Inter
national
J
our
nal
of
Adv
ances
in
A
pplied
Sciences
(IJ
AAS)
V
ol.
15,
No.
1,
March
2026,
pp.
1
∼
18
ISSN:
2252-8814,
DOI:
10.11591/ijaas.v15.i1.pp1-18
❒
1
Extension
of
Hermite-Hadamard
type
inequalities
to
Katugampola
fractional
integrals
Dipak
kr
Das
1
,
Shashi
Kant
Mishra
1
,
P
ankaj
K
umar
1
,
Abdelouahed
Hamdi
2
1
Department
of
Mathematics,
Institute
of
Science,
Banaras
Hindu
Uni
v
ersity
,
V
aranasi,
India
2
Department
of
Mathematics
and
Statistics,
Colle
ge
of
Art
and
Science,
CAS,
Qatar
Uni
v
ersity
,
Doha,
Qatar
Article
Inf
o
Article
history:
Recei
v
ed
Jun
8,
2025
Re
vised
Sep
28,
2025
Accepted
No
v
7,
2025
K
eyw
ords:
Exponentially
(
s,
m
)
-con
v
e
x
function
Fractional
calculus
H
¨
o
lder
inequality
Hermite-Hadamard
inequality
Katug
ampola
fractional
inte
gral
Po
wer
-mean
inequality
ABSTRA
CT
In
this
study
,
we
introduce
se
v
eral
ne
w
Hermite-Hadamard
type
general
inte
gral
inequalities
for
e
xponentially
(
s,
m
)
-con
v
e
x
f
unctions
via
Katug
ampola
frac-
tional
inte
gral.
The
Kat
ug
a
mpola
fractional
inte
gral
is
a
broader
form
of
the
Riemann–Liouville
and
Hadamard
fractional
inte
grals.
W
e
utilized
the
po
wer
-
mean
int
e
gr
al
inequality
,
the
H
¨
o
lder
inequality
and
a
fe
w
additional
generaliza-
tions
to
deri
v
e
these
inequalities.
Numerous
limiting
results
are
deri
v
ed
from
the
main
results
presented
in
the
remarks.
Furthermore,
we
pro
vide
an
e
xample
illustrating
our
theoretical
ndings,
supported
by
a
graphical
representation.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Abdelouahed
Hamdi
Department
of
Mathematics
and
Statistics,
Colle
ge
of
Art
and
Science,
CAS,
Qatar
Uni
v
ersity
Doha,
P
.O.
Box
2713,
Qatar
Email:
abhamdi@qu.edu.qa
1.
INTR
ODUCTION
Ov
er
the
past
century
,
the
concept
of
“con
v
e
xity”
has
g
arnered
considerable
interest
among
math-
ematicians.
This
term
has
played
a
signicant
role
and
has
recei
v
ed
remarkable
attention
from
numerous
researchers
in
the
adv
ancement
of
v
arious
elds
within
pure
and
applied
sciences.
In
nancial
mathematics,
mathematical
statistics,
and
functional
analysis,
the
theory
of
con
v
e
xity
holds
signicant
importance.
Con
v
e
x
function
optimization
has
numerous
real-w
orld
uses,
such
as
controller
design,
circuit
design,
and
modeling.
Due
to
its
broad
rele
v
ance
and
man
y
practical
applications,
con
v
e
xity
has
de
v
eloped
into
a
highly
inuential
and
intellectually
eng
aging
area
for
scientists
and
mathematicians.
W
e
encourage
interested
readers
to
see
the
references
[1]-[6]
for
some
discussion
about
con
v
e
xity
and
its
properties.
Inequalities
and
the
property
of
con
v
e
xity
play
a
crucial
role
in
contemporary
mathematical
res
earch.
These
tw
o
concepts
are
inter
related.
Inequalities
are
essential
in
di
v
erse
areas
lik
e
mechanics,
functional
analy-
sis,
probability
theory
,
numerical
methods,
and
statistical
problems.
From
this
perspecti
v
e,
the
eld
of
inequal-
ities
stands
as
an
independent
discipline
in
mathematical
analysis.
F
or
further
details,
see
[7]-[10].
Research
on
mathematical
inequalities
associated
with
fractional
inte
gral
operators,
including
the
Rie-
mann–Liouville
type,
plays
a
k
e
y
role
in
the
eld
of
fractional
calculus.
These
inequalities
play
an
important
role
in
re
v
ealing
the
beha
vior
of
fractional
inte
grals
and
supporting
their
application
in
di
v
erse
areas,
including
ph
ysics,
engineering,
and
mathematics.
The
Riemann-Liouville
fractional
inte
gral
operator
,
which
generalizes
the
traditional
notion
of
inte
gration
to
non-inte
ger
orders,
is
fundamental
in
formulating
these
inequalities.
De-
v
eloping
inequalities
for
Riemann-Liouville
(
R
−
L
)
fractional
inte
grals
allo
ws
researchers
to
establish
strict
J
ournal
homepage:
http://ijaas.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
2
❒
ISSN:
2252-8814
bounds
and
conditions
that
are
essential
for
solving
fractional
dif
ferential
equations,
helping
to
ensure
the
stability
and
rob
ustness
of
solutions
in
systems
where
con
v
entional
calculus
pro
v
es
inadequate,
see
[11]-[14].
