aainstitutetext: Saha Institute of Nuclear Physics,
Bidhannagar, Kolkata 700064, Indiabbinstitutetext: Homi Bhabha National
Institute,
Anushakti Nagar, Mumbai 400094, Indiaccinstitutetext: Department of Physics,
Indian Institute of Technology Bombay,
Mumbai 400076, India

# Neutrino Oscillation parameter determination at INO-ICAL using track and hit
information from GEANT

Jaydeep Datta, 111Université libre de Bruxelles, Av. Franklin Roosevelt 50,
1050 Bruxelles, Belgium (Present address) c,2 Bana Singh, 222Indian Institute
of Science Education and Research, Mohali, Punjab 140306, India (Present
address) c S.Uma Sankar<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the capability of INO-ICAL to determine the atmospheric neutrino
oscillation parameters. We do not use any generator level information but
instead use only the output of GEANT4 simulation of the atmospheric neutrino
events in the detector. In a similar previous study, by other authors, only
the momentum and direction of the longest track were used. In this study, we
consider a third variable based on additional hits, which arise due to hadrons
in the event. We show that the inclusion of the third variable leads to a
$30\%$ reduction in the uncertainties of $|\Delta m^{2}_{31}|$ for a five-year
run of ICAL. We find that doubling the exposure time leads to a $30\%$
reduction in the uncertainties of both $|\Delta m^{2}_{31}|$ and
$\sin^{2}\theta_{23}$.

##  1 Introduction

The pioneering water Cerenkov detectors, IMB Casper:1990ac ;
BeckerSzendy:1992hq  and Kamiokande Hirata:1992ku ; Fukuda:1994mc , observed a
deficit in the up-going muon events generated by atmospheric neutrinos
relative to down-going events. It was hypothesized that the deficit was caused
by neutrino oscillations. This hypothesis was confirmed by the zenith angle
dependence of event rates observed by Super Kamiokande Fukuda:1998mi .
Analysis of Super Kamiokande data gave

$\Delta m^{2}_{\rm atm}\approx 3\times 10^{-3}~{}{\rm eV}^{2}~{}~{}~{}{\rm
and}~{}~{}~{}\sin^{2}2\theta_{\rm atm}\approx 1,$ (1)

for atmospheric neutrino oscillation parameters. A deficit in solar neutrinos,
observed earlier, was also explained in terms of neutrino oscillations with

$\Delta m^{2}_{\rm sol}\approx(2-20)\times 10^{-5}~{}{\rm eV}^{2}~{}~{}~{}{\rm
and}~{}~{}~{}\sin^{2}\theta_{\rm sol}\approx 0.3,$ (2)

as the solar neutrino oscillation parameters.

It is possible to account for both solar and atmospheric neutrino oscillations
in a three flavor oscillation framework. The three neutrino flavor states,
$\nu_{\alpha}$ ($\alpha=e,\mu,\tau$), mix to form three mass eigenstates
$\nu_{i}$ ($i=1,2,3$) with masses $m_{i}$. There are two independent mass
squared differences. Without loss of generality, we take them to be $\Delta
m^{2}_{21}=m^{2}_{2}-m_{1}^{2}\equiv\Delta m^{2}_{\rm sol}$ and $\Delta
m^{2}_{31}=m^{3}_{2}-m_{1}^{2}\equiv\Delta m^{2}_{\rm atm}$. Since $\Delta
m^{2}_{\rm sol}\ll\Delta m^{2}_{\rm atm}$, we have $\Delta
m^{2}_{32}\approx\Delta m^{2}_{31}$. The unitary mixing matrix (PMNS matrix),
connecting $\nu_{\alpha}$ to $\nu_{i}$, is parametrized in terms of three
mixing angles ($\theta_{12},\theta_{13}$ and $\theta_{23}$) and one CP
violating phase $\delta_{\rm CP}$. Given that $\theta_{13}\ll 1$ CHOOZ:1997cow
, it was shown that $\theta_{12}\approx\theta_{\rm sol}$ and
$\theta_{23}\approx\theta_{\rm atm}$ Narayan:1999ck . Later experiments with
man-made sources have confirmed the three oscillation picture and have done
precision measurements of the mass squared differences and the mixing angles
Esteban:2020cvm . Present data fixes $\Delta m^{2}_{21}$ to be positive, but
does not fix the sign of $\Delta m^{2}_{31}$. The three unknowns of the three
flavor oscillation paradigm are

  * •

the CP violating phase $\delta_{\rm CP}$,

  * •

the sign of $\Delta_{31}$ (mass hierarchy) and

  * •

the octant of $\theta_{23}$.

Atmospheric and accelerator neutrino data measure $\sin^{2}2\theta_{23}$ to be
very close to one which gives rise to the last problem. A number of
experiments are either underway or proposed to determine these unknowns. Among
them, the Iron Calorimeter (ICAL) at India-based Neutrino Observatory (INO)
has a unique ability to determine the mass hierarchy.

ICAL, a magnetized iron calorimeter, consists of iron slabs of thickness 40 mm
and interspersed with resistive plate chambers. A muon neutrino (anti-
neutrino) interacting in the iron slab through charged current (CC)
interaction creates a negative (positive) muon, which leaves a track in the
detector. The magnetic field of the detector bends these tracks, which leads
to an identification of the charge of the leptons. This ability to measure the
neutrino and anti-neutrino event rates separately is crucial in mass hierarchy
determination. In addition, some other physics goals of ICAL are

  * •

observing the minimum in the $(L/E)$ distribution of the muon event rate
Kumar:2020wgz ,

  * •

measuring the density profile of earth Kumar:2021faw ,

  * •

searching for CPT violation Datta:2003dg  and

  * •

searching for sterile neutrinos Behera:2016kwr .

In addition, ICAL can also improve the precision of atmospheric neutrino
oscillation parameters.

The first studies Thakore:2013xqa  of oscillation parameter precision at ICAL
were done with event samples binned in muon energy and muon direction. The
muon kinematic information used in this analysis was taken from the event
generator and smeared with resolution functions. Later studies Devi:2014yaa
did additional binning in a third variable $E_{\rm had}=E_{\nu}-E_{\mu}$ and
showed that the precision improves significantly. The kinematical variables
used in these analyses will not be available in a real experiment. It is
desirable to do a full GEANT4 simulation GEANT4:2002zbu  of atmospheric
neutrino events in ICAL and process the simulation output through the event
reconstruction code to obtain the kinematical variables to be used in the
analysis. In an earlier work, Rebin et al. Rebin:2018fdl  studied the
oscillation parameter precision using the event distributions binned in
reconstructed track energy and track direction. In their study, they used a
simulated five-year event sample as data. For theoretical prediction, they
simulated a 1000-year event sample and scaled it down by 200. Thus, they have
also taken into account the statistical fluctuations present in the data.
Their results are summarized in table 1.

Parameter | Input Value | Reconstructed | 2 $\sigma$ Range | 3 $\sigma$ Range
---|---|---|---|---
|  | Best Fit point |  |
$|\Delta_{31}|$(10-3 eV2) | 2.32 | 2.32 | 2.03 (1.68 - 3.71) | 4.07 (1.40 - 5.47)
$\sin^{2}\theta_{23}$ | 0.5 | 0.496 | 0.38 (0.34 - 0.72) | 0.48 (0.29 - 0.77)
Table 1:  Results of 2-variable analysis taken from figure 5 of ref.
Rebin:2018fdl . A five-year exposure was assumed.

In this work, we do a similar analysis using only kinematical variables
obtained from GEANT4 simulation of atmospheric neutrino events in ICAL. In
this analysis, we include a third variable, which is a measure of hadronic
energy in an atmospheric neutrino event. We find that the inclusion of this
third variable leads to a noticeable improvement in the precision of the mass
squared difference, but the precision of the mixing angle shows little
improvement. We have also studied how the parameter precision improves with
increased exposure.

##  2 Methodology

###  2.1 Data and Theory sample

The analysis starts with 500 years of un-oscillated events generated by NUANCE
Casper:2002sd , with Kamiokande fluxes as input. From this, a five-year event
sample has been isolated. The oscillation probabilities, $P_{\mu\mu}$ and
$P_{\bar{\mu}\bar{\mu}}$, have been calculated using the three flavor
oscillation code nuCraft Wallraff:2014qka . The following values of neutrino
oscillation parameters have been used as inputs: $\sin^{2}\theta_{12}=0.310$,
$\sin^{2}\theta_{13}=0.02240$, $\sin^{2}\theta_{23}=0.5$,
$\Delta_{31}=2.32\times 10^{-3}$ eV2, $\Delta_{21}=7.39\times 10^{-5}$ eV2 and
$\delta_{CP}=0$. The accept/reject method is applied to
$\nu_{\mu}(\bar{\nu}_{\mu})$ CC events using
$P_{\mu\mu}(P_{\bar{\mu}\bar{\mu}})$ to obtain the muon events due to the
survival of $\nu_{\mu}/\bar{\nu}_{\mu}$. Because $\sin^{2}\theta_{13}\ll 1$,
the muon CC events occurring due to the oscillation of
$\nu_{e}\to\nu_{\mu}(\bar{\nu}_{e}\to\bar{\nu}_{\mu})$ form a small fraction
(about <1%) of the total number. Hence, they are not included in this
analysis. This five-year event sample is labeled data.

