Novel Mean-Type Inequalities via Generalized Riemann-Type Fractional Integral for Composite Convex Functions: Some Special Examples

Ci a ion: Mukh a , M.; Yaqoob, M.;
Sam aiz, M.; Shabbi , I.; E emad, S.;
De la Sen, M.; Rezapou , S. No el
Mean-Type Inequali ies ia
Gene alized Riemann-Type
F ac ional In eg al o Composi e
Con ex Func ions: Some Special
Examples. Symme y 2023,15, 479.
h ps://dx.doi.o g/10.3390/
sym15020479
Academic Edi o s: Nicuso Mincule e
and Shige u Fu uichi
Recei ed: 22 Janua y 2023
Re ised: 4 Feb ua y 2023
Accep ed: 7 Feb ua y 2023
Published: 10 Feb ua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
symme y
S
S
A icle
No el Mean-Type Inequali ies ia Gene alized Riemann-Type
F ac ional In eg al o Composi e Con ex Func ions: Some
Special Examples
Muzammil Mukh a 1, Muhammad Yaqoob 1, Muhammad Sam aiz 2, I am Shabbi 1, Sina E emad 3,∗,
Manuel De la Sen 4,∗and Shah am Rezapou 3,5,6,∗
1Depa men o Ma hema ics, The Islamia Uni e si y o Bahawalpu , Bahawalnaga Campus,
Bahawalnaga 63100, Pakis an
2Depa men o Ma hema ics, Uni e si y o Sa godha, Sa godha 40100, Pakis an
3Depa men o Ma hema ics, Aza baijan Shahid Madani Uni e si y, Tab iz 3751-71379, I an
4Ins i u e o Resea ch and De elopmen o P ocesses, Depa men o Elec ici y and Elec onics,
Facul y o Science and Technology, Uni e si y o he Basque Coun y (UPV/EHU), 48940 Leioa, Bizkaia, Spain
5Depa men o Ma hema ics, Kyung Hee Uni e si y, 26 Kyungheedae- o, Dongdaemun-gu,
Seoul 02447, Republic o Ko ea
6Depa men o Medical Resea ch, China Medical Uni e si y Hospi al, China Medical Uni e si y,
Taichung 40402, Taiwan
*Co espondence: sina.e emad@aza uni .ac.i (S.E.); [email p o ec ed] (M.D.l.S.);
sh. ezapou @aza uni .ac.i (S.R.)
Abs ac :
This s udy deals wi h a no el class o mean- ype inequali ies by employing ac ional
calculus and con exi y heo y. The high co ela ion be ween symme y and con exi y inc eases i s
signi icance. In his pape , we i s es ablish an iden i y ha is c ucial in in es iga ing ac ional
mean inequali ies. Then, we es ablish he main esul s in ol ing he e o es ima ion o he He mi e–
Hadama d inequali y o composi e con ex unc ions ia a gene alized Riemann- ype ac ional
in eg al. Such esul s a e e i ied by choosing ce ain composi e unc ions. These esul s gi e well-
known examples in special cases. The main consequences can gene alize many known inequali ies
ha exis in o he s udies.
Keywo ds: mean inequali ies; ac ional in eg al; Hölde ’s inequali y; Minkowski inequali y
MSC: 26A33; 35J05
1. In oduc ion
F ac ional calculus has wide applica ion in ma hema ics as well as in many o he
ields o he mode n sciences, such as bio-enginee ing [
1
–
3
], biological memb anes [
4
],
medicine [
5
–
7
], geophysics [
8
], demog aphy [
9
], he economy [
10
], physics [
11
] and also
in signal p ocessing. O e he pas ew decades, scien is s ha e paid a en ion o he
ac ional heo y o calculus and in es iga ed and modeled many physical eal phenomena
using ac ional calculus heo y; o ins ance, ac ional applica ions in epidemiology [12],
