\documentclass[10pt]{studiamnew}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\sloppy

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}

\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{notation}[theorem]{Notation}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\numberwithin{equation}{section}

\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\newcommand{\e}{\varepsilon}
\renewcommand{\d}{\delta}
\newcommand{\D}{\Delta}
\newcommand{\g}{\gamma}
\newcommand{\G}{\Gamma}
\renewcommand{\th}{\theta}
\renewcommand{\l}{\lambda}
\renewcommand{\L}{\Lambda}
\renewcommand{\O}{\Omega}
\newcommand{\s}{\sigma}
\begin{document}
%
\setcounter{page}{1}
\setcounter{firstpage}{1}
\setcounter{lastpage}{4}
\renewcommand{\currentvolume}{??}
\renewcommand{\currentyear}{??}
\renewcommand{\currentissue}{??}
%
\title{On a generalization of the Wirtinger inequality and some its applications}
\author{Latifa Agamalieva}
\address{Azerbaijan University,  \\
J. Hajibeyli,
71, \\
AZ1007, Baku, Azerbaijan,}
\address{Baku State University, \\
Institute for Physical Problems,\\
Z.Khalilov, 23,\\
AZ1148, Baku, Azerbaijan}
 \email{latifa.agamalieva@gmail.com}
%
\author{Yusif S.
Gasimov}
\address{Azerbaijan University,  \\
J. Hajibeyli,
71, \\
AZ1007, Baku, Azerbaijan,}
\address{Baku State University, \\
Institute for Physical Problems,\\
Z.Khalilov, 23,\\
AZ1148, Baku, Azerbaijan}
\address{Institute of
Mathematics and Mechanics, ANAS\\
B.Vahabzade, 9, \\
AZ1148, Baku, Azerbaijan} \email{gasimov.yusif@gmail.com}
%
\author{Juan E. N\'apoles-Valdes}
\address{UNNE, FaCENA,\\ Ave. Libertad 5450, Corrientes \\
3400, Argentina}
\address{UTN-FRRE,\\
French 414, Resistencia\\
Chaco 3500, Argentina}
 \email{jnapoles@exa.unne.edu.ar}
%
\subjclass{26A33, 26Dxx, 35A23} \keywords{integral operator,
Fractional calculus, Wirtinger inequality}
\begin{abstract}
In this paper, we present generalized versions of the Wirtinger
inequality, which contains as particular cases many of the
well-known versions of this classic isoperimetric inequality. Some
applications and open problems are also presented in the work.
\end{abstract}
\maketitle

\section{Introduction}
It is known that Fractional Calculus has a history practically
similar to that of Ordinary Calculus, however only in the last 40
years has it become one of the most dynamic areas of Mathematics.
Not only the development of the "classic" (global to be more
precise) Fractional Calculus has contributed to this, but since the
1960s generalized differential operators, called local fractional
ones, began to appear, which have shown their usefulness in
different problems of application. However, until 2014 (see
\cite{KHYS}) it is that a formalization of these operators is not
achieved with the appearance of what is called Conformable
Derivative, on the other hand, in 2018, a local derivative of a new
type is presented, called Non conformable [13, 34], which comes to
consolidate this area as one in constant development.

As we said, between the theoretical development and the multiplicity
of applications, a multitude of operators, fractional and
generalized, have appeared, making it practically impossible to
follow these new operators. In \cite {BF} suggests and justifies the
idea of a fairly complete classification of the known operators of
the Fractional Calculus (global or local), on the other hand, in the
work \cite {B} some reasons are presented why new operators linked
to applications and developments theorists appear every day. These
operators had been developed by numerous mathematicians with a
barely specific formulation, for instance, the Riemann-Liouville
(RL), the Weyl, Erdelyi-Kober, Hadamard integrals, and the Liouville
and Katugampola fractional operators and many authors have
introduced new fractional operators generated from general classical
local derivatives.

In addition, Chapter 1 of \cite {Atan} presents a history of
differential operators, both local and global, from Newton to Caputo
and presents a definition of local derivative with new parameter,
providing a large number of applications, with a difference
qualitative between both types of operators, local and global. Most
importantly, Section 1.4 LIMITATIONS AND STRENGTH ..... concludes
``We can therefore conclude that both the Riemann – Liouville and
Caputo operators are not derivatives, and then they are not
fractional derivatives, but fractional operators. We agree with the
result \cite {US} that, the local fractional operator is not a
fractional derivative'' (p.24). As we said before, they are new
tools that have proved their usefulness and potential in the
modeling of different processes and phenomena.


Wirtinger's inequality for real functions was an inequality used in
Fourier analysis. It was named after Wilhelm Wirtinger. It was used
in 1904 to prove the isoperimetric inequality, one of the versions
of this inequality is the following

 \begin{equation*}
 { \pi  }^{ 2 }\int _{ 0 }^{ b }{ { \left| f(x) \right|  }^{ 2 }dx } \le { b }^{ 2 }\int _{ 0 }^{ b }{ { \left| f'(x) \right|  }^{ 2 }dx
 },
 \end{equation*}

 whenever $f$ is a $C^{1}$ function such that $f(0)=f(b)=0$. In this form, Wirtinger's inequality is seen as the one-dimensional version of Friedrichs' inequality. If in the proof of the previous result, the well-known Schwarz Inequality is used, it is reduced to

\begin{equation} \label{e:cd}
\int _{ 0 }^{ b }{ { \left| f(x) \right|  }^{ 2 }dx } \le { b }^{ 2
}\int _{ 0 }^{ b }{ { \left| f'(x) \right|  }^{ 2 }dx },
\end{equation}
where condition $y(b) = 0$ is not needed. It is worth noting that
this inequality is relevant because it gives an estimate of the
function f through its derivative.