To calculate the corresponding theoretical event sample, with test values of
neutrino oscillation parameters, the following method has been followed. A
test value of $|\Delta_{31}|$ from the range $(1,5)\times 10^{-3}$ eV2 and a
test value of the $\sin^{2}\theta_{23}$ in the range $(0,1)$ have been chosen.
The test values of $|\Delta_{31}|$ have been varied systematically in steps of
$0.2\times 10^{-3}$ eV2 in the range $(1,5)\times 10^{-3}$ eV2 and
$\sin^{2}\theta_{23}$ in steps of $0.02$ in the range $(0,1)$. The other
neutrino oscillation parameters have been kept fixed at the input values,
which were used to generate the data event sample. The three flavor muon
neutrino survival probabilities, $P_{\mu\mu}$ and $P_{\bar{\mu}\bar{\mu}}$,
have been calculated using the code nuCraft Wallraff:2014qka . The rest 495
year un-oscillated muon event sample has been converted into an oscillated
event sample through accept-reject method using the above probabilities. This
sample’s normalization is scaled by 99 to obtain a test event sample
corresponding to five years, which is labeled as theory.

###  2.2 Variables for Analysis

Muons, being minimum ionizing particles, pass through many layers of iron,
leaving behind localized hits in the RPCs. Using this hit information, the
track of the muon can be reconstructed. Because of the magnetic field, this
track will be curved and the bending of the track is opposite for negative and
positive muons. Thus, ICAL can distinguish between the CC interactions of
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$. If a track has been reconstructed, the
event has been considered to be a CC interaction of
$\nu_{\mu}/\bar{\nu}_{\mu}$. The charge, the momentum and the initial
direction ($\cos\theta_{track}$), of a reconstructed track, are also
calculated from the track properties Bhattacharya:2014tha . Electrons,
positrons and photons lose their energy very quickly and leave no track. Pions
pass through a few layers, and it is difficult to construct a track for them
unless they are of very high energy. Baryons have very low kinetic energy and
can pass through one or two layers. Therefore, the hadrons produced in these
reactions leave a few $(0-10)$ scattered hits Devi:2013wxa ; Nizam:2019jdi .
Thus, the GEANT4 simulation of a typical $\nu_{\mu}/\bar{\nu}_{\mu}$ CC event
consists of a track and a few hadron hits.

Events for which one or more tracks have been reconstructed are stored along
with the charge, the momentum and the initial direction of the track with the
largest momentum. The leftover hits, that is the hits which are not used for
track reconstruction, have been labelled hadron hits. In dealing with the
hadron hits, the problem of ghost hits needs to be considered. If the actual
hits occurred at $(x_{1},y_{2})$, $(x_{2},y_{1})$ and $(x_{3},y_{1})$ the
simulation only tells us that the $x$-strips $x_{1},x_{2},x_{3}$ and the
$y$-strips $y_{1},y_{2}$ are hit. This leads to the possibility of there being
six hits, three of which are ghost hits. This problem can be solved very
simply by defining the number of hadron hits to be the maximum of (number of
$x$-strips, number of $y$-strips) with hits.

In this analysis, the following variables have been used, which are given as
the output of the GEANT4 simulator and the track reconstruction program. For
about 90% of the events, only one track is reconstructed. In the other 10% of
the events, two or more tracks are reconstructed. This work concentrates only
on the parameters $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ but not on mass
ordering. Therefore, $\nu_{\mu}/\bar{\nu}_{\mu}$ CC events have been
considered together, and the muon charge information has not been used. This
leads to more events in each bin. The time information of the events has not
been used explicitly. The time stamps of hits, however, were used by the
reconstruction program in determining the track direction.

###  2.3 Analysis Methods

The analysis has been done using two different binning schemes. These schemes
are labeled as 2-variable analysis and 3-variable analysis.

  * •

2-variable analysis: In this analysis method, the events have been binned
according to the modulus of the reconstructed track momentum and the
reconstructed track direction. For the events where more than one track has
been reconstructed, the track momentum and direction are those of the longest
track. There are ten momentum bins given by $(0.1,0.5)$, $(0.5,1.0)$,
$(1.0,1.5)$, $(1.5,2.0)$, $(2.0,2.5)$, $(2.5,3.0)$, $(3.0,5.0)$, $(5.0,10.0)$,
$(10.0,20.0)$, $(20.0,100.0)$ GeV/$c$. At higher energies, the number of
atmospheric neutrino events becomes quite small. To have a sufficient number
of events in each bin, larger bin sizes are chosen at higher energies. We
tried a number of plausible binning schemes and selected the one which gave
the best results. There are 40 direction bins spanning the range $(-1,1)$ in
steps of $0.05$. This analysis uses 400 bins in the double differential
distribution of the two variables.

  * •

3-variable analysis: This analysis method incorporates additional information
compared to 2-variable analysis method. We classified events into four classes
based on $E$ and $|\cos\theta_{\rm track}|$. Here, $E$ is the magnitude of the
track momentum for single track events and is the sum of the magnitudes of
track momenta for multitrack events. The variable $\cos\theta_{\rm track}$ is
the direction of the reconstructed track for single track events and is that
of the longest track for multitrack events. The four event classes are:

    1. 1.

High energy, horizontal events with $E~{}>$ 10 GeV and $|\cos\theta_{\rm
track}|~{}<$ 0.3, with 2 bins in $E$ and 12 bins in $|\cos\theta_{\rm track}|$
for a total of 24 bins.

    2. 2.

High energy, vertical events with $E~{}>$ 10 GeV and $|\cos\theta_{\rm
track}|~{}>$ 0.3, with 2 bins in $E$ and 28 bins in $|\cos\theta_{\rm track}|$
for a total of 56 bins.

    3. 3.

Low energy, horizontal events with $E~{}<$ 10 GeV and $|\cos\theta_{\rm
track}|~{}<$ 0.3, with 8 bins in $E$ and 12 bins in $|\cos\theta_{\rm track}|$
for a total of 96 bins.

    4. 4.

Low energy, vertical events with $E~{}<$ 10 GeV and $|\cos\theta_{\rm
track}|~{}>$ 0.3, with 8 bins in $E$ and 28 bins in $|\cos\theta_{\rm track}|$
for a total of 224 bins.

For atmospheric neutrinos, the number of events at high energies is small and
for ICAL the reconstruction ability for horizontal events is poor. Hence, the
number of events in each bin of the first three classes are too small to be
further subdivided based on non-track hits. Hence, the events in these 176
bins have been binned only in the two variables $E$ and $\cos\theta_{\rm
track}$. Binning in the third variable, non-track hits, has been done only for
the 224 bins of low energy, vertical events. The bins used for the third
variable are $(0,4)$, $(5,10)$ and $>10$.

The data and the theory samples have been binned according to the scheme
described above. Each entry of the theory sample is divided by 99. Thus, the
theoretical prediction for a five-year muon event sample has been obtained if
the values of the oscillation parameters were equal to the chosen test values.
Since this theory event sample is generated by scaling a huge generated
sample, it is expected that the effect of fluctuations in each entry is
negligibly small. On the other hand, event number fluctuations are expected to
distort both the track momentum and track direction distributions in the data
event sample.

###  2.4 Systematic Errors and $\chi^{2}$ Calculation

The theory event sample is calculated using standard inputs for atmospheric
neutrino fluxes and cross-sections. However, these inputs have uncertainties
associated with them. They have been taken into account as systematic errors
through the method of pulls. Usually three systematic errors are considered:
a) normalization error, b) energy dependent tilt error and c) zenith angle
dependent error. The second and third errors are defined in terms of neutrino
energy and direction, respectively. But in our analysis, the variables used
are track momentum and direction. Hence, the systematic errors specified as
functions of neutrino energy and direction have to be transformed into
functions of track momentum and track direction. Following the method used in
Nizam:2019eim  the neutrino energy dependent tilt error has been transformed
to track momentum dependent tilt error and neutrino direction dependent error
has been transformed to reconstructed track direction dependent error. These
two errors have been assumed to be independent of each other. As described in
the 3-variable binning scheme, some events have been binned using the non-
track hit information. An overall $5\%$ systematic error has been considered
for each of those bins. This has been assumed to be independent of the
momentum and direction of the longest track. The method of pulls has been used
to apply the systematic errors.

The number of events in bin $ijk$, $N_{ijk}^{\rm test}$ is defined as

$N_{ijk}^{\rm test}=N_{ijk}^{\rm theory}[1+\pi_{ijk}^{l}]\xi_{l},$ (3)

where $\xi_{l}$ is the pull variable for the systematic error $l$, varied in
the range $-3$ to $3$ in steps of 0.5. The pull coefficient $\pi_{ijk}^{l}$ is
for the $l^{\rm th}$ systematic error. The first three systematic errors are
flux normalization dependent error($\pi_{\rm norm}$), track momentum dependent
tilt error ($\pi_{\rm tilt}$), track direction dependent error ($\pi_{\rm
dir}$) and the last one is systematic error ($\pi_{\rm had}$) taken for each
non-track hit information bin in case of 3-variable analysis. A constant value
of 0.2 has been assumed for $\pi_{\rm norm}$ and 0.05 for $\pi_{\rm had}$. The
$\chi^{2}$ between the data and test event samples is

$\chi^{2}=\sum_{ijk}\left[2\left\\{\left(N_{ijk}^{\rm test}-N_{ijk}^{\rm
data}\right)-N_{ijk}^{\rm data}\ln\left(\frac{N_{ijk}^{\rm test}}{N_{ijk}^{\rm
data}}\right)\right\\}\right]+\xi_{l}^{2},$ (4)

where the prior terms $\xi_{l}^{2}$ have been added in the pull variables.
This $\chi^{2}$ is a function of the test values of $|\Delta_{31}|$ and
$\sin^{2}\theta_{23}$. For each pair of test values of $|\Delta_{31}|$ and
$\sin^{2}\theta_{23}$, $\chi^{2}$ is computed, and the minimum $\chi^{2}$ is
evaluated. The test values corresponding to minimum $\chi^{2}$ are the best
fit values of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$.