he A angana-Baleanu e sion o ope a o s in con ex analysis [
13
], impulsi e Lange in
equa ions in ac ional se ings [
14
], he applica ion o ac ional ope a o s in inclusion
heo y [
15
–
17
], quan um calculus [
18
], a iable o de ac ional enginee ing models based
on he mos a con ol [19], e c.
Ma hema ical inequali ies p o ide boundedness and uniqueness o solu ions o bound-
a y alue p oblems, so hey ha e became he backbone o ma hema ical me hods. Due
o hei as use in he ield o ma hema ics as well as in o he mode n ields o science,
hei need and impo ance ha e inspi ed ma hema icians o u n o mo e gene alized and
ad anced inequali ies [
20
–
22
]. Addi ionally, his g oup o inequali ies has been applied
Symme y 2023,15, 479. h ps://doi.o g/10.3390/sym15020479 h ps://www.mdpi.com/jou nal/symme y
Symme y 2023,15, 479 2 o 18
in mos s udies s udying ac ional models, ac ional BVPs and IVPs, e c. A p esen , he
lis o inequali ies is e y long and s ill g owing. S udies by Beckenbach [
23
] a e a good
esou ce o su ey hese inequali ies. The inequali ies wi h gene al ke nels and measu es
can be s udied in he [
24
,
25
]. AlNeme e al. [
26
] and Zaka ya e al. [
27
] es ablished some
Ha dy and Coposn inequali ies, espec i ely. The HH-inequali y [
28
] is conside ed he
undamen al inequali y in he s udy o con exi y. I helps us unde s and he geome ical
aspec s o a con ex unc ion. I can be w i en as:
Theo em 1. I Φ:[c,d]→Ris a con ex unc ion, hen
Φ(c+d
2)≤1
d−cZd
c
Φ(x)dx ≤Φ(c) + Φ(d)
2
holds. Fo he conca e unc ion abo e, inequali y holds in he o he di ec ion.
Taking ad an age o ac ional ope a o s, Fa id e al. u ilized a Riemann–Liou ille
ac ional in eg al o s udy he e o es ima ion o one o he mos basic and amous He mi e–
Hadama d (HH) inequali ies by using he concep o con exi y o s ic ly mono one
mappings [29].
In his pape , o ob ain mo e ad anced esul s, we used a gene alized Riemann–
Liou ille ac ional in eg al [
30
] on HH-inequali y. We es ablish he gene alized iden i ies
and es ima e he e o o HH-inequali y, which is u he used in es ima ing e o s o
mid-poin and apezoidal inequali ies o s ic ly mono onic con ex unc ions.
The inspi a ion behind his pape is he ecen wo k conduc ed by Fa id e al. in [29].
We de elop a gene alized iden i y o Rieman- ype ac ional in eg als and use i o in es i-
ga e apezoid- ype inequali ies o a class o composi e con ex unc ions wi h espec o a
s ic ly mono one unc ion. The basic pu pose o his esea ch is o ob ain mo e ad anced
and e ined esul s han exis in he li e a u e.
The o ganiza ion o he pape is as ollows: he p elimina ies a e s a ed in Sec ion 2;
he main esul s and special cases, in he o m o se e al examples and applica ions, a e
gi en in Sec ion 3; and conclusi e ema ks a e p o ided in Sec ion 4.
2. P elimina ies
We gi e some p elimina ies ha a e necessa y o deal wi h ou main esul s.
De ini ion 1. A eal- alued unc ion Φde ined on [c,d]is called con ex i i sa is ies
Φ(ηx+ (1−η)y)≤ηΦ(x) + (1−η)Φ(y),
whe e 0≤η≤1and x,y∈[c,d].
The HH-inequali y and i s gene aliza ions ha e been s udied by many au ho s in [
31