By
inte
grating
the
concepts
of
fractional
calculus
with
generalized
con
v
e
xity
,
researchers
can
formu-
late
a
wider
range
of
mathematical
inequalities.
These
inequalities
enhance
the
theoretical
comprehension
of
fractional
inte
grals
and
also
hold
practical
s
ignicance
for
analyzing
and
optimizing
comple
x
systems
in
science
and
engineering.
The
interaction
of
generalized
con
v
e
xity
with
fractional
inte
gral
operators
creates
ne
w
research
opportunities
and
mak
es
it
pos
sible
to
create
stronger
mathematical
frame
w
orks
for
fractional
dynamics
modeling,
analysis,
and
problem
solving.
W
ithin
this
a
nalytical
frame
w
ork,
the
fractional
Hermite-Hadamard
inequalities
e
v
olv
e
as
a
k
e
y
e
x-
tension
of
their
classical
counterparts
into
the
fracti
onal
calculus
domain.
The
classical
Hermite-Hadamard
inequality
gi
v
es
estimates
for
the
inte
gral
of
a
con
v
e
x
function,
and
its
fractional
analogue
generalizes
this
idea
to
the
setting
of
fractional
inte
grals,
particularly
those
dened
by
operators
such
as
the
Riemann–Liouville
inte
gral.
Fractional
Hermite-Hadamard
inequalities
apply
the
principles
of
con
v
e
xity
and
fractional
calculus
to
deri
v
e
bounds
for
the
fractional
inte
grals
of
con
v
e
x
functions.
This
inequality
is
used
in
man
y
dif
ferent
areas
of
economics;
for
e
xample,
the
e
xistence
and
uniqueness
of
certain
economic
models
(such
as
general
equilibrium
models
or
compan
y
beha
vior
models)
ar
e
demonstrated
using
this
inequality
.
It
ca
n
also
play
an
important
role
in
v
arious
areas
of
mathematics,
such
as
number
theory
,
comple
x
analysis,
and
numerical
analysis.
This
in-
equality
can
also
apply
to
information
theory
,
engineering,
ph
ysical
science,
biology
,
and
chemistry
.
According
to
this
inequality
,
let
Φ
1
:
ξ
⊂
R
→
R
be
a
con
v
e
x
function
dened
on
the
interv
al
ξ
(
ξ
⊂
R
)
and
w
1
,
v
1
∈
ξ
with
w
1
<
v
1
.
The
follo
wing
double
inequality
Φ
1
w
1
+
v
1
2
⩽
1
v
1
−
w
1
Z
v
1
w
1
Φ
1
(
η
)
dη
⩽
Φ
1
(
w
1
)
+
Φ
1
(
v
1
)
2
(1)
It
is
kno
wn
as
the
Hermite-Hadamard
inequality
for
a
con
v
e
x
function.
That
is,
in
a
set
of
real
numbers,
if
a
function
is
con
v
e
x,
its
weighted
a
v
erage
v
alue
at
the
endpoints
will
be
equal
to
or
greater
than
its
v
alue
at
the
middle
of
an
y
interv
al.
See
[15]-[18]
for
more
impro
v
ement,
e
xtension
and
generalization
about
this
Inequality
(1).
let
Φ
1
:
ξ
⊂
R
→
R
be
con
v
e
x
function
and
w
1
,
v
1
,
∈
ξ
with
0
≤
w
1
<
v
1
such
that
Φ
1
∈
L
[
w
1
,
v
1
]
.
If
Φ
1
is
con
v
e
x
on
L
[
w
1
,
v
1
]
,
then
the
inequality
Φ
1
w
1
+
v
1
2
⩽
Γ(
ν
+
1)
2(
v
1
−
w
1
)
I
ν
v
1
−
Φ
1
(
w
1
)
+
I
ν
w
1
+
Φ
1
(
v
1
)
⩽
Φ
1
(
w
1
)
+
Φ
1
(
v
1
)
2
(2)
with
ν
>
0
,
is
kno
wn
as
the
fractional
Hermite-Hadamard
inequality
,
where
I
+
w
1
and
I
−
v
1
stand
for
the
right-sided
and
the
left-sided
Riemann-Liouville
fractional
inte
grals
of
the
order
ν
.
It
is
note
w
orth
y
that
the
fractional
Hermite-Hadamard
inequality
simplies
to
the
class
ical
Hermite-Hadamard
inequality
when
ν
=
1
in
Inequality
(2).
Researchers
can
create
more
sophisticated
methods
for
e
xamining
and
enhancing
systems
that
e
xhibit
fractional
dynamic
beha
vior
by
connecting
fractional
Hermite-Hadamard
inequalities
to
the
broader
frame
w
ork
of
fractional
inte
gral
operators
and
generalized
con
v
e
xity
.
The
interaction
of
these
ideas
enables
the
creation
of
ne
w
,
more
accurate
mathematical
inequalities
that
can
be
used
in
a
v
ariety
of
intricate
sys-
tems.
This
inte
grated
method
pro
vides
deeper
insights
into
t
he
mathematical
processes
controlling
fractional
calculus
and
its
applications
in
science
and
engineering,
opening
up
ne
w
a
v
enues
for
research;
for
more
details,
see
references
[19]-[21].