##  3 Results

###  3.1 Procedure to calculate best fit point, 2$\sigma$ and 3$\sigma$ range

The procedure described above has been carried out for 25 mutually independent
five-year muon data samples, which are selected randomly. The value of
$\chi^{2}$ for each of the data samples has been plotted as a function of test
$|\Delta_{31}|$ and as a function of test $\sin^{2}\theta_{23}$. From these
plots, the values of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ which minimize
$\chi^{2}$ are obtained for each data sample. These values of the oscillation
parameters have been noted down as the best fit point for that data sample.
The values of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ for which
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}$ is 4 (9) define the 2$\sigma$
(3$\sigma$) range of the oscillation parameters. We have chosen not to define
1$\sigma$ range of the oscillation parameters using a similar definition
because such ranges show a wide variation due to the fluctuations in the data
sample. These ranges are calculated for each data sample, along with the best
fit point. It has been observed that for some samples, the upper bound of
2$\sigma$ and 3$\sigma$ for $|\Delta_{31}|$ lies outside the corresponding
test value range. For those samples it has been assumed that the upper bound
of 2$\sigma$ is $7\times 10^{-3}$ eV2 and that for 3$\sigma$ range is
$10\times 10^{-3}$ eV2. We computed the averages of the best fit parameter
values, 2$\sigma$ ranges and 3$\sigma$ ranges over the 25 sets, for the
purpose of comparing the results of different analysis methods. In the
following subsections, the results for five-year and ten-year exposure time
and 2-variable analysis and 3-variable analysis have been discussed.

###  3.2 Comparison of 2-variable and 3-variable analysis methods

The results for five-year exposure are summarized in tables 2.

Average of 25 Sets
---
Parameter | Binning | $3\sigma$ | $2\sigma$ | Best | $2\sigma$ | $3\sigma$
| Scheme | Lower | Lower | Fit | Upper | Upper
|  | Bound | Bound | Point | Bound | Bound
$|\Delta_{31}|$ | 2-variable | 1.53 | 1.80 | 2.43 | 3.43 | 5.13
$(*10^{-3}eV^{2})$ | 3-variable | 1.67 | 1.99 | 2.53 | 3.19 | 4.13
$\sin^{2}\theta_{23}$ | 2-variable | 0.36 | 0.40 | 0.52 | 0.66 | 0.70
| 3-variable | 0.36 | 0.40 | 0.51 | 0.65 | 0.70
Table 2: Comparison of average best fit point, 2$\sigma$ and 3$\sigma$ allowed
ranges of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ for the 25 mutually
exclusive data sets obtained with 2-variable and 3-variable analysis methods.
The average values are calculated for five-year exposure time.

The results obtained for $\sin^{2}\theta_{23}$ and its allowed ranges in
2$\sigma$ and 3$\sigma$ are comparable for 2-variable analysis and for
3-variable analysis. On the other hand, the allowed ranges for $|\Delta_{31}|$
show a noticeable improvement for 3-variable analysis compared to 2-variable
analysis. For example, the 2-variable analysis does not give an upper limit in
the explored range at 2$\sigma$ in two of the 25 cases, whereas the 3-variable
analysis does so for every case. A comparison between 2-variable analysis and
3-variable analysis methods is shown in the figure 1.

Figure 1: Best fit point, 2$\sigma$ upper and lower limits and 3$\sigma$ upper
and lower limits of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ for the 25
mutually exclusive data sets we considered. The exposure considered is five
years. The top panel is for the 2-variable analysis method and the bottom-
panel is for the 3-variable analysis method. The left panel is for
$|\Delta_{31}|$ and the right panel is for $\sin^{2}\theta_{23}$. Of the 25
sets, 15 sets give smaller allowed range of $|\Delta_{31}|$ for 3-variable
analysis compared to 2-variable analysis, and 12 sets give smaller allowed
range of $\sin^{2}\theta_{23}$ for 3-variable analysis compared to 2-variable
analysis.

If the exposure is increased to ten years, the corresponding numbers are one
and zero respectively as shown in figure 2.

Figure 2: Best fit point, 2$\sigma$ upper and lower limits and 3$\sigma$ upper
and lower limits of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ for the 25
mutually exclusive data sets we considered. The exposure considered is 10
years. The top panel is for the 2-variable analysis method and the bottom-
panel is for the 3-variable analysis method. The left panel is for
$|\Delta_{31}|$ and the right panel is for $\sin^{2}\theta_{23}$. Of the 25
sets, 18 sets give smaller allowed range of $|\Delta_{31}|$ for 3-variable
analysis compared to 2-variable analysis, and 13 sets give smaller allowed
range of $\sin^{2}\theta_{23}$ for 3-variable analysis compared to 2-variable
analysis.

The results for ten-year exposure are summarized in table 3. From table 3, we
note that the average allowed ranges of $|\Delta_{31}|$, both at 2$\sigma$ and
at 3$\sigma$, are smaller for 3-variable analysis compared to 2-variable
analysis.

Average of 25 Sets
---
Parameter | Binning | $3\sigma$ | $2\sigma$ | Best | $2\sigma$ | $3\sigma$
| Scheme | Lower | Lower | Fit | Upper | Upper
|  | Bound | Bound | Point | Bound | Bound
$|\Delta_{31}|$ | 2-variable | 1.74 | 2.01 | 2.35 | 2.92 | 3.54
$(*10^{-3}eV^{2})$ | 3-variable | 1.86 | 2.09 | 2.37 | 2.77 | 3.08
$\sin^{2}\theta_{23}$ | 2-variable | 0.39 | 0.43 | 0.52 | 0.62 | 0.67
| 3-variable | 0.40 | 0.45 | 0.52 | 0.62 | 0.66
Table 3: Comparison of average best fit point, 2$\sigma$ and 3$\sigma$ allowed
ranges of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ for the 25 mutually
exclusive data sets obtained with 2-variable and 3-variable analysis methods.
The average values are calculated for 10-year exposure time.

###  3.3 Analysis with independent $\mu^{-}$ and $\mu^{+}$ data sets

ICAL is a magnetized iron calorimeter, which is capable of determining the
muon charge and hence we can study the oscillations of $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ independently. From CPT conservation, the mass squared
differences and mixing angles of neutrinos and anti-neutrinos are expected to
be the same. Therefore, the analysis described in the previous subsection
binned the events using only track momentum and track direction, but not the
charge ID of the track. However, we wanted to explore if the inclusion of the
charge ID of the track makes any difference to the results obtained. So, we
studied the determination of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ by
binning the events according to (a) track momentum, (b) track direction and
(c) charge ID of the track. This leads to 800 bins as opposed to 400 bins for
the case where the charge ID was not considered. An $\chi^{2}$ analysis was
carried out on event samples of five-year exposure, using the procedure
described in section 3, for this binning where the charge ID is included. The
best fit point, 2$\sigma$ and 3$\sigma$ allowed range for the 25 sets are
shown in figure 3 for analyses with and without charge ID.

Figure 3: Best fit point, 2$\sigma$ upper and lower limits and 3$\sigma$ upper
and lower limits of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ for the 25
mutually exclusive data sets we considered. The exposure considered is five
years. The top panel is for the 2-variable analysis with charge ID and the
bottom-panel is for the 2-variable analysis without charge ID. The left panel
is for $|\Delta_{31}|$ and the right panel is for $\sin^{2}\theta_{23}$. Of
the 25 sets, 15 sets give smaller allowed range of $|\Delta_{31}|$ for with
charge ID compared to without charge ID analysis, and 15 sets give smaller
allowed range of $\sin^{2}\theta_{23}$ for with charge ID analysis compared to
without charge ID analysis.

The comparison between these two analysis methods are presented in 4. We note
that the allowed ranges of parameters, without and with charge ID, are quite
similar. However, the charge ID is crucial for the determination of neutrino
mass hierarchy.

Average of 25 Sets
---
Parameter | Binning | $3~{}\sigma$ | $2~{}\sigma$ | Best | $2~{}\sigma$ | $3~{}\sigma$
| Scheme | Lower | Lower | Fit | Upper | Upper
|  | Bound | Bound | Point | Bound | Bound
$|\Delta_{31}|$ | Without charge ID | 1.53 | 1.80 | 2.43 | 3.43 | 5.13
$(*10^{-3}eV^{2})$ | With charge ID | 1.63 | 1.93 | 2.56 | 3.22 | 4.69
$\sin^{2}\theta_{23}$ | Without charge ID | 0.36 | 0.40 | 0.52 | 0.66 | 0.70
| With charge ID | 0.37 | 0.42 | 0.54 | 0.66 | 0.70
Table 4: Comparison of average best fit point, 2$\sigma$ and 3$\sigma$ allowed
ranges of $|\Delta_{31}|$ and $\sin^{2}\theta_{23}$ for the 25 mutually
exclusive data sets obtained with 2-variable and 3-variable analysis methods.
The average values are calculated for 10-year exposure time.