–
33
].
Due o ad ancemen and enhancemen o e ec i eness ope a o s, ma hema icians a e
s uggling o in en new e icien mechanisms and ex end he exis ing s udies.
The con exi y o a unc ion w. . . a s ic ly mono one mapping gi en in [
34
] is p e-
sen ed as ollows:
De ini ion 2.
The unc ion
Φ
is con ex w. . . a s ic ly mono one mapping
i he composi e
unc ion Φ◦ −1is con ex.
The ollowing heo em gi es he desc ip ion o HH-inequali y unde a con ex unc ion
w. . . a s ic ly mono one mapping [35].
Symme y 2023,15, 479 3 o 18
Theo em 2.
Suppose
I1
and
I2
a e sub-in e als o
(−∞
,
+∞)
,
:I2⊃[c
,
d]→R
is a mapping
wi h s ic mono onici y p ope y and Φ:[c,d]⊂I1→R is a con ex unc ion w. . . . Then
Φ −1 (c) + (d)
2!≤1
(d)− (c)Z (d)
(c)
Φ( −1(ν))dν≤Φ(c) + Φ(d)
2.
The ollowing de ini ion is an ex ension o he classical Gamma unc ion. Fo mo e
de ails, see [36].
De ini ion 3. The k-Gamma unc ion deno ed by Γkis o mula ed as
Γk(z) = limn→∞n!kn(nk)z
k−1
(z)n,kwhe e k >0and z ∈C Z−.
Ano he o m is
Γk(z) = R∞
0e−νk
kνz−1dν,z∈Cand Re(z)>0.
One can easily obse e ha νΓk(ν) = Γk(ν+k).
De ini ion 4
([
37
])
.
The le and igh sided ac ional RL-in eg als (Riemann-Liou ille) o
G
wi h
o de w a e gi en as
Iw
c+G(x) = 1
Γ(w)Zx
c(x−ν)w−1G(ν)dν,x>c,
Iw
d−G(x) = 1
Γ(w)Zd
x(ν−x)w−1G(ν)dν,x<d.
The gene alized RL-in eg als in oduced in [30] a e as ollows:
De ini ion 5. The le and igh gene alized RL-in eg als o G wi h o de w a e gi en as:
kIw
c+G(x) = 1
kΓk(w)Zx
c(x−ν)w
k−1G(ν)dν,x>c,
kIw
d−G(x) = 1
kΓk(w)Zd
x(ν−x)w
k−1G(ν)dν,x<d.
No e ha he ob ained esul s o he cu en manusc ip a e connec ed wi h he
indings o [38–40].
3. Main Resul s
This sec ion consis s o se e al no el mean- ype inequali ies in ol ing he gene alized
Riemann–Liou ille ac ional in eg als. The ollowing lemma gi es an in eg al iden i y
ha will be help ul o s udy he e o es ima ion (lowe and uppe bounds es ima ion) o
HH-inequali y.
Lemma 1.
Conside a eal unc ion
Φ
and a s ic ly mono one eal unc ion
de ined on
[a1
,
a2]
wi h a2>a1s. . (Φ◦ −1)is di e en iable and (Φ◦ −1)0∈L[a1,a2]. In his case,
Φ(a1) + Φ(a2)
2−Γk(u+k)
2 (a2)− (a1)u
k kIu
(a1)+Φ(a2) + kIu
(a2)−Φ(a1)!
= (a2)− (a1)
2Z1
0(1−ν)u
k−νu
k(Φ◦ −1)0ν (a1) + (1−ν) (a2)dν. (1)
Symme y 2023,15, 479 4 o 18
P oo . Fi s we e alua e he in eg al
Z1
0(1−ν)u
k(Φ◦ −1)0ν (a1) + (1−ν) (a2)dν
=
(1−ν)u
k(Φ◦ −1)ν (a1) + (1−ν) (a2)
(a1)− (a2)
1
o
+u
kZ1
0
(1−ν)u
k−1(Φ◦ −1)ν (a1) + (1−ν) (a2)dν
(a1)− (a2).
=Φ(a2)
(a2)− (a1)−
u
k
(a2)− (a1)Z1
0(1−ν)u
k−1(Φ◦ −1)ν (a1) + (1−ν) (a2)dν.
=Φ(a2)
(a2)− (a1)−
u
k
 (a2)− (a1)u
k+1Z( a2)
(a1)(z− (a))u
k−1(Φ◦ −1)(z)dz.
=Φ(a2)
(a2)− (a1)−Γk(u+k)
 (a2)− (a1)u
k+1 kIu
(a2)−Φ(a1)!. (2)
Simila ly, in eg a ing by pa s, we ob ain
Z1
0νu
k(Φ◦ −1)0ν (a1) + (1−ν) (a2)dν
=−Φ(a2)
(a2)− (a1)+Γk(u+k)
 (a2)− (a1)u
k+1 kIu
(a1)+Φ(a2)!. (3)
By subs i u ing (2) and (3) in he (1), we can ob ain he desi ed esul .
We de i e he ollowing e o es ima e o Theo em 2wi h he help o Lemma 1.
Theo em 3.
Conside a eal unc ion
Φ
and a s ic ly mono one unc ion
de ined on
[a1
,
a2]
wi h a2>a1s. . Φ◦ −1is di e en iable and (Φ◦ −1)0∈L[a1,a2]. Then