T
o
the
best
of
our
kno
wledge,
this
paper
pro
vides
a
no
v
el
and
in-depth
analysis
of
e
xpon
e
ntially
(
s,
m
)
-con
v
e
x
functions
about
Katug
ampola
fractional
inte
grals.
Antczak
[22]
introduced
the
notion
of
e
xpo-
nentially
con
v
e
x
functions,
which
can
be
seen
as
a
substantial
generalization
of
con
v
e
x
functions.
Exponentially
con
v
e
x
functions
play
a
signicant
role
in
di
v
erse
areas,
including
mathematical
programming,
information
ge-
ometry
,
big
data
analysis,
machine
learning,
statistics,
sequential
prediction,
and
stochastic
o
pt
imization
[23],
[24].
Moreo
v
er
,
Rashid
et
al
.
[25]
established
some
trapezoid-type
i
n
e
qu
a
lities
for
generalized
fractional
in-
te
grals
and
related
inequalities
via
e
xponentially
con
v
e
x
functions.
Rashid
et
al
.
[26]
deri
v
ed
a
ne
w
inte
gral
identity
in
v
olving
Riemann–Liouville
fractional
inte
grals
and
obtained
ne
w
fractional
bounds
for
the
functions
ha
ving
the
e
xponential
con
v
e
xity
property
.
Rashid
et
al
.
[27]
introduced
some
ne
w
generalizations
for
e
xponen-
tially
s
-con
v
e
x
functions
and
inequalities
via
fractional
operators.
Recognizing
the
signicance
of
fractional
Int
J
Adv
Appl
Sci,
V
ol.
15,
No.
1,
March
2026:
1–18
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Adv
Appl
Sci
ISSN:
2252-8814
❒
3
inte
grals
in
numerous
areas
of
pure
and
applied
science,
researchers
ha
v
e
e
xpanded
their
concept
in
v
arious
w
ays,
which
leads
to
the
de
v
elopment
of
ne
w
inte
gral
inequalities
for
these
generalized
fractional
inte
grals.
Recently
,
Khan
et
al
.
[28]
established
some
ne
w
de
v
elopments
of
Hermite–Hadamard-type
i
nequal-
ities
via
s
-con
v
e
xity
and
fractional
inte
grals.
Inspired
by
the
mentioned
research
ef
fort
and
notions,
we
study
the
concept
of
e
xponentially
(
s,
m
)
-con
v
e
xity
to
deri
v
e
inequalities
of
fractional
Hermite-Hadamard
type
e
x-
ponentially
(
s,
m
)
-con
v
e
x
functions
and
some
generalizations
associ
ated
with
these
inequalities.
The
primary
object
i
v
e
of
this
study
is
to
de
v
elop
Hermite-Hadamard
type
inequalities
to
Kat
ug
am
pola
fractional
inte
grals.
T
o
this
end,
we
rst
introduce
a
ne
w
inte
gral
identity
on
which
we
base
the
establishment
of
se
v
eral
Hermite-Hadamard
type
inequalities
for
functions
with
e
xtended
(
s,
m
)-con
v
e
x
rst-order
deri
v
a-
ti
v
es.
Subsequently
,
we
present
an
e
xample
that
includes
graphical
repre
sentations
to
conrm
the
v
alidity
of
our
results.
Hermite-Hadamard
inequalities
are
po
werful
tools
for
establishing
bounds
for
symmetric
e
xpres-
sions.
When
paired
with
(
s,
m
)-con
v
e
x
functions,
these
inequalities
become
more
v
ersatile
and
yield
sharper
estimates.
The
proposed
w
ork
is
structured
as
follo
ws:
In
section
2,
we
gi
v
e
some
essential
denitions,
impor
-
tant
theorems,
and
a
generalized
lemma
that
are
required
for
our
major
res
ult.
In
section
3,
we
state
and
pro
v
e
our
k
e
y
results
utilizing
the
generalized
lemma
and
theorems,
as
well
as
deri
ving
se
v
eral
ne
w
corollaries
and
gi
ving
some
important
remarks.
In
section
4,
a
conclusion
is
dra
wn.
2.
RESEARCH
METHOD
In
this
section,
we
collect
some
notations,
basic
denitions
and
essential
results
required
in
the
sequel
of
the
paper
.
Denition
1.
[29]
A
function
Φ
1
:
I
→
R
is
said
to
be
a
con
v
e
x
function
if
Φ
1
(
η
w
1
+
(1
−
η
)
v
1
)
⩽
η
Φ
1
(
w
1
)
+
(1
−
η
)
Φ
1
(
v
1
)
holds
for
all
w
1
,
v
1
∈
I
and
η
∈
[0
,
1]
.
Denition
2.
[30]
Φ
1
:
[0
,
b
]
→
R
is
said
to
be
a
m-con
v
e
x
function
if
Φ
1
(
η
w
1
+
m
(1
−
η
)
v
1
)
≤
η
Φ
1
(
w
1
)
+
m
(1
−
ν
)
Φ
1
(
v
1
)
holds
for
all
w
1
,
v
1
∈
[0
,
b
]
,
η
∈
[0
,
1]
,
and
m
∈
(0
,
1]
.
In
[30],
T
oader
introduced
the
abo
v
e
concept
of
an
m
-con
v
e
x
function.
Denition
3.