##  4 Conclusion

Method | Parameter | Input | Reconstructed | 2$\sigma$ | 3$\sigma$
---|---|---|---|---|---
|  | Value | Best Fit point | Range | Range
Ref. Rebin:2018fdl  |  $|\Delta_{31}|$(10-3 eV2) | 2.32 | 2.32 | 2.03 (1.68 - 3.71) | 4.07 (1.40 - 5.47)
Ref. Rebin:2018fdl  | $\sin^{2}\theta_{23}$ | 0.5 | 0.496 | 0.38 (0.34 - 0.72) | 0.48 (0.29 - 0.77)
5-yr, 2-var |  $|\Delta_{31}|$(10-3 eV2) | 2.32 | 2.43 | 1.63 (1.80 - 3.43) | 3.60 (1.53 - 5.13)
5-yr, 2-var | $\sin^{2}\theta_{23}$ | 0.5 | 0.52 | 0.26 (0.40 - 0.66) | 0.34 (0.36 - 0.70)
5-yr, 3-var |  $|\Delta_{31}|$(10-3 eV2) | 2.32 | 2.53 | 1.20 (1.99 - 3.19) | 2.46 (1.67 - 4.13)
5-yr, 3-var | $\sin^{2}\theta_{23}$ | 0.5 | 0.51 | 0.25 (0.40 - 0.65) | 0.34 (0.36 - 0.70)
10-yr, 2-var |  $|\Delta_{31}|$(10-3 eV2) | 2.32 | 2.35 | 0.91 (2.01 - 2.92) | 1.8 (1.74 - 3.54)
10-yr, 2-var | $\sin^{2}\theta_{23}$ | 0.5 | 0.52 | 0.19 (0.43 - 0.62) | 0.28 (0.39 - 0.67)
10-yr, 3-var |  $|\Delta_{31}|$(10-3 eV2) | 2.32 | 2.37 | 0.68 (2.09 - 2.77) | 1.24 (1.86 - 3.10)
10-yr, 3-var | $\sin^{2}\theta_{23}$ | 0.5 | 0.51 | 0.17 (0.45 - 0.62) | 0.26 (0.40 - 0.66)
Table 5:  Comparison of uncertainty ranges from different methods.

In this work, we have estimated the ability of ICAL at INO to determine the
atmospheric neutrino oscillation parameters. Most of the previous studies of
this topic used generator level kinematic information, smeared by resolution
functions. In this work, we performed full GEANT4 simulation of atmospheric
neutrino events. Using the ICAL reconstruction code Bhattacharya:2014tha  on
this simulated data, we extracted the kinematical variables of tracks and
performed a $\chi^{2}$ analysis with the events binned according to track
variables. In addition to doing a 2-variable analysis with binning only in
track momentum and track direction, we also did a 3-variable analysis where
the number of non-track hits is introduced as a third variable.

From table 5, we see that both the 2$\sigma$ and 3$\sigma$ ranges of
$|\Delta_{31}|$ from the 3-variable analysis are smaller by a factor of 1.5
compared to the corresponding ranges of the 2-variable analysis for a five-
year exposure. On the other hand, the ranges of $\sin^{2}\theta_{23}$, both
2$\sigma$ and 3$\sigma$, are the same for 2 and 3-variable analyses. When the
exposure is increased to 10 years, the parameter ranges are smaller compared
to the ranges of five-year exposure but the relation between the ranges of 2
and 3-variable analyses are similar to those of five-year exposure. The
precision in $|\Delta_{31}|$ is greatly improved by the inclusion of the third
variable. We also found that an analysis, where the binning is done based on
the charge ID of the track also, gives essentially the same allowed ranges of
oscillation parameters as the analysis without charge ID. In conclusion, we
find that the precision in $\sin^{2}\theta_{23}$ improves by a negligible
amount by the inclusion of the third variable. The precision on
$|\Delta_{31}|$ on the other hand improves by 32% when the third variable is
included. The 3-variable analysis method takes care of the energy of the other
particles produced in the neutrino CC interactions. The parameter
$\Delta_{31}$ being dependent on the value of L/Eν can be measured more
precisely if we have better reconstruction of neutrino energy Eν. On the other
hand $\sin^{2}\theta_{23}$ is measured from the oscillation dip. The better
the dip is reconstructed, the more precise it will become. This has been
observed when we increased the exposure time to 10 years.

We now compare the results of our five-year, 2-variable analysis with those of
ref. Rebin:2018fdl . There are two possible reasons.

  * •

The above reference considered $\cos\theta_{\rm track}$ bin size of 0.2. On
the other hand, we have taken a much finer bin size of 0.05. A more accurate
reconstruction of $\cos\theta_{\rm track}$ gives a more accurate estimate of
the distance traveled by the neutrino. This, in turn, leads to a better
estimate of $|\Delta_{31}|$.

  * •

The above reference considered only those events whose tracks were
reconstructed with $\chi^{2}/{\rm ndf}<10$, where ${\rm ndf}=2N_{\rm hits}-5$.
The event sample in this work is larger because we have considered all the
events with a reconstructed track. The larger event sample also led to some
improvement in our results.

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# FRACTIONAL INTEGRAL INEQUALITIES VIA ATANGANA-BALEANU OPERATORS FOR CONVEX
AND CONCAVE FUNCTIONS

Ahmet Ocak Akdemir1 1DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES AND ARTS,
AǦ RI UNIVERSITY, AǦRI, TURKEY<EMAIL_ADDRESS>,  Ali Karaoǧlan2
2DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES AND ARTS, ORDU UNIVERSITY,
ORDU, TURKEY<EMAIL_ADDRESS>,  Maria Alessandra Ragusa3
3DIPARTIMENTO DI MATEMATICA E INFORMATICA, UNIVERSITÁ DI CATANIA, CATANIA,
ITALY - RUDN UNIVERSITY, MOSCOW, RUSSIA<EMAIL_ADDRESS>and
Erhan Set4 1DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES AND ARTS, ORDU
UNIVERSITY, ORDU, TURKEY<EMAIL_ADDRESS>
###### Abstract.

Recently, many fractional integral operators were introduced by different
mathematicians. One of these fractional operators, Atangana-Baleanu fractional
integral operator, was defined by Atangana and Baleanu in [2]. In this study,
firstly, a new identity by using Atangana-Baleanu fractional integral
operators are proved. Then, new fractional integral inequalities have been
obtained for convex and concave functions with the help of this identity and
some certain integral inequalities.

###### Key words and phrases:

Convex functions, Hölder inequality, Young inequality, power mean inequality,
Atangana-Baleanu fractional derivative, Atangana-Baleanu fractinal derivative

###### 2010 Mathematics Subject Classification:

26A33, 26A51, 26D10.

##  1\. Introduction

Mathematics is a tool that serves pure and applied sciences with its deep-
rooted history as old as human history, and sheds light on how to express and
then solve problems. Mathematics uses various concepts and their relations
with each other while performing this task. By defining spaces and algebraic
structures built on spaces, mathematics creates structures that contribute to
human life and nature. The concept of function is one of the basic structures
of mathematics, and many researchers have focused on new function classes and
made efforts to classify the space of functions. One of the types of functions
defined as a product of this intense effort is the convex function, which has
applications in statistics, inequality theory, convex programming and
numerical analysis. This interesting class of functions is defined as follows.

######  Definition 1.1.

The mapping
$f:[\theta_{1},\theta_{2}]\subseteq\mathbb{R}\rightarrow\mathbb{R}$, is said
to be convex if

(1.1) $f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)$

is valid for all $x,y\in[\theta_{1},\theta_{2}]$ and $\lambda\in[0,1]$.