An interesting survey on Wirtinger's and related inequalities can be
found in a recently published monograph \cite{MPF}, which represents
numerous extensions, refinements, variants, discrete analogues and
applications of \eqref{e:cd}, and provides more than 200 references
on this subject. Some interesting applications of this inequality
can be found in [4, 7, 14, 23, 24, 27, 35, 36, 39].

The word calculus comes from the Latin calculus, which means having
stones. The Integral Calculus is a branch of mathematics with so
many ramifications and applications, that the sole intention of
enumerating them makes the task practically impossible. It was used
initially by, Aristoteles, Descartes, Newton and Barrow with the
contributions of Newton, if we refer only to the case of integral
inequalities present in the alliterature, there are different types
of these, which involve certain properties of the functions
involved, from generalizations of the known Mean Value Theorem of
classical Integral Calculus, to varied inequalities in norm, going
through the inequality of Wirtinger presented above.

In this article, a Wirtinger-type inequality is studied, in the
context of the generalized derivative that we will define in the
following section, some remarks will be presented that will show the
strength of our results, having as particular cases, several of
those reported in the literature. In particular we will deal with
real integral operators defined on $\mathbb{R}$.


\section{A general integral operator}

We assume that the reader is familiar with the classic definition of
the Riemann Integral, so we will not present it. In \cite{NGL2} was
presented an integral operator generalized, whcih contain as
particular cases, many of the well-known integral operators, both
integer order and not. First we will present the definition of
generalized derivative (see \cite{NGLK}) which was defined in the
following way.

\begin{definition} \label{d:1}
Given a function $f:[0,+\infty )\rightarrow \mathbb{R}$. Then the
N-derivative of $f$ of order $\alpha $\ is defined by

\begin{equation}\label{e:d}
N_{F}^{\alpha }f(x)=\underset{\varepsilon \rightarrow 0}{\lim
}\frac{f(x+\varepsilon F(x,{\alpha }))-f(x)}{\varepsilon }
\end{equation}
for all $x>0$, $\alpha\in (0,1)$ being $F(\alpha,t)$ is some
function. Here we will use some cases of $F$ defined in function of
$E_{a,b}(.)$ the classic definition of Mittag-Leffler function with
$Re(a),Re(b)>0$. Also we consider $E_{a,b}(x^{-\alpha })_{k}$ is the
k-nth term of $E_{a,b}(.)$.

If $f$ is $\alpha -$differentiable in some $(0,\alpha )$, and
$\underset{t\rightarrow 0^{+}}{\lim }N_{F}^{\alpha}f(x)$ exists,
then define $N_{F}^{\alpha}f(0)=\underset{t\rightarrow 0^{+}}{\lim }
N_{F}^{\alpha}f(x)$. Note that if $f$ is differentiable, then
$N_{F}^{\alpha}f(x)=F(x,\alpha)f'(x)$, where $f'(x)$ is the ordinary
derivative.
\end{definition}

The function $E_{a,b}(z)$ was defined and studied by Mittag-Leffler
in the year $1903$. It is a direct generalization of the exponential
function. This generalization was studied by Wiman in $1905$,
Agarwal in $1953$ and Humbert and Agarwal in $1953$, and others. In
this address the reader can check [3, 8, 9, 10, 12, 16-18, 20, 21]
where several fractional calculus operators have been introduced and
investigated

\bigskip

We consider the following examples:

I) $F(x,\alpha) \equiv 1$, in this case we have the ordinary
derivative.

II) $F(x,\alpha )={ E }_{ 1,1 }({ x }^{ -\alpha  })$. In this case
we obtain, from Definition \ref{d:1}, the non conformable derivative
${ N }_{ 1 }^{ \alpha  }f(x)$ defined in \cite{GLLMN} (see also
\cite{NGL1}).

III) $F(x,\alpha)={ E }_{ 1,1 }((1-\alpha )x)={ e }^{ (1-\alpha)x
}$, this kernel satisfies that $F(x,\alpha)\rightarrow 1$ as $\alpha
\rightarrow 1$, a conformable derivative used in \cite{FMNS}.

IV)$F(x,\alpha)={ { E }_{ 1,1 }({ x }^{ 1-\alpha  }) }_{1}={ x
}^{1-\alpha}$ with this kernel we have  $F(x,\alpha)\rightarrow 0$
as $\alpha \rightarrow 1$ (see \cite{KHYS}), a conformable
derivative.

V)$F(x,\alpha)={ { E }_{ 1,1 }({ x }^{ -\alpha  }) }_{1}={ x
}^{\alpha}$ with this kernel we have  $F(x,\alpha)\rightarrow x$ as
$\alpha \rightarrow 1$ (see \cite{NRS}). It is clear that since it
is a non-conformable derivative, the results will differ from those
obtained previously, which enhances the study of these cases.

VI)  $F(x,\alpha)={ { E }_{ 1,1 }({ x }^{ -\alpha  }) }_{1}={ x
}^{-\alpha}$ with this kernel we have  $F(x,\alpha)\rightarrow
x^{-1}$ as $\alpha \rightarrow 1$ This is the derivative ${ N }_{ 3
}^{ \alpha  }$ studied in \cite{MMN}. As in the previous case, the
results obtained have not been reported in the literature.

Now, we give the definition of a general fractional integral.
Throughout the work we will consider that the integral operator
kernel $T$ defined below is an absolutely continuous function.