[31]
Φ
1
:
[0
,
b
]
→
R
is
said
to
be
a
s
-con
v
e
x
function
if
Φ
1
(
η
w
1
+
(1
−
η
)
v
1
)
≤
η
s
Φ
1
(
w
1
)
+
(1
−
η
)
s
Φ
1
(
v
1
)
holds
for
all
w
1
,
v
1
∈
[0
,
b
]
,
η
∈
[0
,
1]
,
and
s
∈
(0
,
1]
.
Denition
4.
[32]
A
function
Φ
1
:
[0
,
η
]
→
R
is
said
to
be
an
(
s,
m
)
-con
v
e
x
function,
where
(
s,
m
)
∈
[0
,
1]
2
and
η
>
0
,
if
f
∀
w
1
,
v
1
∈
[0
,
η
]
and
η
∈
[0
,
1]
if
Φ
1
η
w
1
+
m
(1
−
η
)
v
1
≤
η
s
Φ
1
(
w
1
)
+
m
(1
−
η
)
s
Φ
1
(
v
1
)
Denition
5.
A
positi
v
e
real-v
alued
function
Φ
1
:
I
⊆
R
→
(0
,
∞
)
is
said
to
be
e
xponentially
con
v
e
x
on
K
,
if
e
Φ
1
(
η
w
1
+(1
−
η
)
v
1
)
≤
η
e
Φ
1
(
k
1
)
+
(1
−
η
)
e
Φ
1
(
v
1
)
Exponentially
con
v
e
x
functions
are
utilized
in
statistical
learning,
sequential
prediction,
and
stochastic
opti-
mization.
Denition
6.
[33]
A
function
Φ
1
:
I
→
R
is
said
to
be
a
e
xponentially
s
-
con
v
e
x
f
un
c
tion
in
the
rst
sence,
if
the
follo
wing
inequality
holds:
e
Φ
1
(
w
1
η
+
v
1
(1
−
η
))
≤
η
s
e
Φ
1
(
w
1
)
+
(1
−
η
)
s
e
Φ
1
(
v
1
)
,
∀
s
∈
[0
,
1]
,
w
1
,
v
1
∈
I
,
η
∈
[0
,
1]
.
Extension
of
Hermite-Hadamar
d
type
inequalities
to...
(Dipak
Kr
Das)
Evaluation Warning : The document was created with Spire.PDF for Python.
4
❒
ISSN:
2252-8814
F
or
η
=
1
2
,
we
get
e
Φ
1
(
w
1
+
v
1
2
)
≤
e
Φ
1
(
w
1
)+Φ
1
(
v
1
)
2
s
,
∀
w
1
,
v
1
∈
I
,
which
is
called
e
xponentially
Jensen-con
v
e
x
function.
Denition
7.
Let
s
∈
[0
,
1]
and
I
⊆
[0
,
∞
)
.
A
function
Φ
:
I
→
R
is
said
to
b
e
e
xponentially
(
s
,
m
)
−
con
v
e
x
function
in
the
second
sense
if
Φ
1
(
η
w
1
+
m
(1
−
η
)
v
2
)
≤
η
s
Φ
1
(
w
1
)
e
ξ
1
w
1
+
m
(1
−
η
)
s
Φ
1
(
w
1
)
e
ξ
1
v
1
,
holds
for
all
w
1
,
v
1
∈
I
,
m
∈
[0
,
1]
and
ξ
1
∈
R
.
Denition
8.
[33]
Let
Φ
1
∈
£
v
1
w
1
(
w
1
,
v
1
)
.
The
left
and
right-sided
Katug
ampola
fractional
inte
grals
of
order
α
∈
C
with
R
e
(
α
)
>
0
and
σ
′
>
0
are
dened
by
σ
′
I
ν
w
+
1
Φ
1
(
x
)
=
σ
′
(1
−
ν
)
Γ(
µ
)
Z
x
w
1
η
σ
′
−
1
Φ
1
(
η
)
(
x
σ
′
−
η
σ
′
)
1
−
ν
dη
,
x
>
w
1
,
σ
′
I
ν
v
−
1
Φ
1
(
x
)
=
σ
′
(1
−
ν
)
Γ(
ν
)
Z
v
1
x
η
σ
′
−
1
Φ
1
(
η
)
(
η
σ
′
−
x
σ
′
)
1
−
ν
dη
,
v
1
>
x.
Where
£
v
1
w
1
(
w
1
,
v
1
)(
w
1
∈
R
and
1
≤
v
1
≤
∞
)
denotes
the
space
of
all
comple
x-v
alued
Lebesgue
measurable
functions
Φ
1
for
which
||
Φ
1
||
£
v
1
w
1
<
∞
and
the
norm
is
dened
by
||
Φ
1
||
£
v
1
w
1
=
Z
v
1
w
1
|
η
w
1
g
(
η
)
|
v
1
1
v
1
f
or
1
≤
v
1
<
∞
and
for
r=
∞
||
Φ
1
||
£
∞
w
1
=
ess
sup
w
1
≤
η
≤
v
1
|
η
w
1
Φ
1
(
η
)
|
.
Denition
9.
[34]
Let
Φ
1
∈
L
′
[
w
1
,
v
1
]
.