Many inequalities have been obtained by using this unique function type and
varieties in inequality theory, which is one of the most used areas of convex
functions. We will continue by introducing the Hermite-Hadamard inequality
that generate limits on the mean value of a convex function and the famous
Bullen inequality as follows.
Assume that $f:I\subseteq\mathbb{R}\rightarrow\mathbb{R}$ is a convex mapping
defined on the interval $I$ of $\mathbb{R}$ where $\theta_{1}<b.$ The
following statement;

(1.2)
$f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\leq\frac{1}{\kappa_{2}-\theta_{1}}\int\limits_{\theta_{1}}^{\theta_{2}}f(x)dx\leq\frac{f(\theta_{1})+f(\theta_{2})}{2}$

holds and known as Hermite-Hadamard inequality. Both inequalities hold in the
reversed direction if $f$ is concave.
The Bullen’s integral inequality can be presented as

$\frac{1}{\theta_{2}-\theta_{1}}\int_{\theta_{1}}^{\theta_{2}}f\left(x\right)dx\leq\frac{1}{2}\left[f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)+\frac{f\left(\theta_{1}\right)+f\left(\theta_{2}\right)}{2}\right]$

where $f:I\subset\mathbb{R}\rightarrow\mathbb{R}$ is a convex mapping on the
interval $I$ of $\mathbb{R}$ where $\kappa_{1},\theta_{2}\in I$ with
$\theta_{1}<\theta_{2}$.
To provide detail information on convexity, let us consider some earlier
studies that have been performed by many researchers. In [17], Jensen
introduced the concept of convex function to the literature for the first time
and drew attention to the fact that it seems to be the basis of the concept of
incremental function. In [19], Beckenbach has mentioned about the concept of
convexity and emphasized several features of this useful function class. In
[18], the authors have focused the relations between convexity and Hermite-
Hadamard’s inequality. This study has led many researchers to the link between
convexity and integral inequalities, which has guided studies in this field.
Based on these studies, many papers have been produced for different kinds of
convex functions. In [13], Akdemir et al. have proved several new integral
inequalities for geometric-arithmetic convex functions via a new integral
identity. Several new Hadamard’s type integral inequalities have been
established with applications to special means by Kavurmaci et al. in [14].
Therefore, a similar argument has been carried out by Zhang et al. but now for
$s-$geometrically convex functions in [16]. On all of these, Xi et al. have
extended the challenge to $m-$ and $(\alpha,m)-$convex functions by providing
Hadamard type inequalities in [15].
Although fractional analysis has been known since ancient times, it has
recently become a more popular subject in mathematical analysis and applied
mathematics. The adventure that started with the question of whether the
solution will exist if the order is fractional in a differential equation, has
developed with many derivative and integral operators. By defining the
derivative and integral operators in fractional order, the researchers who
aimed to propose more effective solutions to the solution of physical
phenomena have turned to new operators with general and strong kernels over
time. This orientation has provided mathematics and applied sciences several
operators with kernel structures that differ in terms of locality and
singularity, as well as generalized operators with memory effect properties.
The struggle that started with the question of how the order in the
differential equation being a fraction would have consequences, has now
evolved into the problem of how to explain physical phenomena and find the
most effective fractional operators that will provide effective solutions to
real world problems. Let us introduce some fractional derivative and integral
operators that have broken ground in fractional analysis and have proven their
effectiveness in different fields by using by many researchers.
We will remember the Caputo-Fabrizio derivative operators. Also, we would like
to note that the functions belong to Hilbert spaces denoted by
$H^{1}(0,\theta_{2})$

######  Definition 1.2.

[3] Let $f\in H^{1}(0,\theta_{2})$, $\theta_{2}>\theta_{1}$, $\alpha\in[0,1]$
then, the definition of the new Caputo fractional derivative is:

(1.3)
${}^{CF}D^{\alpha}f(\tau_{1})=\frac{M(\alpha)}{1-\alpha}\int_{\kappa_{1}}^{\tau_{1}}f^{\prime}(s)exp\left[-\frac{\alpha}{(1-\alpha)}(\tau_{1}-s)\right]ds$

where $M(\alpha)$ is normalization function.

Depends on this interesting fractional derivative operator, the authors have
defined the Caputo-Fabrizio fractional integral operator as follows.

######  Definition 1.3.

[4] Let $f\in H^{1}(0,\theta_{2})$, $\theta_{2}>\theta_{1}$, $\alpha\in[0,1]$
then, the definition of the left and right side of Caputo-Fabrizio fractional
integral is:

$\left({}_{:::\theta_{1}}^{CF}I^{\alpha}\right)(\tau_{1})=\frac{1-\alpha}{B(\alpha)}f(\tau_{1})+\frac{\alpha}{B(\alpha)}\int_{\theta_{1}}^{\tau_{1}}f(y)dy,$

and

$\left({}^{CF}I_{\theta_{2}}^{\alpha}\right)(\tau_{1})=\frac{1-\alpha}{B(\alpha)}f(\tau_{1})+\frac{\alpha}{B(\alpha)}\int_{\tau_{1}}^{\kappa_{2}}f(y)dy$

where $B(\alpha)$ is normalization function.

The Caputo-Fabrizio fractional derivative, which is used in dynamical systems,
physical phenomena, disease models and many other fields, is a highly
functional operator by definition, but has a deficiency in terms of not
meeting the initial conditions in the special case $\alpha=1$. The improvement
to eliminate this deficiency has been provided by the new derivative operator
developed by Atangana-Baleanu, which has versions in the sense of Caputo and
Riemann. In the sequel of this paper, we will denote the normalization
function with $B(\alpha)$ that the same properties with the $M(\alpha)$ which
defined in Caputo-Fabrizio definition.

######  Definition 1.4.

[2] Let $f\in H^{1}(\theta_{1},\theta_{2})$, $\theta_{2}>\kappa_{1}$,
$\alpha\in[0,1]$ then, the definition of the new fractional derivative is
given:

(1.4)
${}_{::::::\theta_{1}}^{ABC}D_{\tau_{1}}^{\alpha}\left[f(\tau_{1})\right]=\frac{B(\alpha)}{1-\alpha}\int_{a}^{\tau_{1}}f^{\prime}(x)E_{\alpha}\left[-\alpha\frac{(\tau_{1}-x)^{\alpha}}{(1-\alpha)}\right]dx.$

######  Definition 1.5.

[2] Let $f\in H^{1}(\theta_{1},\theta_{2})$, $\theta_{2}>\kappa_{1}$,
$\alpha\in[0,1]$ then, the definition of the new fractional derivative is
given:

(1.5)
${}_{:::::::\theta_{1}}^{ABR}D_{\tau_{1}}^{\alpha}\left[f(\tau_{1})\right]=\frac{B(\alpha)}{1-\alpha}\frac{d}{d\tau_{1}}\int_{\theta_{1}}^{\tau_{1}}f(x)E_{\alpha}\left[-\alpha\frac{(\tau_{1}-x)^{\alpha}}{(1-\alpha)}\right]dx.$

Equations (1.4) and (1.5) have a non-local kernel. Also in equation (1.5) when
the function is constant we get zero.

The associated fractional integral operator has been defined by Atangana-
Baleanu as follows.

######  Definition 1.6.

[2] The fractional integral associate to the new fractional derivative with
non-local kernel of a function $f\in H^{1}(\kappa_{1},\theta_{2})$ as defined:

${}_{::::\theta_{1}}^{AB}I^{\alpha}\left\\{f(\tau_{1})\right\\}=\frac{1-\alpha}{B(\alpha)}f(\tau_{1})+\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\theta_{1}}^{\tau_{1}}f(y)(\tau_{1}-y)^{\alpha-1}dy$

where $\theta_{2}>\theta_{1},\alpha\in[0,1].$

In [1], Abdeljawad and Baleanu introduced right hand side of integral operator
as following; the right fractional new integral with ML kernel of order
$\alpha\in[0,1]$ is defined by

$\left({}^{AB}I_{\theta_{2}}^{\alpha}\right)\left\\{f(\tau_{1})\right\\}=\frac{1-\alpha}{B(\alpha)}f(\tau_{1})+\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\tau_{1}}^{\theta_{2}}f(y)(y-\tau_{1})^{\alpha-1}dy.$

In [4], Abdeljawad and Baleanu has presented some new results based on
fractional order derivatives and their discrete versions. Conformable integral
operators have been defined by Abdeljawad in [5]. This useful operator has
been used to prove some new integral inequalities in [10]. Another important
fractional operator -Riemann-Liouville fractional integral operators- have
been used to provide some new Simpson type integral inequalities in [12].
Ekinci and Özdemir have proved several generalizations by using Riemann-
Liouville fractional integral operators in [9] and the authors have
established some similar results with this operator in [20]. In [6], Akdemir
et al. have presented some new variants of celebrated Chebyshev inequality via
generalized fractional integral operators. The argument has been proceed with
the study of Rashid et al. (see [7]) that involves new investigations related
to generalized $k-$fractional integral operators. In [8], Rashid et al. have
presented some motivated findings that extend the argument to the Hilbert
spaces. For more information related to different kinds of fractional
operators, we recommend to consider [11]. The applications of fractional
operators have been demonsrated by several researchers, we suggest to see the
papers [21]-[23].
The main motivation of this paper is to prove an integral identity that
includes the Atangana-Baleanu integral operator and to provide some new Bullen
type integral inequalities for differentiable convex and concave functions
with the help of this integral identity. Some special cases are also
considered.

##  2\. Main Results

We will start with a new integral identity that will be used the proofs of our
main findings:

######  Lemma 2.1.

Let $f:[\theta_{1},\theta_{2}]\rightarrow\mathbb{R}$ be differentiable
function on $(\theta_{1},\theta_{2})$ with $\kappa_{1}<\theta_{2}$. Then we
have the following identity for Atangana-Baleanu fractional integral operators

$\displaystyle\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\left((1-\tau_{1})^{\alpha}-\tau_{1}^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}$
$\displaystyle+\int_{0}^{1}\left(\tau_{1}^{\alpha}-(1-\tau_{1})^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\kappa_{1}\right)d\tau_{1}$

where $\alpha,\tau_{1}\in[0,1]$, $\Gamma(.)$ is Gamma function and $B(\alpha)$
is normalization function.