Let $I$ be an interval $I \subseteq \mathbb{R}$, $a,x \in I$ and
$\alpha \in \mathbb{R}$. The integral operator $J_{T,a}^{\alpha}$,
right and left, is defined for every locally integrable function $f$
on $I$ as

\begin{equation}\label {e:oig+}
J_{T,a+}^{\alpha}(f)(x)=\int _{ a }^{ x } \frac { f(s) }{ T(s,\alpha
) } ds, x>a,
\end{equation}

\begin{equation}\label {e:oig-}
J_{T,b-}^{\alpha}(f)(x)=\int _{ x }^{ b } \frac { f(s) }{ T(s,\alpha
) } ds, b<x.
\end{equation}

Sometimes the kernel of the integral operator is exactly the kernel
of the generalized derivative.

\begin{remark}
It is easy to see that the case of the $J_T^\alpha$ operator defined
above contains, as particular cases, the integral operators obtained
from conformable and non-conformable local derivatives. However, we
will see that it goes much further by containing the cases listed at
the beginning of the work. So, we have
\newline
1) If $F(x,\alpha )={ x }^{ 1-\alpha  },\quad T(x,\alpha )=\Gamma
(\alpha )F(x-x,\alpha )$, from \eqref{e:oig+} we have the right side
Riemann-Liouville fractional integrals $(R_{a+}^{\alpha}f)(x)$,
similarly from \eqref{e:oig-} we obtain the left derivative of
Riemann-Liouville. Then its corresponding right differential
operator is
\begin{equation*}
{ { (_{  }^{ RL }{ { D_{ a^{ + } }^{ \alpha  } } } } }f)(x)=\frac {
d }{ dx } (R_{ a+ }^{ 1-\alpha  }f)(x),
\end{equation*}
analogously we obtain the left. \newline 2) With $F(x,\alpha )={ x
}^{ 1-\alpha  },\quad T(x-x,\alpha )=\Gamma (\alpha
)F(lnt-lnx,\alpha )t$, we obtain the right Hadamard integral from
\eqref{e:oig+}, the left Hadamard integral is obtained similarly
from \eqref{e:oig-}. The right derivative is
\begin{equation*}
{ { (_{  }^{ H }{ { D_{ a^{ + } }^{ \alpha  } } } } }f)(x)=x\frac {
d }{ dx } (H_{ a+ }^{ 1-\alpha  }f)(x),
\end{equation*}
in a similar way we can obtain the left. \newline 3) The right
Katugampola integral is obtained from \eqref{e:oig+} making
\begin{equation*}
F(x,\alpha )={ x }^{ 1-\alpha  },\quad e(x)={ x }^{ \varrho  },\quad
T(x,\alpha )=\frac { \Gamma (\alpha ) }{ F(\rho ,\alpha ) } \frac {
F(e(x)-e(x),\alpha ) }{ e'(x) },
\end{equation*}
analogously for the integral left fractional. In this case, the
right derivative is
\begin{equation*}
{ (_{  }^{ K }{ { D_{ a^{ + } }^{ \alpha ,\rho  } } } }f)(x)={ x }^{
1-\rho  }\frac { d }{ dx } { { K }_{ { a }^{ + } }^{ 1-\alpha ,\rho
} }f(x)=F(x,\rho )\frac { d }{ dx } { { K }_{ { a }^{ + } }^{
1-\alpha ,\rho  } }f(x),
\end{equation*}
and we can obtain the left derivative in the same way. \newline 4)
The solution of equation ${ (-\Delta ) }^{ -\frac { \alpha  }{ 2 }
}\phi (u)=-f(u)$ called Riesz potential, is given by the expression
$\phi ={ C }_{ n }^{ \alpha  }\int _{ { R }^{ n } }^{  }{ \frac {
f(v) }{ { \left| u-v \right|  }^{ n-\alpha  } }  } dv$, where ${ C
}_{ n }^{ \alpha  }$ is a constant (see \cite{C,H,MSP}). Obviously,
this solution can be expressed in terms of the operator \eqref
{e:oig+} very easily.\newline 5) Obviously, we can define the
lateral derivative operators (right and left) in the case of our
generalized derivative, for this it is sufficient to consider them
from the corresponding integral operator. To do this, just make use
of the fact that if $f$ is differentiable, then
$N_{F}^{\alpha}f(x)=F(x,\alpha)f'(x)$ where $f'(x)$ is the ordinary
derivative. For the right derivative we have $\left( { N }_{ F,a+
}^{ \alpha  }f \right) (x)={ N }_{ F }^{ \alpha  }\left[ { J }_{
T,a+ }^{ \alpha  }(f)(x) \right] =\frac { d }{ dx } \left[ { J }_{
T,a+ }^{ \alpha  }(f)(x) \right] F(x,\alpha )$, similarly to the
left. \newline 6) It is clear then, that from our definition, new
extensions and generalizations of known integral operators can be
defined. \newline 7) We can define the function space $L_{\alpha}^{
p}[a,b]$ as the set of functions over $[a,b]$ such that
$({J}_{F,a+}^{\alpha}{[f(x)]^p}(b))<+\infty$.
\end{remark}

The following results are generalizations of the known results of
the integer order Calculus.

\begin{proposition} \label{p:fundamental}
Let $I$ be an interval $I \subseteq \mathbb{R}$, $a \in I$,
$0<\alpha \le 1$ and $f$ a $\alpha$-differentiable function on $I$
such that $f'$ is a locally integrable function on $I$. Then, we
have for all $x \in I$
$$
J_{F,a+}^{\alpha} \big({ N }_{ F }^{ \alpha  }(f)\big)(x)=f(x)-f(a).
$$
\end{proposition}

\begin{proposition} \label{p:inverse}
Let $I$ be an interval $I \subseteq \mathbb{R}$, $a \in I$ and
$\alpha \in (0,1]$.
$$
N_F^{\alpha} \big( J_{F,a+}^{\alpha}(f)\big) (x)= f(x),
$$
for every continuous function $f$ on $I$ and $a,t\in I$.
\end{proposition}