Φ(a1) + Φ(a2)
2−Γk(u+k)
2[ (a2)− (a1)]u
k
(kIu
(a1)+Φ(a2) +kIu
(a2)−Φ(a1))
≤| (a2)− (a1)|
2(u
k+1)1−1
2u
k 
(Φ◦ −1)0( (a1))
+
(Φ◦ −1)0( (a2))!, (4)
holds whene e |(Φ◦ −1)0|is con ex.
Symme y 2023,15, 479 5 o 18
P oo .
F om Lemma 1wi h he p ope ies o he absolu e alue unc ion, he abo e in-
equali y can be es ima ed by

Φ(a1) + Φ(a2)
2−Γk(u+k)
2 (a2)− (a1)u
k kIu
(a1)+Φ(a2) +kIu
(a2)−Φ(a1)!
≤| (a2)− (a1)|
2Z1
0
(1−ν)u
k−νu
k
(Φ◦ −1)0(ν (a1) + (1−ν) (a2))
dν. (5)
Since
|(Φ◦ −1)0|
is con ex, he e o e using his on he igh -hand side o (5) will imply
he ollowing:

Φ(a1) + Φ(a2)
2−Γk(u+k)
2 (a2)− (a1)u
k kIu
(a1)+Φ(a2) +kIu
(a2)−Φ(a1)!
≤| (a2)− (a1)|
2Z1
0
(1−ν)u
k−νu
kν
(Φ◦ −1)0( (a1))
+ (1−ν)
(Φ◦ −1)0( (a2))dν
≤| (a2)− (a1)|
2 Z1
2
0
(1−ν)u
k−νu
kν
(Φ◦ −1)0( (a1))
+ (1−ν)
(Φ◦ −1)0( (a2))dν
+Z1
1
2
(1−ν)u
k−νu
kν
(Φ◦ −1)0( (a1))
+ (1−ν)
(Φ◦ −1)0( (a2))dν!
=| (a2)− (a1)|
2 
(Φ◦ −1)0( (a1))Z1
2
0(ν(1−ν)u
k−νu
k+1)dν
+
(Φ◦ −1)0( (a2))Z1
2
0((1−ν)u
k+1−νu
k(1−ν))dν
+
(Φ◦ −1)0( (a1))Z1
1
2
(νu
k+1−ν(1−ν)u
k)dν
+
(Φ◦ −1)0( (a2))Z1
1
2
(νu
k(1−ν)−(1−ν)u
k+1)dν!.
Nex , some calcula ions will imply ou desi ed esul .
Now, we p esen some special cases in he con ex o se e al examples, all o which
ha e been p o ed in p e ious s udies.

Symme y 2023,15, 479 6 o 18
Example 1. By se ing (x) = 1/x in (4), we ob ain

Φ(a1) + Φ(a2)
2−Γk(u+k)
2a1a2
a2−a1u
kkIu
(1
a1)−Φ◦g(1
a2
) + kIu
(1
a2)+Φ◦g(1
a1
)
≤|a1−a2|
2|a1a2|(u
k+1)1−1
2u
ka2
1
Φ0(a1)
+a2
2
Φ0(a2),
whe e g(ν) = 1
ν.
Example 2. By se ing (x) = 1
xand u
k=1in (4), we ob ain

Φ(a1)+Φ(a2)
2−ka1a2
a2−a1Z1
a1
1
a2
(Φ◦g)(ν)dν
≤|a1−a2|
8|a1a2|a2
1
Φ0(a1)
+a2
2
Φ0(a2),
whe e g(ν) = 1
ν.
Example 3. By se ing (x) = x whe e 6=0in (4), we ob ain