###### Proof.

By adding $I_{1}$ and $I_{2}$, we have

$\displaystyle I_{1}+I_{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\left((1-\tau_{1})^{\alpha}-\tau_{1}^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\kappa_{2}\right)d\tau_{1}$
$\displaystyle+\int_{0}^{1}\left(\tau_{1}^{\alpha}-(1-\tau_{1})^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)d\tau_{1}.$

By using integration, we have

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\left((1-\tau_{1})^{\alpha}-\tau_{1}^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\kappa_{2}\right)d\tau_{1}$
$\displaystyle=$
$\displaystyle\frac{\left((1-\tau_{1})^{\alpha}-\tau_{1}^{\alpha}\right)f\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}}{\frac{\theta_{1}-\theta_{2}}{2}}\bigg{|}_{1}^{0}$
$\displaystyle-\frac{2\alpha}{\kappa_{2}-\theta_{1}}\int_{0}^{1}\left((1-\tau_{1})^{\alpha-1}+\tau_{1}^{\alpha-1}\right)f\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\kappa_{2}\right)d\tau_{1}$
$\displaystyle=$
$\displaystyle-\frac{2}{\theta_{1}-\theta_{2}}f(\theta_{1})-\frac{2}{\kappa_{1}-\theta_{2}}f(\frac{\theta_{1}+\theta_{2}}{2})$
$\displaystyle-\frac{2\alpha}{\kappa_{2}-\theta_{1}}\int_{0}^{1}(1-\tau_{1})^{\alpha-1}f\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}$
$\displaystyle-\frac{2\alpha}{\theta_{2}-\theta_{1}}\int_{0}^{1}\tau_{1}^{\alpha-1}f\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}$
$\displaystyle=$
$\displaystyle\frac{2}{\theta_{2}-\theta_{1}}\left(f(\theta_{1})+f(\frac{\kappa_{1}+\theta_{2}}{2})\right)-\frac{2^{\alpha+1}\alpha}{(\kappa_{2}-\theta_{1})^{\alpha+1}}\int_{\theta_{1}}^{\frac{\theta_{1}+\kappa_{2}}{2}}\left(x-\theta_{1}\right)^{\alpha-1}f(x)dx$
$\displaystyle-\frac{2^{\alpha+1}\alpha}{(\theta_{2}-\theta_{1})^{\alpha+1}}\int_{\theta_{1}}^{\frac{\theta_{1}+\theta_{2}}{2}}\left(\frac{\kappa_{1}+\theta_{2}}{2}-x\right)^{\alpha-1}f(x)dx.$

Multiplying both side of (2.1) identity by
$\frac{(\kappa_{2}-\theta_{1})^{\alpha+1}}{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}$,
we have

$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha+1}}{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}I_{1}$
$\displaystyle=$
$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha}}{2^{\alpha}B(\alpha)\Gamma(\alpha)}\left(f(\theta_{1})+f(\frac{\theta_{1}+\kappa_{2}}{2})\right)-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\kappa_{1}}^{\frac{\theta_{1}+\theta_{2}}{2}}\left(x-\theta_{1}\right)^{\alpha-1}f(x)dx$
$\displaystyle-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\theta_{1}}^{\frac{\theta_{1}+\theta_{2}}{2}}\left(\frac{\theta_{1}+\theta_{2}}{2}-x\right)^{\alpha-1}f(x)dx.$

Similarly, by using integration, we get

$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\left(\tau_{1}^{\alpha}-(1-\tau_{1})^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\kappa_{1}\right)d\tau_{1}$
$\displaystyle=$
$\displaystyle\frac{\left(\tau_{1}^{\alpha}-(1-\tau_{1})^{\alpha}\right)f\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)d\tau_{1}}{\frac{\theta_{2}-\theta_{1}}{2}}\bigg{|}_{1}^{0}$
$\displaystyle-\frac{2\alpha}{\kappa_{2}-\theta_{1}}\int_{0}^{1}\left(\tau_{1}^{\alpha-1}+(1-\tau_{1})^{\alpha-1}\right)f\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\kappa_{1}\right)d\tau_{1}$
$\displaystyle=$
$\displaystyle\frac{2}{\theta_{2}-\theta_{1}}\left(f(\theta_{2})+f(\frac{\kappa_{1}+\theta_{2}}{2})\right)-\frac{2^{\alpha+1}\alpha}{(\kappa_{2}-\theta_{1})^{\alpha+1}}\int_{\frac{\theta_{1}+\theta_{2}}{2}}^{\theta_{2}}\left(x-\frac{\theta_{1}+\theta_{2}}{2}\right)^{\alpha-1}f(x)dx$
$\displaystyle-\frac{2^{\alpha+1}\alpha}{(\theta_{2}-\theta_{1})^{\alpha+1}}\int_{\frac{\theta_{1}+\theta_{2}}{2}}^{\theta_{2}}\left(\theta_{2}-x\right)^{\alpha-1}f(x)dx.$

Multiplying both side of (2.1) identity by
$\frac{(\theta_{2}-\kappa_{1})^{\alpha+1}}{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}$,
we get

$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha+1}}{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}I_{2}$
$\displaystyle=$
$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha}}{2^{\alpha}B(\alpha)\Gamma(\alpha)}\left(f(\theta_{2})+f(\frac{\theta_{1}+\kappa_{2}}{2})\right)-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\frac{\theta_{1}+\theta_{2}}{2}}^{\theta_{2}}\left(x-\frac{\theta_{1}+\kappa_{2}}{2}\right)^{\alpha-1}f(x)dx$
$\displaystyle-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\frac{\theta_{1}+\kappa_{2}}{2}}^{\theta_{2}}\left(\theta_{2}-x\right)^{\alpha-1}f(x)dx.$

By adding identity (2.1) and (2.1), we obtain

$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha+1}}{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}[I_{1}+I_{2}]$
$\displaystyle=$
$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha}\Gamma(\alpha)}{2^{\alpha}B(\alpha)\Gamma(\alpha)}\left[f(\theta_{1})+f(\frac{\theta_{1}+\theta_{2}}{2})\right]$
$\displaystyle-\frac{1-\alpha}{B(\alpha)}f(\theta_{1})-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\theta_{1}}^{\frac{\theta_{1}+\theta_{2}}{2}}\left(x-\theta_{1}\right)^{\alpha-1}f(x)dx$
$\displaystyle-\frac{1-\alpha}{B(\alpha)}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\theta_{1}}^{\frac{\theta_{1}+\theta_{2}}{2}}\left(\frac{\theta_{1}+\theta_{2}}{2}-x\right)^{\alpha-1}f(x)dx$
$\displaystyle+$
$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha}\Gamma(\alpha)}{2^{\alpha}B(\alpha)\Gamma(\alpha)}\left[f(\theta_{2})+f(\frac{\theta_{1}+\theta_{2}}{2})\right]$
$\displaystyle-\frac{1-\alpha}{B(\alpha)}f(\theta_{2})-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\frac{\kappa_{1}+\theta_{2}}{2}}^{\theta_{2}}\left(\theta_{2}-x\right)^{\alpha-1}f(x)dx$
$\displaystyle-\frac{1-\alpha}{B(\alpha)}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)-\frac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\frac{\kappa_{1}+\theta_{2}}{2}}^{\theta_{2}}\left(x-\frac{\theta_{1}+\theta_{2}}{2}\right)^{\alpha-1}f(x)dx.$

Using the definition of Atangana-Baleanu fractional integral operators, we get

$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha+1}}{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}\Bigg{[}\int_{0}^{1}\left((1-\tau_{1})^{\alpha}-\tau_{1}^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}$
$\displaystyle+\int_{0}^{1}\left(\tau_{1}^{\alpha}-(1-\tau_{1})^{\alpha}\right)f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)d\tau_{1}\Bigg{]}$
$\displaystyle=$
$\displaystyle\frac{(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha}\Gamma(\alpha)}{2^{\alpha}B(\alpha)\Gamma(\alpha)}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\kappa_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right].$

∎

######  Theorem 2.1.

Let $f:[\theta_{1},\theta_{2}]\rightarrow\mathbb{R}$ be differentiable
function on $(\theta_{1},\theta_{2})$ with $\kappa_{1}<\theta_{2}$ and
$f^{\prime}\in L_{1}[\theta_{1},\theta_{2}]$. If $|f^{\prime}|$ is a convex
function, we have the following inequality for Atangana-Baleanu fractional
integral operators

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{2\left[\left|f^{\prime}(\theta_{1})\right|+\left|f^{\prime}(\theta_{2})\right|\right]}{\alpha+1}$

where $\alpha\in[0,1]$, $B(\alpha)$ is normalization function.