In \cite{KHYS} it is defined the integral operator
$J_{F,a}^{\alpha}$ for the choice of the function $F$ given by
$F(x,{\alpha})= x^{1-{\alpha}}$, and \cite[Theorem 3.1]{KHYS} shows
$$
N^{\alpha} J_{x^{1-{\alpha}}\!,\,a}^\a(f)(x)= f(x),
$$
for every continuous function $f$ on $I$, $a,x \in I$ and $\a \in
(0,1]$. Hence, Proposition \ref{p:inverse} extends to any $F$ this
important equality.

\begin{theorem} \label{t:pos}
Let $I$ be an interval $I \subseteq \RR$, $a,b \in I$ and $\a \in
\RR$. Suppose that $f,g$ are locally integrable functions on $I$,
and $k_1,k_2\in \RR$. Then we have

$(1)$ $J_{T,a}^\a \big( k_1 f+k_2 g \big) (x) =k_1 J_{T,a}^\a f(x) +
k_2 J_{T,a}^\a g(x),$

$(2)$ if $f \ge g$, then $J_{T,a}^\a f(x)\ge J_{T,a}^\a g(x)$ for
every $t\in I$ with $t \ge a$,

$(3)$ $\left| J_{T,a}^\a f(x) \right| \le J_{T,a}^\a \left| f
\right| (x)$ for every $t\in I$ with $t \ge a$,

$(4)$ $\int _{ a }^{ b } \frac { f(s) }{ T(s,\alpha ) } ds=J_{ T,a
}^{ \alpha  }f(x)-J_{ T,b }^{ \alpha  }f(x)=J_{ T,a }^{ \alpha
}f(x)(b)$ for every $t \in I$.
\end{theorem}

Let $C^{1}[a,b]$ be the set of functions f with first ordinary
derivative continuous on $[a,b]$, we consider the following norms on
$C^{1}[a,b]$:

\begin{equation*}
{ \left\| F \right\|  }_{ C }=\max _{ [a,b] }{ \left| f(x) \right| }
,\quad { \left\| F \right\|  }_{ { C }^{ 1 } }=\left\{ \max _{ [a,b]
}{ \left| f(x) \right|  } +\max _{ [a,b] }{ \left| f'(x) \right|  }
\right\}.
\end{equation*}

\begin{theorem}
The fractional derivatives ${ N }_{ F,a+ }^{ \alpha  }f(x)$ and $ {
N }_{ F,b- }^{ \alpha  }f(x)$  are bounded operators from
$C^{1}[a,b]$ to $C[a,b]$ with

\begin{equation}
\left| { N }_{ F,a+ }^{ \alpha  }f(x) \right| \le K{ \left\| F
\right\|  }_{ C }{ \left\| f \right\|  }_{ { C }^{ 1 } },
\end{equation}

\begin{equation}
\left| { N }_{ F,b- }^{ \alpha  }f(x) \right| \le K{ \left\| F
\right\|  }_{ C }{ \left\| f \right\|  }_{ { C }^{ 1 } },
\end{equation}
where the constant K, may be depend of derivative frame considered.
\end{theorem}

\begin{remark}
From previous results we obtain that the derivatives ${ N }_{ F,a+
}^{ \alpha  }f(x)$ and $ { N }_{ F,b- }^{ \alpha  }f(x)$ are well
defined.
\end{remark}

\begin{theorem} (Integration by parts) \label {t:parts}
Let $f,g:[a,b] \rightarrow \mathbb{R}$ be differentiable functions
and $\alpha \in (0,1]$. Then, the following property hold

\begin{equation}
J_{ F,a+ }^{ \alpha  }((f)({ N }_{ F,a+ }^{ \alpha  }g(x)))={ \left[
f(x)g(x) \right]  }_{ a }^{ b }-J_{ F,a+ }^{ \alpha  }((g)({ N }_{
F,a+ }^{ \alpha  }f(x))).
\end{equation}
\end{theorem}

\section{The generalized Wirtinger inequality}

In this section, we will state Wirtinger type inequalities using
generalized integral operator defined in the previous section.

First, we will give a generalized version of the inequality \eqref
{e:cd}.

\begin{theorem} \label{t:1}
For any function f, N-differentiable on $[a,b]$ with $a<b$ such that
$f(a)=0$, $\alpha \in (0,1]$, we have

\begin{equation} \label{e:gw1}
{ { J }_{ F,a+ }^{ \alpha  }(f^{ 2 }(x))(b) }\le { { \left[
\mathcal{ F }(b) \right]  }^{ 2 } }\quad  { { J }_{ F,a+ }^{ \alpha
}\left( { N }_{ F }^{ \alpha  }f(x) \right) (b) }
\end{equation}
with $\mathcal{ F }(b)={ { J }_{ F,a+ }^{ \alpha  }\left( 1 \right)
(b) }$ and $F(x,\alpha)>0$.
\end{theorem}
\begin{proof}
From Proposition \ref{p:fundamental} we have $J_{F,a+}^\a \big({ N
}_{ F }^{ \alpha  }(f)\big)(x)=f(x)-f(a)$, and using the fact that
$f(a)=0$ we have $f(x)=J_{F,a+}^\a \big({ N }_{ F }^{ \alpha
}(f)\big)(x)$ and so, from Property (3) of Theorem \ref{t:pos}

\begin{equation}\label{e:4}
\left|f(x) \right|=\left| J_{F,a+}^\a { N }_{ F }^{ \alpha  }(f)
(x)\right| \le J_{F,a+}^\a \left| { N }_{ F }^{ \alpha  }(f) \right|
(x)
\end{equation}
for every $x\in [a,b]$. Applying Schwarz inequality to the right
side of \eqref{e:4} we obtain

\begin{align*}
&\left| f(x) \right| \le { \left( J_{ F,a+ }^{ \alpha  }(1)(x)
\right)  }^{ \frac { 1 }{ 2 }  }{ \left( J_{ F,a+ }^{ \alpha  }{
\left( { N }_{ F }^{ \alpha  }(f) \right)  }^{ 2 }(x) \right)  }^{
\frac { 1 }{ 2 }  }
\\
&\le { \left( \mathcal{ F }(x) \right)  }^{ \frac { 1 }{ 2 }  }{
\left( J_{ F,a+ }^{ \alpha  }{ \left( { N }_{ F }^{ \alpha  }(f)
\right)  }^{ 2 }(x) \right)  }^{ \frac { 1 }{ 2 }  }
\\
&\le { \left( \mathcal{ F }(b) \right)  }^{ \frac { 1 }{ 2 }  }{
\left( J_{ F,a+ }^{ \alpha  }{ \left( { N }_{ F }^{ \alpha  }(f)
\right)  }^{ 2 }(b) \right)  }^{ \frac { 1 }{ 2 }  }.
\end{align*}