Φ(a1)+Φ(a2)
2− u
kΓk(u+k)
2(a
2−a
1)µ
k
kIu
a1+Φ(ν)+
kIu
a2−Φ(ν)
≤|a
2−a
1|
2| |(u
k+1)1−1
2u
ka1−
1
Φ0(a1)
+a1−
2
Φ0(a2).
Example 4. By se ing (x) = x whe e 6=0and u
k=1in (4), we ob ain

Φ(a1)+Φ(a2)
2−k
2(a
2−a
1)Za2
a1
ν −1 (ν)dν
≤|a
2−a
1|
8| |a1−
1
Φ0(a1)
+a1−
2
Φ0(a2).
Example 5. By se ing (x) = logex in (4), we ob ain

Φ(a1)+Φ(a2)
2−Γk(u+k)
2(ln(a2)−ln(a1))u
kkIu
ln(a1)+Φ(a2)+kIu
ln(a2)−Φ(a1)
≤|ln(a2)−ln(a1)|
2(u
k+1)1−1
2u
ka1
Φ0(a1)
+a2
Φ0(a2).
Example 6. By se ing (x) = logex wi h u
k=1in (4), we ob ain

Φ(a1)+Φ(a2)
2−k
ln(a2)−ln(a1)Za2
a1
Φ(u)
udu
≤ln(a2)−ln(a1)
8a1
Φ0(a1)
+a2
Φ0(a2).
Nex we p esen he ollowing heo em.
Symme y 2023,15, 479 7 o 18
Theo em 4.
Conside a eal unc ion
Φ
and a s ic ly mono one unc ion
de ined on
[a1
,
a2]
wi h a2>a1s. . Φ◦ −1is di e en iable and (Φ◦ −1)0∈L[a1,a2]. Then

Φ(a1) + Φ(a2)
2−Γk(u+k)
2 (a2)− (a1)u
kkIu
(a1)+Φ(a2) +kIu
(a2)−Φ(a1)
≤| (a2)− (a1)|
21
q(u
k+1)1−1
2u
k
(Φ◦ −1)0( (a1))
q
+
(Φ◦ −1)0( (a2))
q1
q
, (6)
whene e |(Φ◦ −1)0|q,q≥1is con ex.
P oo . In wo cases, he p oo will be comple ed:
Case(i). Fo q=1.
Via he con exi y o
|(Φ◦ −1)0|
and he p ope ies o he absolu e alue unc ion in
Lemma 1, he abo e inequali y can be ob ained.
Case (ii): Fo q>1.
We use he powe mean inequali y and he p ope ies o he absolu e alue unc ion o
R.H.S o Lemma 1. We ha e

Φ(a1) + Φ(a2)
2−Γk(u+k)
2 (a2)− (a1)u
kkIu
(a1)+Φ(a2) +kIu
(a2)−Φ(a1)
≤| (a2)− (a1)|
2Z1
0
(1−ν)u
k−νu
k1−1
q
(7)
×Z1
0
(1−ν)u
k−νu
k
(Φ◦ −1)0ν (a1) + (1−ν) (a2)
q
dν1
q
.
This can be w i en as
Z1
0
(1−ν)u
k−νu
k
dν=Z1
2
0(1−ν)u
k−νu
kdν+Z1
1
2νu
k−(1−ν)u
kdν
=2
(u
k+1)1−1
2u
k. (8)
Since |(Φ◦ −1)0|qis con ex, he e o e
Z1
0
(1−ν)u
k−νu
k
(Φ◦ −1)0ν (a1) + (1−ν) (a2)
q
dν
≤Z1
2
0(1−ν)u
k−νu
kν
(Φ◦ −1)0( (a1))
q
+ (1−ν)
(Φ◦ −1)0( (a2))
qdν
+Z1
1
2νu
k−(1−ν)u
kν
(Φ◦ −1)0( (a1))
q
+ (1−ν)
(Φ◦ −1)0( (a2))
qdν.(9)
=
(Φ◦ −1)0( (a1))
q Z1
2
oν(1−ν)u
k−νu
k+1dν+Z1
1
2
ννu
k−(1−ν)u
kd !
Symme y 2023,15, 479 8 o 18
+|(Φ◦ −1)0( (a2))|q Z1
2
o(1−ν)u
k+1−νu
k(1−ν)dν+Z1
1
2νu
k(1−ν)−(1− )u
k+1dν!
=
(Φ◦ −1)0( (a1))
q1
(u
k+1)1−1
2u
k+
(Φ◦ −1)0( (a2))
q1
(u
k+1)1−1
2u
k.(10)
Nex , some calcula ions wi h he use o (10), (9) and (8) in (7) will imply ou desi ed
esul .
Now, he ollowing examples show he applica ion o he conclusion o he abo e
heo em.
Example 7. By se ing (x) = x in (6), we ob ain