The
fractional
Riemann-Liouville
inte
grals
ȷ
ν
w
+
1
Φ
1
and
ȷ
ν
v
−
1
Φ
1
of
order
ν
are
dened
by
ȷ
ν
w
+
1
Φ
1
(
x
)
=
1
Γ(
ν
)
Z
x
w
1
(
x
−
η
)
ν
−
1
Φ
1
(
η
)
dη
,
x
>
w
1
,
ȷ
ν
v
−
1
g
2
(
x
)
=
1
Γ(
ν
)
Z
v
1
x
(
η
−
x
)
ν
−
1
Φ
1
(
η
)
dη
,
v
1
>
x,
where
Γ(
ν
)
is
a
g
amma
function.
Denition
10.
[35]
The
left
and
right-sided
Hadamard’
s
fractional
inte
gral
operators
of
order
ν
>
0
are
dened
by
H
ν
w
+
1
Φ
1
(
x
)
=
1
Γ(
ν
)
Z
x
w
1
(
l
n
x
−
l
n
η
)
ν
−
1
Φ
1
(
η
)
η
dη
,
x
>
w
1
,
H
ν
v
−
1
g
2
(
x
)
=
1
Γ(
ν
)
Z
v
1
x
(
l
n
η
−
l
n
x
)
ν
−
1
Φ
1
(
η
)
η
dη
,
v
1
>
x.
Int
J
Adv
Appl
Sci,
V
ol.
15,
No.
1,
March
2026:
1–18
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Adv
Appl
Sci
ISSN:
2252-8814
❒
5
Theor
em
1.
Let
ν
>
0
and
σ
′
>
0
.
Then,
for
x
>
w
1
,
1.
l
im
σ
′
σ
′
→
1
ȷ
ν
w
+
1
Φ
1
(
x
)
=
I
ν
w
−
1
Φ
1
(
x
)
,
2.
lim
σ
′
→
0
+
ȷ
ν
w
+
1
Φ
1
(
x
)
=
H
ν
w
−
1
Φ
1
(
x
)
W
e
recall
the
special
functions
that
are
kno
wn
as
Gamma
function
and
beta
function,
respecti
v
ely
.
Γ(
x
)
=
Z
∞
0
e
−
η
η
x
−
1
dη
,
B
(
x,
y
)
=
Z
1
0
η
x
−
1
(1
−
η
)
(
y
−
1)
dη
=
Γ(
x
)Γ(
y
)
Γ(
x
+
y
)
,
x,
y
>
0
,
where
B
(
.,
.
)
denotes
beta
as
a
special
functi
on.
The
Katug
ampola
fractional
inte
gral
is
a
po
werful
fractional
calculus
tool
that
unies
both
the
Ri
e-
mann–Liouville
and
Hadamard
fractional
inte
grals.
This
w
ork
presents
a
Hermite-Hadamard
type
inequality
that
incorporates
the
properties
of
e
xponentially
(
s,
m
)
con
v
e
x
functions
to
e
xtend
and
strengthen
classical
results.
3.
RESUL
TS
AND
DISCUSSION
In
this
section,
we
rst
establish
an
identity
in
v
olving
the
Katug
ampola
fractional
inte
gral.
Then
we
establish
an
inte
gral
inequality
in
v
olving
beta
function.
W
e
be
gin
with
the
follo
wing
lemma
which
is
used
to
e
xplore
inte
gral
inequality
.
Lemma
2.
Let
σ
′
>
0
,
ν
>
0
and
Φ
1
:
[
w
σ
′
1
,
v
σ
′
1
]
→
R
be
dif
ferentiable
function
on
[
w
σ
′
1
,
v
σ
′
1
]
with
0
≤
w
σ
′
1
<
v
σ
′
1
,
and
Φ
1
′
∈
L
1
[
w
σ
′
1
,
v
σ
′
1
]
.
Then
the
follo
wing
equality
holds:
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
=
(
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
))
2
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
dη
+
(
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
))
2
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
dη
.
Pr
oof
Let
H
1
=
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
dη
,
and
H
2
=
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
dη
.
I
nt
eg
r
eating
by
par
ts
H
1
,
w
e
g
et
H
1
=
η
σ
′
ν
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
1
0
−
ν
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
Z
1
0
η
σ
′
ν
−
1
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
dη
=
Φ
1
(
k
σ
′
1
)
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
−
ν
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+1
Z
k
1
mw
1
(
u
σ
′
−
m
σ
′
w
σ
′
1
)
ν
−
1
u
ν
−
1
Φ
1
(
u
σ
′
)
du
=
Φ
1
(
k
σ
′
1
)
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
−
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+1
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
.
N
ow
,
(
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
))
H
1
=
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
.
(3)
Extension
of
Hermite-Hadamar
d
type
inequalities
to...
(Dipak
Kr
Das)
Evaluation Warning : The document was created with Spire.PDF for Python.
6
❒
ISSN:
2252-8814
Similarly
,
we
ha
v
e
H
2
=
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
dη
=
η
σ
′
ν
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
1
0
−
ν
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
Z
1
0
η
σ
′
ν
−
1
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
dη
=
Φ
1
(
k
σ
′
1
)
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
+
ν
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
+1
Z
mv
1
k
1
(
m
σ
′
v
σ
′
1
−
u
σ
′
)
ν
−
1
u
ν
−
1
Φ
1
(
u
σ
′
)
du
=
Φ
1
(
k
σ
′
1
)
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
+1
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
.