From this last inequality, squareing and N-integrating between a and
b, we obtain the desired result.
\end{proof}

\begin{remark}
If in the previous result we have $a \equiv 0$, the kernel $
F(x,\alpha) \equiv 1$, that is, the classic Riemann integral, we get
the inequality \eqref {e:cd}.
\end{remark}

\begin{remark}
Similarly, if we use the kernel $F(x,\alpha)=x^{1-\alpha}$, that is,
in the case of the conformable derivative $T_{\alpha}$ of
\cite{KHYS}, we obtain the inequality of Theorem 3.1 of \cite{A}.
\end{remark}

\begin{remark}
It's easy to get new versions of the classic Wirtinger Inequality
using other kernels, for example, if we take
$F(x,\alpha)=x^{\alpha}$ we get, from \eqref{e:gw1} the following

\begin{equation} \label{e:gw2}
{ { J }_{ F,a+ }^{ \alpha  }f(x)(b) }\le \frac { { ({ b }^{ 1-\alpha
}-a^{ 1-\alpha  }) }^{ 2 } }{ { (1-\alpha ) }^{ 2 } } { { J }_{ F,a+
}^{ \alpha  }\left( { N }_{ F }^{ \alpha  }f(x) \right) (b) }.
\end{equation}
\end{remark}

\section{Some applications}

In \cite{HL} the authors gave a generalized Wirtinger type
inequality using an auxiliary function. Thus, the following result
is a generalization of this result.

\begin{theorem}
For any positive continuous funtion $M(x)$ on $[a,b]$ with ${ N }_{
F }^{ \alpha  }M(x)>0$ or ${ N }_{ F }^{ \alpha  }M(x)<0$ on
$[a,b]$, $\alpha \in (0,1]$, we have

\begin{equation} \label{e:gw3}
{ { J }_{ F,a+ }^{ \alpha  }\left( \left( { N }_{ F }^{ \alpha
}M(x) \right)  \right) { (y }^{ 2 }(x))(b) }\le 4{ { J }_{ F,a+ }^{
\alpha  }\left[ \left( \frac { { M }^{ 2 }(x) }{ { N }_{ F }^{
\alpha  }M(x) }  \right) { \left( { N }_{ F }^{ \alpha  }y(x)
\right)  }^{ 2 } \right] (b) },
\end{equation}
for all continuous function $y(x)$ defined on $[a,b]$ with
$y(a)=y(b)=0$.
\end{theorem}
\begin{proof}
We consider the case ${ N }_{ F }^{ \alpha  }M(x)>0$ then,
N-integrating by parts (see Theorem \ref{t:parts}), we have
\[
{ J }_{ 1 }={ J }_{ F,a+ }^{ \alpha  }\left( \left( { N }_{ F }^{
\alpha  }M(x) \right)  \right) { (y }^{ 2 }(x))(b)=\] \[M(b){ y }^{
2 }(b)-M(a){ y }^{ 2 }(a)-2{ { J }_{ F,a+ }^{ \alpha  }(M(x)){ ({ N
}_{ F }^{ \alpha  }y(x))(y }(x))(b) }.
\]
From here, using the Schwarz inequality we get

\begin{align*}
&{ J }_{ 1 }={ J }_{ F,a+ }^{ \alpha  }\left( \left( { N }_{ F }^{
\alpha  }M(x) \right)  \right) { (y }^{ 2 }(x))(b)=
\\
&-2{ { J }_{ F,a+ }^{ \alpha  }(M(x)){ ({ N }_{ F }^{ \alpha  }y(x))
}y(x))(b) } \le
\\
&2{ { J }_{ F,a+ }^{ \alpha  }\left[ \sqrt { \frac { { (M(x)) }^{ 2
} }{ { N }_{ F }^{ \alpha  }M(x) }  }  \right] { \left| { N }_{ F
}^{ \alpha  }y(x) \right| \sqrt { { N }_{ F }^{ \alpha  }M(x) }
\left| y(x) \right|  }(b) }\le
\\
&2\sqrt { { J }_{ 1 }{ J }_{ 2 } }
\end{align*}
with ${ J }_{ 2 }={ J }_{ F,a+ }^{ \alpha  }\left[ \left( \frac { {
M }^{ 2 }(x) }{ { N }_{ F }^{ \alpha  }M(x) }  \right) { ({ N }_{ F
}^{ \alpha  }y(x)) }^{ 2 } \right]$. From the above inequality we
have then ${ J }_{ 1 }=2\sqrt { { J }_{ 1 }{ J }_{ 2 } }$, where the
desired conclusion is reached.
\end{proof}

\begin{remark}
If in the previous result we have $a \equiv 0$, the kernel $
F(x,\alpha) \equiv 1$, that is, the classic Riemann integral, we get
the Lemma 1 of \cite{HL}.
\end{remark}

\begin{remark}
Similarly, if we use the kernel $F(x,\alpha)=x^{1-\alpha}$, that is,
in the case of the conformable derivative $T_{\alpha}$ of
\cite{KHYS}, we obtain the inequality of Theorem 3.2 of \cite{A}.
\end{remark}