Φ(a1) + Φ(a2)
2−Γk(u+k)
2(a2−a1)u
kkIu
(a1)+Φ(a2) + kIu
(a2)−Φ(a1)
≤|a2−a1|
21
q(u
k+1)1−1
2u
k
Φ0(a1)
q
+
Φ0(a2)
q1
q
.
Example 8. By se ing Φ(x) = 1
xin (6), we ob ain

Φ(a1)+Φ(a2)
2−Γk(u+k)
2a1a2
a2−a1u
kkI
u
k
1
a1
−Φ◦g(1
a2
) + kI
u
k
1
a2
+Φ◦g(1
a1
)
≤|a1−a2|
21
q|a1a2|(u
k+1)1−1
2u
ka2q
1
Φ0 (a1)
q
+a2q
2
Φ0 (a2)
q1
q
.
Example 9. By se ing (x) = ln(x)in (6), we ob ain

Φ(a1)+Φ(a2)
2−Γk(u+k)
2(lna2−lna1)u
kkIu
lna1+Φ(a2) + kIu
lnb−Φ(a1)
≤|ln(a2)−ln(a1)|
21
q(u
k+1)1−1
2u
kaq
1
Φ0 (a1)
q
+aq
2
Φ0 (a2)
q1
q
.
Example 10. By se ing (x) = x whe e 6=0in (6), we ob ain

Φ(a1) + Φ(a2)
2− u
kΓk(u+k)
2(a
2−a
1)u
kk
Iu
a+
1
Φ(a2) + kIu
a−
2
Φ(a1)
≤|a
2−a
1|
21
q| |(u
k+1)1−1
2u
ka(1− )q
1
Φ0 (a1)
q
+a(1− )q
2
Φ0 (a2)
q1
q
.
Fo he nex heo em, he ollowing Lemma will be help ul.
Lemma 2 ([41]).Fo y >x≥0and α∈(0, 1), we ha e

xα−yα
≤y−xα
.
Symme y 2023,15, 479 9 o 18
Theo em 5.
Conside a eal unc ion
Φ
and a s ic ly mono one eal unc ion
de ined on
[a1
,
a2]
wi h
a2>a1
such ha
Φ◦ −1
is di e en iable and
(Φ◦ −1)0∈L[a1
,
a2]
I
|(Φ◦ −1)0|q
,
q≥1is con ex, hen

Φ(a1) + Φ(a2)
2−Γk(u+k)
2 (a2)− (a1)u
kkIu
(a1)+Φ(a2) +kIu
(a2)−Φ(a1)
≤| (a2)− (a1)|
21+1
qup
k+11
p
(Φ◦ −1)0( (a1))
q
+
(Φ◦ −1)0( (a2))
q1
q
, (11)
s. . 1
q+1
p=1.
P oo .
The absolu e alue unc ion along wi h he Holde ’s inequali y on R.H.S o Lemma 1,
gi e