N
ow
,
(
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
))
H
2
=
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
.
(4)
No
w
adding
(3)
and
(4),
we
get
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
=
(
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
))
2
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
dη
+
(
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
))
2
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
dη
.
Thus
the
proof
is
completed.
Remark
1
.
If
we
put
m
=
σ
′
=
1
in
Lemma
2,
then
we
get
Lemma
(2)
of
[28].
Theor
em
3.
Let
σ
′
>
0
,
ν
>
0
and
Φ
1
:
[
w
σ
′
1
,
v
σ
′
1
]
→
R
be
a
positi
v
e
function
with
0
≤
w
σ
′
1
<
k
σ
′
1
<
v
σ
′
1
,
and
Φ
1
′
∈
L
1
[
w
σ
′
1
,
v
σ
′
1
]
.
If
Φ
1
is
e
xponentially
(
s,
m
)
-con
v
e
x
function
on
[
w
σ
′
1
,
v
σ
′
1
]
,
then
the
follo
wing
inequality
for
fractional
inte
grals
holds:
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
≤
ν
σ
′
2
B
(
ν
+
s,
1)
Φ
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
+
B
(
ν
,
s
+
1)
m
σ
′
Φ
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
+
Φ
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
,
for
all
m,
s
∈
(0
,
1]
and
ξ
1
∈
R
.
Pr
oof
Applying
e
xponentially
(
s,
m
)
con
v
e
xity
of
Φ
1
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
+
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
≤
η
σ
′
s
Φ
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
s
Φ
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
+
η
σ
′
s
Φ
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
s
Φ
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
(5)
Int
J
Adv
Appl
Sci,
V
ol.
15,
No.
1,
March
2026:
1–18
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Adv
Appl
Sci
ISSN:
2252-8814
❒
7
Multiply
both
sides
of
(5)
by
η
σ
′
ν
−
1
and
inte
grate
w
.r
.t
η
o
v
er
[0,1]
Z
1
0
η
σ
′
ν
−
1
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
dη
+
Z
1
0
η
σ
′
ν
−
1
Φ
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
dη
≤
2
Z
1
0
η
σ
′
ν
−
1
η
σ
′
s
Φ
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
dη
+
Z
1
0
η
σ
′
ν
−
1
m
σ
′
(1
−
η
σ
′
)
s
Φ
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
dη
+
Z
1
0
η
σ
′
ν
−
1
m
σ
′
(1
−
η
σ
′
)
s
Φ
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
dη
(6)
=
2
σ
′
(
ν
+
s
)
Φ
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
+
m
σ
′
σ
′
Φ
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
+
Φ
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
,
s
+
1)
.
No
w
consider
t
σ
′
=
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
in
rst
term
of
L.H.S
of
(6)
and
r
σ
′
=
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
in
second
term
of
L.H.S
of
(6).
Z
k
1
mw
t
σ
′
−
m
σ
′
w
σ
′
1
k
σ
′
1
−
m
σ
′
w
σ
′
1
ν
−
1
t
σ
′
−
1
Φ
1
(
t
σ
′
)
dt
k
σ
′
1
−
m
σ
′
w
σ
′
1
+
Z
mw
k
1
m
σ
′
v
σ
′
1
−
r
σ
′
m
σ
′
v
σ
′
1
−
k
σ
′
1
ν
−
1
r
σ
′
−
1
Φ
1
(
r
σ
′
)
dr
m
σ
′
v
σ
′
1
−
k
σ
′
1
≤
Φ
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
2
σ
′
(
ν
+
s
)
+
m
σ
′
σ
′
Φ
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
+
Φ
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
,
s
+
1)
.
B
y
mul
tipl
y
ing
both
sides
by
ν
,
w
e
g
et
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
≤
ν
σ
′
2
B
(
ν
+
s,
1)
Φ
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
+
m
σ
′
Φ
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
+
Φ
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
,
s
+
1)
.
Thus
the
proof
is
completed.
Remark
2
.
If
we
put
m
=
σ
′
=
1
and
ξ
1
=
0
in
Theorem
3,
then
we
get
Theorem
(4)
of
[28].
In
[28]
at
Theorem
(4),
researchers
use
the
concept
of
(
R
−
L
)
fractional
Hermite-Hadamard
inte
gral
inequality
by
using
s
−
con
v
e
xity
.
In
Theorem
3,
we
e
xtended
the
result
using
the
Katug
ampola
fractional
Hermite-Hadamard
inte
gral
inequality
with
e
xponential
(
s,
m
)
−
con
v
e
xit
y
,
which
becomes
more
v
ersatile
and
yields
sharper
estimates.
F
or
better
understanding,
we
pro
vide
an
e
xample
illustrating
our
theoretical
ndings,
supported
by
a
graphical
representation.
Example
1.
If
we
choose
s
=
m
=
1
∈
(0
,
1]
,
ξ
1
=
0
∈
R
,
σ
′
=
1
,
η
=
1
2
∈
[0
,
1]
,
k
1
=
5
2
,
and
v
1
=
3
in
Theorem
3,
then
Φ
1
(
t
)
=
t
4
is
an
e
xponentially
(
s,
m
)
-con
v
e
x,
as
Theorem
3
satisfying
the
follo
wing
estimation:
195
.