\begin{remark}
Of course we can also generate new generalizations considering other
kernels, a simple matter that we leave to the reader.
\end{remark}

\begin{theorem}
For any function f, N-differentiable on $[a,b]$ with $a<b$ such that
$f(a)=0$, $p \ge 1$, $\alpha \in (0,1]$, we have

\begin{equation} \label{e:gw4}
{ { J }_{ F,a+ }^{ \alpha  }{ \left| f(x) \right|  }^{ p }(b) }\le
\frac { { \left[\mathcal { F }(b) \right]  }^{ p } }{p}\quad { { J
}_{ F,a+ }^{ \alpha  }\left( { \left| { N }_{ F }^{ \alpha  }f(x)
\right|  }^{ p } \right) (b) }
\end{equation}
with $\mathcal{ F }(b)={ { J }_{ F,a+ }^{ \alpha  }\left( 1 \right)
(b) }$ and $F(x,\alpha)>0$.
\end{theorem}
\begin{proof}
As before, from Proposition \ref{p:fundamental} we have $J_{F,a+}^\a
\big({ N }_{ F }^{ \alpha  }(f)\big)(x)=f(x)-f(a)$, and using the
fact that $f(a)=0$ we have $f(x)=J_{F,a+}^\a \big({ N }_{ F }^{
\alpha  }(f)\big)(x)$ and so, from Property (3) of Theorem
\ref{t:pos}

\begin{equation*}
\left|f(x) \right|=\left| J_{F,a+}^\a { N }_{ F }^{ \alpha  }(f)
(x)\right| \le J_{F,a+}^\a \left| { N }_{ F }^{ \alpha  }(f) \right|
(x)
\end{equation*}
for every $x\in [a,b]$. Using the Holder inequality with $p$ and
$\frac{p}{p-1}$ we obtain

\begin{equation*}
\left| f(x) \right| \le { \left( J_{ F,a+ }^{ \alpha  }(1)(x)
\right)  }^{ \frac { p-1 }{ p }  }{ \left( J_{ F,a+ }^{ \alpha  }{
\left( { N }_{ F }^{ \alpha  }(f) \right)  }^{ p }(x) \right)  }^{
\frac { 1 }{ p }  },
\end{equation*}
so we have

\begin{equation*}
{ \left| f(x) \right|  }^{ p }\le { \left( J_{ F,a+ }^{ \alpha
}(1)(x) \right)  }^{ p-1 }{ \left( J_{ F,a+ }^{ \alpha  }{ \left( {
N }_{ F }^{ \alpha  }(f) \right)  }^{ p }(x) \right)  },
\end{equation*}

N-integrating by parts  (see Theorem \ref{t:parts}), the conclusion
of the Theorem is obtained.
\end{proof}

\begin{remark}
If in the previous result we have $a \equiv 0$, the kernel $
F(x,\alpha) \equiv 1$, that is, the classic Riemann integral, we get
a light variant of result of \cite{B}.
\end{remark}

\begin{remark}
Similarly, if we use the kernel $F(x,\alpha)=x^{1-\alpha}$, that is,
in the case of the conformable derivative $T_{\alpha}$ of
\cite{KHYS}, we obtain the inequality of Theorem 6 of \cite{SB} (see
also Theorem 2.2 of \cite{Sa}). Theorem 2.2 is obtained in a
different way, and its conclusion differs from that presented here,
it is necessary that $f(b)=0$, however the interested reader can
obtain it without any difficulty and have, instead of \eqref {e:gw4}
the following expression

\begin{equation*}
{ { J }_{ F,a+ }^{ \alpha  }{ \left| f(x) \right|  }^{ p }(b) }\le
\frac { { \left[ \mathcal{ F }(b) \right]  }^{ p } }{ { 2 }^{ p-1 }p
} \quad { { J }_{ F,a+ }^{ \alpha  }\left( { \left| { N }_{ F }^{
\alpha  }f(x) \right|  }^{ p } \right) (b) }.
\end{equation*}

\end{remark}

\begin{remark}
New inequalities of type Wirtinger, generalizations of
\eqref{e:gw4}, can be obtained using other kernels.
\end{remark}

Following the ideas of Theorem \ref{t:1} we can obtain a weighted
version of Wirtinger's Inequality as follows.

\begin{theorem}
Let f and g N-differentiables functions on $[a,b]$ with
$f(a)=g(a)=0$ and $f,g \in L_{\alpha}^{2}[a,b]$ then, we have the
following inequality

\begin{equation*}
{ J }_{ F,a+ }^{ \alpha  }(\left| f \right| \left| g \right| )(b)\le
\frac { \mathcal{ F }^{ 2 }(b) }{ 2 } { J }_{ F,a+ }^{ \alpha
}\left( { \left( { N }_{ F }^{ \alpha  }f \right)  }^{ 2 }+{ \left(
{ N }_{ F }^{ \alpha  }g \right)  }^{ 2 } \right).
\end{equation*}
\end{theorem}
\begin{proof}
From assumptions and proceeding as in Theorem \ref {t:1} we get
easily

\begin{equation*}
{ J }_{ F,a+ }^{ \alpha  }(\left| f \right| \left| g \right| )(b)\le
{\mathcal F}^{2}(b){ { \left[ { J }_{ F,a+ }^{ \alpha  }\left( {
\left( { N }_{ F }^{ \alpha  }f \right)  }^{ 2 } \right)  \right]
}^{ \frac { 1 }{ 2 }  } }{ { \left[ { J }_{ F,a+ }^{ \alpha  }\left(
{ \left( { N }_{ F }^{ \alpha  }g \right)  }^{ 2 } \right)  \right]
}^{ \frac { 1 }{ 2 }  } }
\end{equation*}

Applying the known inequality $\sqrt { AB } \le \frac { A+B }{ 2 }$
with $A,B>0$, the desired conclusion is obtained.
\end{proof}

\section{Conclusion}
In this paper, some generalized extensions of classical Wirtinger
type inequality are obtained, using less restrictive conditions on
the function $f$, for example, the condition $f(b)=0$ is not used.
In addition, several known results in this topic are obtained as
particular cases of our results, in addition to raising several
possibilities of future work in this address.

