{33}  \label{33.}
\end{equation}%
$\forall $ $x\in \left[ a,b\right] ,$

in particular when $h\left( t\right) :=f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) $ and $\left( f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) \right) ^{\left( i\right) }|_{t=a}=0$, $i=1,...n-1
$, we get:
\begin{equation*}
\int_{a}^{x}\left\vert f\left( z\left( t\right) \right) \right\vert
\left\vert \left( D_{\ast a}^{\nu }\left( f\left( z\left( t\right) \right)
z^{\prime }\left( t\right) \right) \right) \right\vert \left\vert z^{\prime
}\left( t\right) \right\vert dt\leq
\end{equation*}%
\begin{equation}
\frac{\left( x-a\right) ^{\nu -1+\frac{2}{p}}}{2^{\frac{1}{q}}\Gamma \left(
\nu \right) \left( \left( p\left( \nu -1\right) +1\right) \left( p\left( \nu
-1\right) +2\right) \right) ^{\frac{1}{p}}}\left( \int_{a}^{x}\left\vert
D_{\ast a}^{\nu }\left( f\left( z\left( t\right) \right) z^{\prime }\left(
t\right) \right) \right\vert ^{q}dt\right) ^{\frac{2}{q}},  \tag{34}
\label{34.}
\end{equation}%
$\forall $ $x\in \left[ a,b\right] .$
\end{theorem}

\begin{proof}
By Theorem \ref{t10}.
\end{proof}

We finish with Hilbert-Pachpatte left $\mathbb{C}$-fractional inequalities:

\begin{theorem}
\label{t22} Let $p,q>1:\frac{1}{p}+\frac{1}{q}=1$, and $\nu _{1}>\frac{1}{q}$%
, $\nu _{2}>\frac{1}{p}$, $n_{i}:=\left\lceil \nu _{i}\right\rceil $, $i=1,2$%
. Let $h_{i}\in C^{n_{i}}\left( \left[ a_{i},b_{i}\right] ,\mathbb{C}\right)
$, $i=1,2$. Assume $h_{i}^{\left( k_{i}\right) }\left( a_{i}\right) =0$, \ $%
k_{i}=0,1,...,n_{i}-1;$ $i=1,2.$ Then
\begin{equation*}
\int_{a_{1}}^{b_{1}}\int_{a_{2}}^{b_{2}}\frac{\left\vert h_{1}\left(
t_{1}\right) \right\vert \left\vert h_{2}\left( t_{2}\right) \right\vert
dt_{1}dt_{2}}{\left( \frac{\left( t_{1}-a_{1}\right) ^{p\left( \nu
_{1}-1\right) +1}}{p\left( p\left( \nu _{1}-1\right) +1\right) }+\frac{%
\left( t_{2}-a_{2}\right) ^{q\left( \nu _{2}-1\right) +1}}{q\left( q\left(
\nu _{2}-1\right) +1\right) }\right) }\leq
\end{equation*}%
\begin{equation}
\frac{\left( b_{1}-a_{1}\right) \left( b_{2}-a_{2}\right) }{\Gamma \left(
\nu _{1}\right) \Gamma \left( \nu _{2}\right) }\left\Vert D_{\ast
a_{1}}^{\nu _{1}}h_{1}\right\Vert _{L_{q}\left( \left[ a_{1},b_{1}\right] ,%
\mathbb{C}\right) }\left\Vert D_{\ast a_{2}}^{\nu _{2}}h_{2}\right\Vert
_{L_{p}\left( \left[ a_{2},b_{2}\right] ,\mathbb{C}\right) },  \tag{35}
\label{35.}
\end{equation}%
in particular when $h_{1}\left( t_{1}\right) :=f_{1}\left( z_{1}\left(
t_{1}\right) \right) z_{1}^{\prime }\left( t_{1}\right) $ and $h_{2}\left(
t_{2}\right) :=f_{2}\left( z_{2}\left( t_{2}\right) \right) z_{2}^{\prime
}\left( t_{2}\right) $, with $h_{i}^{\left( k_{i}\right) }\left(
a_{i}\right) =0$, $k_{i}=0,1,...n_{i}-1$; $i=1,2,$ we get:
\begin{equation*}
\int_{a_{1}}^{b_{1}}\int_{a_{2}}^{b_{2}}\frac{\left\vert f_{1}\left(
z_{1}\left( t_{1}\right) \right) z_{1}^{\prime }\left( t_{1}\right)
\right\vert \left\vert f_{2}\left( z_{2}\left( t_{2}\right) \right)
z_{2}^{\prime }\left( t_{2}\right) \right\vert dt_{1}dt_{2}}{\left( \frac{%
\left( t_{1}-a_{1}\right) ^{p\left( \nu _{1}-1\right) +1}}{p\left( p\left(
\nu _{1}-1\right) +1\right) }+\frac{\left( t_{2}-a_{2}\right) ^{q\left( \nu
_{2}-1\right) +1}}{q\left( q\left( \nu _{2}-1\right) +1\right) }\right) }%
\leq \frac{\left( b_{1}-a_{1}\right) \left( b_{2}-a_{2}\right) }{\Gamma
\left( \nu _{1}\right) \Gamma \left( \nu _{2}\right) }\cdot
\end{equation*}%
\begin{equation}
\left\Vert D_{\ast a_{1}}^{\nu _{1}}\left( f_{1}\left( z_{1}\left(
t_{1}\right) \right) z_{1}^{\prime }\left( t_{1}\right) \right) \right\Vert
_{L_{q}\left( \left[ a_{1},b_{1}\right] ,\mathbb{C}\right) }\left\Vert
D_{\ast a_{2}}^{\nu _{2}}\left( f_{2}\left( z_{2}\left( t_{2}\right) \right)
z_{2}^{\prime }\left( t_{2}\right) \right) \right\Vert _{L_{p}\left( \left[
a_{2},b_{2}\right] ,\mathbb{C}\right) }.  \tag{36}  \label{36}
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t11}.
\end{proof}

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\end{document}