31
−
2
w
5
1
25
−
10
w
1
+
58
.
13
≤
39
.
06
+
w
4
1
+
81
2
(7)
W
e
ha
v
e
sho
wn
the
graphical
representation
of
Inequality
(7)
using
MA
TLAB
R2019a
softw
are.
Remark
3
.
From
Figure
1,
we
observ
e
that
in
the
Inequality
(7),
the
left
hand
side
gi
v
es
more
accurate
estimate
than
the
right
hand
side
of
Theorem
3
graphically
.
In
Figure
1,
the
v
ertical
axis
is
represented
by
w
,
and
the
horizontal
axis
is
represented
by
w
1
.
Extension
of
Hermite-Hadamar
d
type
inequalities
to...
(Dipak
Kr
Das)
Evaluation Warning : The document was created with Spire.PDF for Python.
8
❒
ISSN:
2252-8814
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
50
100
150
200
250
300
350
400
Figure
1.
Graphical
description
of
Inequality
(7),
which
pro
vides
better
understanding
and
v
alidity
of
Theorem
3
Cor
ollary
1.
If
we
choose
s
=
1
in
Theorem
3,
we
deduc
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
≤
ν
σ
′
2
B
(
ν
+
1
,
1)Φ
1
(
k
σ
′
1
)
e
−
(
ξ
1
k
σ
′
1
)
+
B
(
ν
,
2)
m
σ
′
Φ
1
(
w
σ
′
1
)
e
−
(
ξ
1
w
σ
′
1
)
+
Φ
1
(
v
σ
′
1
)
e
−
(
ξ
1
v
σ
′
1
)
Cor
ollary
2.
If
we
choose
s
=
0
in
Theorem
3,
we
deduce
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
≤
1
σ
′
2Φ
1
(
k
σ
′
1
)
e
−
(
ξ
1
k
σ
′
1
)
+
m
σ
′
Φ
1
(
w
σ
′
1
)
e
−
(
ξ
1
w
σ
′
1
)
+
Φ
1
(
v
σ
′
1
)
e
−
(
ξ
1
v
σ
′
1
)
Cor
ollary
3.
If
we
choose
ξ
1
=
0
in
Theorem
3,
we
deduce
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
≤
ν
σ
′
2
B
(
ν
+
s,
1)Φ
1
(
k
σ
′
1
)
+
B
(
ν
,
s
+
1)
m
σ
′
Φ
1
(
w
σ
′
1
)
+
Φ
1
(
v
σ
′
1
)
,
Cor
ollary
4.
If
we
choose
m
=
1
in
Theorem
3,
we
deduce
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
w
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
v
σ
′
1
)
≤
ν
σ
′
2
B
(
ν
+
s,
1)Φ
1
(
k
σ
′
1
)
e
−
(
ξ
1
k
σ
′
1
)
+
B
(
ν
,
s
+
1)
Φ
1
(
w
σ
′
1
)
e
−
(
ξ
1
w
σ
′
1
)
+
Φ
1
(
v
σ
′
1
)
e
−
(
ξ
1
v
σ
′
1
)
Theor
em
4.
Let
σ
′
>
0
,
ν
>
0
and
Φ
1
:
I
→
R
be
a
dif
ferentiable
mapping
on
I
o
,
and
w
σ
′
1
,
v
σ
′
1
∈
I
o
with
w
σ
′
1
<
k
σ
′
1
<
v
σ
′
1
such
that
Φ
1
′
∈
L
1
[
w
σ
′
1
,
v
σ
′
1
]
.
If
Φ
′
1
is
e
xponentially
(
s,
m
)
-con
v
e
x
function
on
[
w
σ
′
1
,
v
σ
′
1
]
,
then
the
follo
wing
inequality
for
fractional
inte
grals
holds:
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
≤
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
s
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
B
(
ν
+
1
,
s
+
1)
+
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
s
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
+
1
,
s
+
1)
,
holds
∀
m,
s
∈
(0
,
1]
and
ξ
1
∈
R
.
Int
J
Adv
Appl
Sci,
V
ol.
15,
No.
1,
March
2026:
1–18
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Adv
Appl
Sci
ISSN:
2252-8814
❒
9
pr
oof
By
taking
absolute
v
alue
in
Lemma
2,
we
deduce
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
≤
(
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
))
2
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
w
σ
′
1
)
dη
+
(
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
))
2
Z
1
0
η
σ
′
ν
η
σ
′
−
1
Φ
′
1
(
η
σ
′
k
σ
′
1
+
m
σ
′
(1
−
η
σ
′
)
v
σ
′
1
)
dη
,
since
|
Φ
′
1
|
is
exponential
l
y
(
s,
m
)
con
v
e
xity
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
≤
(
σ
′
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
))
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
Z
1
0
η
σ
′
(
ν
+
s
+1)
−
1
dη
+
m
σ
′
Φ
′
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
Z
1
0
η
σ
′
(
ν
+1)
−
1
(1
−
η
σ
′
)
s
dν
+
(
σ
′
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
))
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
Z
1
0
η
σ
′
(
ν
+
s
+1)
−
1
dη
+
m
σ
′
Φ
′
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
Z
1
0
η
σ
′
(
ν
+1)
−
1
(1
−
η
σ
′
)
s
dν
=
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
s
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
B
(
ν
+
1
,
s
+
1)
+
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
s
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
+
1
,
s
+
1)
.