Taking into account the ideas of \cite {MN} we can define
generalized partial derivatives as follows.

\begin{definition}\label{d:gpd}
Given a real function $f:\mathbb{R}^{n} \rightarrow
\mathbb{\mathbb{R}}$ and $\overrightarrow { a } =(a_{ 1 },…,a_{ n
})\in {\mathbb{R}}^{n}$ a point whose i-th component is positive.
Then the generalized partial N-derivative of order $\alpha $ of $f$
en el punto $\overrightarrow { a }=(a_{ 1 },…,a_{ n })$  is
defined by
\begin{equation}
N_{F_{i},t_{i}}^{\alpha }f(\overrightarrow { a
})=\underset{\varepsilon \rightarrow 0}{\lim }{ \frac { f(a_{ 1
},..,a_{ i }+\varepsilon F_{i} (a_{ i },\alpha),…,a_{ n })-f(a_{ 1
},…,a_{i}, ...,a_{ n })) }{ \varepsilon  }  }
\end{equation}
if it exists, it is denoted by ${ N }_{ F_{i},t_{ i } }^{ \alpha
}f(\overrightarrow { a })$, and is called the i-th generalized
partial derivative of order $\alpha \in (0,1]$ of $f$ in
$\overrightarrow { a }$.
\end{definition}

\begin{remark}
If a real function $f$ multivariable, has all the partial
derivatives of order $\alpha \in (0,1]$ in $\overrightarrow { a }$,
with $a_{i}>0$, then the generalized $\a$-gradient of $f$ of order
$\a \in (0,1]$ in $\overrightarrow { a }$ is

\begin{equation}
\nabla_{ N }^{ \alpha  }f(\overrightarrow { a })=(N_{ t_{ 1 } }^{
\alpha }f(\overrightarrow { a }),…,N_{ t_{ n } }^{ \alpha
}f(\overrightarrow { a }))
\end{equation}
\end{remark}

On this basis, the results of \cite {RSB} can be generalized to a
generalized context without much difficulty.





\begin{thebibliography}{99}
\bibitem{A}  Asliyüce, S. \emph{Wirtinger type inequalities via fractional integral operators}, Stud. Univ. Babes-Bolyai Math.
\textbf{64}(2019), no. 1, 35-42.

\bibitem{Atan}  Atangana, A. \emph{Derivative with a New Parameter Theory, Methods and Applications}, Academic Press, 2016.

\bibitem{AK} Aziz, R., Kumawat, Y. \emph{Marichev-Saigo-Maeda Fractional Operators with extended Mittag-Leffler function and generalized Galue type
 Struve function}, Adv. Math. Models Appl.,
\textbf{4}(2019), no.3, 210-223.

\bibitem{B}Baleanu, D. \emph{Comments On: Ortigueira M., Martynyuk V., Fedula M., Machado J.A.T., The
failure of certain fractional calculus operators in two physical
models}, in Fract. Calc. Appl. Anal. \textbf{22}(2019), no.2, Fract.
Calc. Appl. Anal., \textbf{23}(2020), no.1.

\bibitem{BF} Baleanu, D., Fernandez, A. \emph{On Fractional Operators and Their Classifications}, Mathematics, \textbf{7}(2019),
830.

\bibitem{B}  Beesack, P.R. \emph{Integral inequalities of the Wirtinger type}, Duke Math. J., \textbf{25}(1958), 477-498.

\bibitem{C}  C{\i}nar, I. \emph{On Some Properties of Generalized Riesz Potentials}, Intern. Math. J., \textbf{3}(2003), no. 12,
1393-1397.

\bibitem{CT}  Capelas de Oliveira, E., Tenreiro Machado, J.A.
 \emph{A Review of Definitions for Fractional Derivatives and Integral}, Math. Probl.
 Eng.,
Volume 2014, Article ID 238459, 6 pages
http://dx.doi.org/10.1155/2014/238459

\bibitem{DM}  Diaz, J.B.,  Metcalf, F.T. \emph{Variations of Wirtinger’s inequality, Inequalities,} Academic Press,
New York, 1967, 79-103.

\bibitem{ES}  Ertugral, F.,  Sarikaya, M.Z. \emph{Some trapezoid type inequalities for generalized fractional integral}, 2018, preprint.

\bibitem{FMNS}  Fleitas, A.,  M\'{e}ndez, J.A.,  N\'{a}poles, J.E., Sigarreta, J.M. \emph{On the some classical systems of Li\'{e}nard in general context},
Rev. Mexicana F\'{i}s., \textbf{65}(2019), no.6, 618-625.

\bibitem{FNRS}  Fleitas, A.,  N\'apoles Vald\'es, J.E.,  Rodr\'{\i}guez, J.M., Sigarreta, J M.  \emph{On some fractional differential equations}. Submitted.

\bibitem{GLLMN}  Guzman, P.M. Langton, G.,  Lugo, L.M.,  Medina J.,  N\'{a}poles Vald\'es, J.E.
\emph{A new definition of a fractional derivative of local type}, J.
Math. Anal. {\bf 9}(2018), no.2, 88--98.

\bibitem{GLN}  Guzman, P.M., Lugo Motta Bittencurt, L.M.,  N\'{a}poles Valdes, J.E. \emph{A Note on Stability of Certain Lienard
Fractional Equation,} Int. J. Math. Comput. Sci., \textbf{14}(2019),
no. 2, 301-315.