###### Proof.

By using Lemma 2.1, we can write

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}$
$\displaystyle+\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}$
$\displaystyle+\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}$
$\displaystyle+\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}.$

By using convexity of $|f^{\prime}|$, we get

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}(1-\tau_{1})^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{1})\right|+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|\right]d\tau_{1}$
$\displaystyle+\int_{0}^{1}\tau_{1}^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{1})\right|+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|\right]d\tau_{1}$
$\displaystyle+\int_{0}^{1}\tau_{1}^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\kappa_{1})\right|\right]d\tau_{1}$
$\displaystyle+\int_{0}^{1}(1-\tau_{1})^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\theta_{1})\right|\right]d\tau_{1}.$

By computing the above integral, we obtain

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{2\left[\left|f^{\prime}(\theta_{1})\right|+\left|f^{\prime}(\theta_{2})\right|\right]}{\alpha+1}$

and the proof is completed. ∎

######  Corollary 2.1.

In Theorem 2.1, if we choose $\alpha=1$ we obtain

$\displaystyle\bigg{|}\frac{f(\theta_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\kappa_{2}}{2}\right)}{\theta_{2}-\theta_{1}}-\frac{4}{(\theta_{2}-\kappa_{1})^{2}}\int_{\theta_{1}}^{\theta_{2}}f(x)dx\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{\left|f^{\prime}(\theta_{1})\right|+\left|f^{\prime}(\kappa_{2})\right|}{2}.$

######  Theorem 2.2.

Let $f:[\theta_{1},\theta_{2}]\rightarrow\mathbb{R}$ be differentiable
function on $(\theta_{1},\theta_{2})$ with $\kappa_{1}<\theta_{2}$ and
$f^{\prime}\in L_{1}[\theta_{1},\theta_{2}]$. If $|f^{\prime}|^{q}$ is a
convex function, then we have the following inequality for Atangana-Baleanu
fractional integral operators:

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\frac{2}{(\alpha
p+1)^{\frac{1}{p}}}\left[\left(\frac{3\left|f^{\prime}(\theta_{1})\right|^{q}+\left|f^{\prime}(\kappa_{2})\right|^{q}}{4}\right)^{\frac{1}{q}}+\left(\frac{3\left|f^{\prime}(\theta_{2})\right|^{q}+\left|f^{\prime}(\kappa_{1})\right|^{q}}{4}\right)^{\frac{1}{q}}\right]$

where $p^{-1}+q^{-1}=1$, $\alpha\in[0,1]$, $q>1$, $B(\alpha)$ is normalization
function.

###### Proof.

By using the identity that is given in Lemma 2.1, we have

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}+\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}$
$\displaystyle+\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}+\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}.$

By applying Hölder inequality, we have

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\left(\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\kappa_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}.$

By using convexity of $|f^{\prime}|^{q}$, we obtain

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\left(\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\kappa_{1})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\kappa_{2})\right|^{q}\right]dv\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\kappa_{1})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\kappa_{2})\right|^{q}\right]d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\kappa_{2})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\kappa_{1})\right|^{q}\right]d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\kappa_{2})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\kappa_{1})\right|^{q}\right]d\tau_{1}\right)^{\frac{1}{q}}.$

By calculating the integrals that is in the above inequalities, we get desired
result. ∎

######  Corollary 2.2.

In Theorem 2.2, if we choose $\alpha=1$ we obtain

$\displaystyle\bigg{|}\frac{f(\theta_{1})+f(\theta_{2})+2f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)}{\theta_{2}-\theta_{1}}-\frac{4}{(\kappa_{2}-\theta_{1})^{2}}\int_{\theta_{1}}^{\theta_{2}}f(x)dx\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{(p+1)^{\frac{1}{p}}}\bigg{[}\left(\frac{3\left|f^{\prime}(\theta_{1})\right|^{q}+\left|f^{\prime}(\kappa_{2})\right|^{q}}{4}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\frac{3\left|f^{\prime}(\theta_{2})\right|^{q}+\left|f^{\prime}(\theta_{1})\right|^{q}}{4}\right)^{\frac{1}{q}}\bigg{]}.$

######  Theorem 2.3.

Let $f:[\theta_{1},\theta_{2}]\rightarrow\mathbb{R}$ be differentiable
function on $(\theta_{1},\theta_{2})$ with $\kappa_{1}<\theta_{2}$ and
$f^{\prime}\in L_{1}[\theta_{1},\theta_{2}]$. If $|f^{\prime}|^{q}$ is a
convex function, then we have the following inequality for Atangana-Baleanu
fractional integral operators

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{\alpha+1}\right)^{1-\frac{1}{q}}\Bigg{[}\left(\frac{\alpha+3}{2(\alpha+1)(\alpha+2)}\left|f^{\prime}(\kappa_{1})\right|^{q}+\frac{1}{2(\alpha+2)}\left|f^{\prime}(\kappa_{2})\right|^{q}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\frac{2\alpha+3}{2(\alpha+1)(\alpha+2)}\left|f^{\prime}(\theta_{1})\right|^{q}+\frac{1}{2(\alpha+1)(\alpha+2)}\left|f^{\prime}(\theta_{2})\right|^{q}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\frac{2\alpha+3}{2(\alpha+1)(\alpha+2)}\left|f^{\prime}(\theta_{2})\right|^{q}+\frac{1}{2(\alpha+1)(\alpha+2)}\left|f^{\prime}(\theta_{1})\right|^{q}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\frac{\alpha+3}{2(\alpha+1)(\alpha+2)}\left|f^{\prime}(\theta_{2})\right|^{q}+\frac{1}{2(\alpha+2)}\left|f^{\prime}(\theta_{1})\right|^{q}\right)^{\frac{1}{q}}\Bigg{]}$

where $\alpha\in[0,1]$, $q\geq 1$, $B(\alpha)$ is normalization function.

###### Proof.

By Lemma 2.1, we get

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}+\int_{0}^{1}t^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\kappa_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}$
$\displaystyle+\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}+\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}.$

By applying power mean inequality, we get

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}.$

By using convexity of $|f^{\prime}|^{q}$, we obtain

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{1})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|^{q}\right]d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}\tau_{1}^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{1})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|^{q}\right]d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}\tau_{1}^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\theta_{1})\right|^{q}\right]d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}\right)^{1-\frac{1}{q}}\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}\left[\frac{1+\tau_{1}}{2}\left|f^{\prime}(\theta_{2})\right|^{q}+\frac{1-\tau_{1}}{2}\left|f^{\prime}(\theta_{1})\right|^{q}\right]d\tau_{1}\right)^{\frac{1}{q}}$

By computing the above integrals, the proof is completed. ∎

######  Corollary 2.3.

In Theorem 2.3, if we choose $\alpha=1$ we obtain

$\displaystyle\bigg{|}\frac{2\left[f(\theta_{1})+f(\theta_{2})+2f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right]}{\theta_{2}-\theta_{1}}-\frac{8}{(\theta_{2}-\theta_{1})^{2}}\int_{\theta_{1}}^{\theta_{2}}f(x)dx\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{2}\right)^{1-\frac{1}{q}}\Bigg{[}\left(\frac{2\left|f^{\prime}(\theta_{1})\right|^{q}+\left|f^{\prime}(\theta_{2})\right|^{q}}{6}\right)^{\frac{1}{q}}+\left(\frac{5\left|f^{\prime}(\theta_{1})\right|^{q}+\left|f^{\prime}(\theta_{2})\right|^{q}}{12}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\frac{5\left|f^{\prime}(\theta_{2})\right|^{q}+\left|f^{\prime}(\theta_{1})\right|^{q}}{12}\right)^{\frac{1}{q}}+\left(\frac{2\left|f^{\prime}(\theta_{2})\right|^{q}+\left|f^{\prime}(\theta_{1})\right|^{q}}{6}\right)^{\frac{1}{q}}\Bigg{]}.$

######  Theorem 2.4.

Let $f:[\theta_{1},\theta_{2}]\rightarrow\mathbb{R}$ be differentiable
function on $(\theta_{1},\theta_{2})$ with $\kappa_{1}<\theta_{2}$ and
$f^{\prime}\in L_{1}[\theta_{1},\theta_{2}].$ If $|f^{\prime}|^{q}$ is a
convex function, then we have the following inequality for Atangana-Baleanu
fractional integral operators

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\frac{4}{p(\alpha
p+1)}+\frac{2\left[\left|f^{\prime}(\kappa_{1})\right|^{q}+\left|f^{\prime}(\theta_{2})\right|^{q}\right]}{q}$

where $p^{-1}+q^{-1}=1$, $\alpha\in[0,1]$, $q>1$, $B(\alpha)$ is normalization
function.

###### Proof.

By using identity that is given in Lemma 2.1, we get

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}+\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|d\tau_{1}$
$\displaystyle+\int_{0}^{1}\tau_{1}^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}+\int_{0}^{1}(1-\tau_{1})^{\alpha}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|d\tau_{1}.$

By using the Young inequality as $xy\leq\frac{1}{p}x^{p}+\frac{1}{q}y^{q}$

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\frac{1}{p}\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}+\frac{1}{q}\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}$
$\displaystyle+\frac{1}{p}\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}+\frac{1}{q}\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}$
$\displaystyle+\frac{1}{p}\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}+\frac{1}{q}\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}$
$\displaystyle+\frac{1}{p}\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}+\frac{1}{q}\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}.$

By using convexity of $|f^{\prime}|^{q}$ and by a simple computation, we have
the desired result. ∎

######  Corollary 2.4.

In Theorem 2.4, if we choose $\alpha=1$ we obtain

$\displaystyle\bigg{|}\frac{f(\theta_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\kappa_{2}}{2}\right)}{\theta_{2}-\theta_{1}}-\frac{4}{(\theta_{2}-\kappa_{1})^{2}}\int_{\theta_{1}}^{\theta_{2}}f(x)dx\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{2}{p^{2}+p}+\frac{\left|f^{\prime}(\theta_{1})\right|^{q}+\left|f^{\prime}(\theta_{2})\right|^{q}}{q}.$

######  Theorem 2.5.

Let $f:[\theta_{1},\theta_{2}]\rightarrow\mathbb{R}$ be differentiable
function on $(\theta_{1},\theta_{2})$ with $\kappa_{1}<\theta_{2}$ and
$f^{\prime}\in L_{1}[\theta_{1},\theta_{2}]$. If $|f^{\prime}|$ is a concave
for $q>1$, then we have

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{\alpha+1}\right)\Bigg{[}\left|f^{\prime}\left(\frac{\theta_{1}(\alpha+3)+\theta_{2}(\alpha+1)}{2(\alpha+2)}\right)\right|+\left|f^{\prime}\left(\frac{\theta_{1}(2\alpha+3)+\theta_{2}}{2(\alpha+2)}\right)\right|$
$\displaystyle+\left|f^{\prime}\left(\frac{\theta_{2}(2\alpha+3)+\theta_{1}}{2(\alpha+2)}\right)\right|+\left|f^{\prime}\left(\frac{\kappa_{2}(\alpha+3)+\theta_{1}(\alpha+1)}{2(\alpha+2)}\right)\right|\Bigg{]}$

where $\alpha\in[0,1]$ and $B(\alpha)$ is normalization function.