Thus
the
proof
is
completed.
Remark
4
.
If
we
put
m
=
σ
′
=
1
and
ξ
1
=
0
in
Theorem
4,
then
we
get
Theorem
(5)
of
[28].
In
[28]
at
Theorem
(5),
researchers
use
the
concept
of
(
R
−
L
)
fracti
onal
Hermite-Hadamard
inte-
gral
inequality
by
using
s
−
con
v
e
xity
and
the
property
of
modulus.
In
Theorem
4,
we
e
xtended
the
result
using
Katug
ampola
fractional
Hermite-Hadamard
inte
gral
inequality
with
e
xponential
(
s,
m
)
−
con
v
e
xity
and
the
property
of
modulus,
which
becomes
more
v
ersatile
and
yields
sharper
estimates.
F
or
bet
ter
understanding,
we
pro
vide
an
e
xample
illustrating
our
theoretical
ndings,
supported
by
a
graphical
representation.
Example
2.
If
we
choose
s
=
m
=
1
∈
(0
,
1]
,
ξ
1
=
0
∈
R
,
σ
′
=
1
,
η
=
1
2
∈
[0
,
1]
,
k
1
=
5
2
,
and
v
1
=
3
in
Theorem
4,
then
Φ
1
(
t
)
=
t
4
is
an
e
xponentially
(
s,
m
)
-con
v
e
x,
as
Theorem
4
satisfying
the
follo
wing
estimation:
10
−
195
.
31
−
2
w
5
1
50
−
20
w
1
≤
5
−
2
w
1
4
20
.
83
+
2
3
w
3
1
−
9
.
70
.
(8)
W
e
ha
v
e
sho
wn
the
graphical
representation
of
Inequality
(8)
using
MA
TLAB
R2019a
softw
are.
Remark
5
.
From
Figure
2,
we
observ
e
that
in
the
Inequality
(8),
the
left
hand
side
gi
v
es
more
accurate
estimate
than
the
right
hand
side
of
Theorem
4
graphically
.
In
Figure
2,
the
v
ertical
axis
is
represented
by
w
,
and
the
horizontal
axis
is
represented
by
w
1
.
Extension
of
Hermite-Hadamar
d
type
inequalities
to...
(Dipak
Kr
Das)
Evaluation Warning : The document was created with Spire.PDF for Python.
10
❒
ISSN:
2252-8814
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
4
6
8
10
12
14
16
Figure
2.
Graphical
description
of
Inequality
(8),
which
pro
vides
better
understanding
and
v
alidity
of
Theorem
4
Cor
ollary
5.
If
we
choose
s
=
1
in
Theorem
4,
we
deduce
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
≤
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
2)
,
1)
+
m
σ
′
Φ
′
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
B
(
ν
+
1
,
2)
+
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
2)
,
1)
+
m
σ
′
Φ
′
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
+
1
,
2)
.
Cor
ollary
6.
If
we
choose
s
=
0
in
Theorem
4,
we
deduce
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
≤
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
B
(
ν
+
1
,
1)
+
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
+
1
,
1)
.
Cor
ollary
7.
If
we
choose
ξ
1
=
0
in
Theorem
4,
we
deduce
Φ
1
(
k
σ
′
1
)
−
(
σ
′
)
ν
Γ(
ν
+
1)
2
σ
′
I
ν
k
−
1
Φ
1
(
m
σ
′
w
σ
′
1
)
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
ν
+
σ
′
I
ν
k
+
1
Φ
1
(
m
σ
′
v
σ
′
1
)
(
m
σ
′
v
σ
′
1
−
k
σ
′
1
)
ν
≤
(
k
σ
′
1
−
m
σ
′
w
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
B
((
ν
+
s
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
w
σ
′
1
)
B
(
ν
+
1
,
s
+
1)
+
(
k
σ
′
1
−
m
σ
′
v
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
B
((
ν
+
s
+
1)
,
1)
+
m
σ
′
Φ
′
1
(
v
σ
′
1
)
B
(
ν
+
1
,
s
+
1)
.
Cor
ollary
8.
If
we
choose
m
=
1
in
Theorem
4,
we
deduce
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
k
σ
′
1
−
w
σ
′
1
)
ν
σ
′
I
ν
k
−
1
Φ
1
(
w
σ
′
1
)
+
(
σ
′
)
ν
−
1
Γ(
ν
+
1)
(
v
σ
′
1
−
k
σ
′
1
)
ν
σ
′
I
ν
k
+
1
Φ
1
(
v
σ
′
1
)
≤
(
k
σ
′
1
−
w
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
s
+
1)
,
1)
+
Φ
′
1
(
w
σ
′
1
)
e
ξ
1
w
σ
′
1
B
(
ν
+
1
,
s
+
1)
+
(
k
σ
′
1
−
v
σ
′
1
)
2
Φ
′
1
(
k
σ
′
1
)
e
ξ
1
k
σ
′
1
B
((
ν
+
s
+
1)
,
1)
+
Φ
′
1
(
v
σ
′
1
)
e
ξ
1
v
σ
′
1
B
(
ν
+
1
,
s
+
1)
.
Int
J
Adv
Appl
Sci,
V
ol.
15,
No.
1,
March
2026:
1–18
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