\bibitem{NGL2}  Guzman, P.M.,  Lugo, L.M.  N\'{a}poles, J.E.,  Vivas-Cortez, M. \emph{On a New Generalized Integral Operator and Certain
Operating Properties}, Axioms, \textbf{9}(2020), no.2, 69.

\bibitem{GM} Gorenflo, R.,  Mainardi, F. \emph{Fractional Calculus: Integral and Differential Equations of Fractional Order}, 223-276.
Springer, Wien (1997).

\bibitem{HLP}  Hardy, G.H., Littlewood, J.E.,  Polya, G. \emph{Inequalities}, 2nd ed., Cambridge University Press,
London and New York, 1959.

\bibitem{H}  Helms, L.L. \emph{Introduction To Potential Theory}, New York: Wiley-Interscience, 1969.

\bibitem{HL} Hinton, D.B., Lewis, R.T. \emph{Discrete spectra criteria for singular differential operators
with middle terms}, Math. Proc. Cambridge Philos. Soc.,
\textbf{77}(1975), 337-347.

\bibitem{K1}  Katugampola, U.N. \emph{New approach to a generalized fractional integral}, Appl. Math. Comput., \textbf{218}(2011), 860-865.

\bibitem{K2}  Katugampola, U.N. \emph{A new approach to generalized fractional derivatives}, Bull. Math. Anal. App., \textbf{6}(2014), 1-15.

\bibitem{KHYS}  Khalil, R.,  Al Horani, M.,  Yousef, A.,  Sababheh,  M. \emph{A new definition of fractional derivative}. J. Comput. Appl. Math.,
{\bf 264}(2014), 65-70.

\bibitem{MBGY}  Ma, M.,  Baleanu, D.,  Gasimov, Y.S.,  Yang, X.-J. \emph{New results for multidimensional diffusion equations in fractal dimensional space}.
 Rom. J. Phys., \textbf{61}(2016), no.5-6, 784-794.

\bibitem{MKM} Machado, J.T.,  Kiryakova, V.,  Mainardi, F. \emph{Recent history of fractional calculus}. Commun. Nonlinear Sci. Numer. Simul.,
 \textbf{16}(2011), 1140-1153.

\bibitem{MMN}  Mart\'inez, F., Mohammed, P.O., N\'apoles V J.E. \emph{Non Conformable Fractional Laplace Transform}, Kragujevac J.
Math., \textbf{46}(2022), no.3, 341-354.

\bibitem{MN}  Mart\'inez, F.,   N\'apoles V., J.E. \emph{Towards a Non-Conformable Fractional Calculus of N-variables}.
J. Math. Appl., to appear.

\bibitem{MSP}  Martinez, C.,  Sanz, M.,  Periogo, F. \emph{Distributional Fractional Powers of Laplacian, Riesz Potential}. Studia Math., \textbf{135}(1999),
no.3, 253-271.

\bibitem{MPF}  Mitrinović, D.S.,  Pecarić, J.E.,  Fink, A.M. \emph{Inequalities involving functions and their integrals and derivatives}. Dordrecht, 1991.

\bibitem{NGL1}  N\'{a}poles Vald\'es, J.E., Guzman, P.M.,  Lugo, L.M. \emph{Some New Results on Nonconformable Fractional Calculus},
Adv. Dyn. Syst. Appl., \textbf{13}(2018), no.2, 167-175.

\bibitem{NGLK}  N\'{a}poles, J.E.,  Guzman, P.M.,  Lugo, L.M., Kashuri, A. \emph{The local non conformable derivative and Mittag Leffler function},
Sigma J. Eng. Nat. Sci., \textbf{38}(2020), no.2, 1007-1017.

\bibitem{NRS}  N\'apoles Vald\'es, J.E.,  Rodr\'{\i}guez, J.M.,  Sigarreta, J.M. \emph{On Hermite-Hadamard type inequalities for non-conformable integral operators},
 Symmetry,\textbf{ 11}(2019), no.9, 1108.

\bibitem{RSB}  Reis, J. C., Da Silva, P.N.,  Boldrini, J.L. \emph{Uma desigualdade do tipo Wirtinger}, Cadernos do IME - Série Matemática,
\textbf{8}(2014).

\bibitem{Sa}  Sarıkaya, M.Z. \emph{On the New Wirtinger Type Inequalities}, Konuralp J. Math., \textbf{7}(2019), no.1, 112-116.

\bibitem{SB}  Sarikaya, M.Z.,  Bilisik, C.C., \emph{Wirtinger type inequalities for conformable fractional integrals,} preprint.

\bibitem{SE} Sarikaya, M.Z., Ertugral, F., \emph{On the generalized Hermite-Hadamard inequalities,} 2017, submitted.

\bibitem{SY}  Sarikaya, M.Z., Yildirim,H.,  \emph{On generalization of the Riesz potential}, Indian J. Math. Math. Sci., \textbf{3}(2007), no.2, 231-235.

\bibitem{S}  Swanson, C.A., \emph{Wirtinger's Inequality,} SIAM J. Math. Anal.,\textbf{ 9}(1978), no.3, 484-491.

\bibitem {US} Umarov, S.,  Steinberg, S.,  \emph{Variable order differential equations with piecewise constant order-function
and diffusion with changing modes}, Z. Anal. Anwend.,
\textbf{28}(2009), no.4, 431-450.

\bibitem {YGGA} Yang, X.-J., Gasimov, Y.S., Gao,F., Allahverdiyeva, N.A., \emph{Travelling-wave solutions for Klein-Gordon and Helmholtz equations on Cantor sets,}
 \emph{Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb}., 43(2017), no.1, 123-131.


\end{thebibliography}

\end{document}