###### Proof.

From Lemma 2.1 and the Jensen integral inequality, we have

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}\right)\left|f^{\prime}\left(\frac{\int_{0}^{1}(1-\tau_{1})^{\alpha}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}}{\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}}\right)\right|$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}\right)\left|f^{\prime}\left(\frac{\int_{0}^{1}\tau_{1}^{\alpha}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}}{\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}}\right)\right|$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}\right)\left|f^{\prime}\left(\frac{\int_{0}^{1}\tau_{1}^{\alpha}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)d\tau_{1}}{\int_{0}^{1}\tau_{1}^{\alpha}d\tau_{1}}\right)\right|$
$\displaystyle+\left(\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}\right)\left|f^{\prime}\left(\frac{\int_{0}^{1}(1-\tau_{1})^{\alpha}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)d\tau_{1}}{\int_{0}^{1}(1-\tau_{1})^{\alpha}d\tau_{1}}\right)\right|.$

By computing the above integrals we have

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{\alpha+1}\right)\Bigg{[}\left|f^{\prime}\left(\frac{\theta_{1}(\alpha+3)+\theta_{2}(\alpha+1)}{2(\alpha+2)}\right)\right|+\left|f^{\prime}\left(\frac{\theta_{1}(2\alpha+3)+\theta_{2}}{2(\alpha+2)}\right)\right|$
$\displaystyle+\left|f^{\prime}\left(\frac{\theta_{2}(2\alpha+3)+\theta_{1}}{2(\alpha+2)}\right)\right|+\left|f^{\prime}\left(\frac{\kappa_{2}(\alpha+3)+\theta_{1}(\alpha+1)}{2(\alpha+2)}\right)\right|\Bigg{]}.$

So, the proof is completed. ∎

######  Corollary 2.5.

In Theorem 2.5, if we choose $\alpha=1$ we obtain

$\displaystyle\bigg{|}\frac{f(\theta_{1})+f(\theta_{2})+2f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)}{\theta_{2}-\theta_{1}}-\frac{4}{(\kappa_{2}-\theta_{1})^{2}}\int_{\theta_{1}}^{\theta_{2}}f(x)dx\bigg{|}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{4}\right)\Bigg{[}\left|f^{\prime}\left(\frac{2\theta_{1}+\theta_{2}}{3}\right)\right|+\left|f^{\prime}\left(\frac{5\theta_{1}+\theta_{2}}{6}\right)\right|$
$\displaystyle+\left|f^{\prime}\left(\frac{5\theta_{2}+\theta_{1}}{6}\right)\right|+\left|f^{\prime}\left(\frac{2\theta_{2}+\theta_{1}}{3}\right)\right|\Bigg{]}.$

######  Theorem 2.6.

Let $f:[\theta_{1},\theta_{2}]\rightarrow\mathbb{R}$ be differentiable
function on $(\theta_{1},\theta_{2})$ with $\kappa_{1}<\theta_{2}$ and
$f^{\prime}\in L_{1}[\theta_{1},\theta_{2}]$. If $|f^{\prime}|^{q}$ is a
concave function, we have

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\frac{2}{(\alpha
p+1)^{\frac{1}{p}}}\left[\left|f^{\prime}\left(\frac{3\theta_{1}+\theta_{2}}{4}\right)\right|+\left|f^{\prime}\left(\frac{3\theta_{2}+\theta_{1}}{4}\right)\right|\right]$

where $p^{-1}+q^{-1}=1$, $\alpha\in[0,1]$, $q>1$.

###### Proof.

By using the Lemma 2.1 and Hölder integral inequality, we can write

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\left(\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\kappa_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}\tau_{1}^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}$
$\displaystyle+\left(\int_{0}^{1}(1-\tau_{1})^{\alpha
p}d\tau_{1}\right)^{\frac{1}{p}}\left(\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}\right)^{\frac{1}{q}}.$

By using concavity of $|f^{\prime}|^{q}$ and Jensen integral inequality, we
get

$\displaystyle\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\tau_{1}^{0}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)\right|^{q}d\tau_{1}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}\tau_{1}^{0}d\tau_{1}\right)\left|f^{\prime}\left(\frac{\int_{0}^{1}\tau_{1}^{0}\left(\frac{1+\tau_{1}}{2}\theta_{1}+\frac{1-\tau_{1}}{2}\theta_{2}\right)d\tau_{1}}{\int_{0}^{1}\tau_{1}^{0}d\tau_{1}}\right)\right|^{q}$
$\displaystyle=$
$\displaystyle\left|f^{\prime}\left(\frac{3\theta_{1}+\theta_{2}}{4}\right)\right|^{q}.$

Similarly

$\int_{0}^{1}\left|f^{\prime}\left(\frac{1+\tau_{1}}{2}\theta_{2}+\frac{1-\tau_{1}}{2}\theta_{1}\right)\right|^{q}d\tau_{1}\leq\left|f^{\prime}\left(\frac{3\theta_{2}+\theta_{1}}{4}\right)\right|^{q}$

so, we obtain

$\displaystyle\bigg{|}\frac{2(\theta_{2}-\theta_{1})^{\alpha}+(1-\alpha)2^{\alpha+1}\Gamma(\alpha)}{(\theta_{2}-\theta_{1})^{\alpha+1}}\left[f(\kappa_{1})+f(\theta_{2})+2f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]$
$\displaystyle-\frac{2^{\alpha+1}B(\alpha)\Gamma(\alpha)}{(\theta_{2}-\kappa_{1})^{\alpha+1}}\left[{}^{AB}I_{\frac{\theta_{1}+\theta_{2}}{2}}^{\alpha}f(\theta_{1})+_{::::\theta_{1}}^{AB}I^{\alpha}f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)\right.$
$\displaystyle\left.+_{::\frac{\theta_{1}+\theta_{2}}{2}}^{AB}I^{\alpha}f(\theta_{2})+^{AB}I_{\theta_{2}}^{\alpha}f\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\right]\bigg{|}$
$\displaystyle\leq$ $\displaystyle\frac{2}{(\alpha
p+1)^{\frac{1}{p}}}\left[\left|f^{\prime}\left(\frac{3\theta_{1}+\theta_{2}}{4}\right)\right|+\left|f^{\prime}\left(\frac{3\theta_{2}+\theta_{1}}{4}\right)\right|\right].$

∎

######  Corollary 2.6.

In Theorem 2.6, if we choose $\alpha=1$ we obtain

$\displaystyle\bigg{|}\frac{f(\theta_{1})+f(\theta_{2})+2f\left(\frac{\kappa_{1}+\theta_{2}}{2}\right)}{\theta_{2}-\theta_{1}}-\frac{4}{(\kappa_{2}-\theta_{1})^{2}}\int_{\theta_{1}}^{\theta_{2}}f(x)dx\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{(p+1)^{\frac{1}{p}}}\Bigg{[}\left|f^{\prime}\left(\frac{3\theta_{1}+\theta_{2}}{4}\right)\right|+\left|f^{\prime}\left(\frac{3\theta_{2}+\theta_{1}}{4}\right)\right|\Bigg{]}.$

##  3\. Conclusion

In this study, an integral identity including Atangana-Baleanu integral
operators has been proved. Some integral inequalities are established by using
Hölder inequality, Power-mean inequality, Young inequality and convex
functions with the help of Lemma 2.1 which has the potential to produce Bullen
type inequalities. Some special cases of the results in this general form have
been pointed out. Researchers can establish new equations such as the integral
identity in the study and reach similar inequalities of these equality-based
inequalities.

##  4\. Acknowledgments

The publication has been prepared with the support of GNAMPA 2019 and the RUDN
University Strategic Academic Leadership Program.

## Funding

GNAMPA 2019 and the RUDN University Strategic Academic Leadership Program.

## Availability of data and materials

Data sharing is not applicable to this paper as no datasets were generated or
analyzed during the current study.

## Competing interests

The authors declares that there is no conflict of interests regarding the
publication of this paper.

## Author’s contributions

All authors jointly worked on the results and they read and approved the final
manuscript.

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