Φ(a1) + Φ(a2)
2−Γk(u+k)
2 (a2)− (a1)u
kkIu
(a1)+Φ(a2) +kIu
(a2)−Φ(a1)
≤| (a2)− (a1)|
2Z1
0
(1−ν)u
k−νu
k
p1
p
×Z1
0
(Φ◦ −1)0ν (a1) + (1−ν) (a2)
q
dν1
q
,
We apply Lemma 2, o ob ain he ollowing:
Z1
0
(1−ν)u
k−νu
k
p
dν≤Z1
0
1−2ν
up
k
dν
=Z1
2
01−2νup
k
dν+Z1
1
22ν−1up
k
dν
=1
up
k+1.
Now con exi y o |(Φ◦ −1)0|qimplies ha
Z1
0
(Φ◦ −1)0ν (a1) + (1−ν) (a2)
q
dν
≤Z1
0 ν
(Φ◦ −1)
0
( (a1))
q
+ (1−ν)
(Φ◦ −1)0( (a2)
q!dν
=
(Φ◦ −1)0( (a1))
q
+
(Φ◦ −1)0( (a2)
q
2.
Hence by using he compu a ions abo e, we can ob ain he desi ed esul (11).
Symme y 2023,15, 479 16 o 18
Example 20. By se ing (x) = 1
xin (17), we ob ain

2u
k−1(Γk(u+k))(a1a2)u
k
(a2−a1)u
kk
Iu
(a1+a2
2a1a2)−Φ◦g(1
a2
)+kIu
(a1+a2
2a1a2)+Φ◦g(1
a1
)−Φ2a1a2
a1+a2
≤|a1−a2|
41−1
p(up
k+1)1
p|a1a2|a2
1
Φ0(a1)
+a2
2
Φ0(a2),
whe e g(ν) = 1
ν.
Example 21. By se ing (x) = x in (17), we ob ain

2u
k−1(Γk(u+k))( )u
k
(a
2−a
1)u
kkIu
(a
1+a
2
2)1
+
Φ(a2)+kIu
(a
1+a
2
2)1
−
Φ(a1)−Φa
1+a
2
21

≤|a
2−a
1|
41−1
p| |(up
k+1)1
pa1−
1
Φ0(a1)
+a1−
2
Φ0(a2).
Example 22. By se ing (x) = logex in (17), we ob ain

2u
k−1Γk(u+k)
ln(a2)−ln(a1)kIu
(ln(a1)+ln(a2)
2)+Φ(a2) +kIu
(ln(a1)+ln(a2)
2)−Φ(a1)−Φexpln(a1) + ln(a2)
2
≤|ln(a2)−ln(a1)|
41−1
p(up
k+1)1
pa1
Φ0(a1)
+a2
Φ0(a2).
4. Conclusions
The ac ional calculus heo y and in eg al ope a o s ha e been used o yield mo e
gene alized inequali ies. In his a icle, we u ilized he gene alized o m o he Riemann-
ype ac ional in eg al o ob ain mean- ype inequali ies. The main esul s we e based on
iden i y. The consequences we e e i ied o co espond o di e en choices o ce ain
unc ions. The indings o his esea ch educed o he indings o [
29
] jus by eplacing
k=
1. Simila ly, some o he esul s ha exis in he li e a u e we e ec ea ed. The p o en
esul s in his esea ch a e hope ully help ul in he ield o modi ied scien i ic. In he u u e,
we a e commi ed o ob aining mo e gene alized and e ined inequali ies o ac ional
ope a o s.
Au ho Con ibu ions:
Concep ualiza ion, M.M., M.Y. and I.S.; o mal analysis, M.M., M.Y., M.S., I.S.
and M.D.l.S.; Funding acquisi ion, M.D.l.S.; me hodology, M.S., I.S., S.E. and S.R.; so wa e, S.E. and
S.R.; All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
The six h au ho is g a e ul o he Basque Go e nmen o i s suppo h ough G an s
IT1555-22 and KK-2022/00090 and o MCIN/AEI 269.10.13039/501100011033 o G an PID2021-
1235430B-C21/C22.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen :
Da a sha ing is no applicable o his a icle as no da ase s we e
gene a ed no analyzed du ing he cu en s udy.
Acknowledgmen s:
The i h and se en h au ho s would like o hank Aza baijan Shahid Madani
Uni e si y.

Symme y 2023,15, 479 17 o 18
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1. Magin, R.L. F ac ional Calculus in Bio-Enginee ing; Begell House Inc. Publishe s: Danbu y, CT, USA, 2006.
2.
Baleanu, D.; E emad, S.; Rezapou , S. A hyb id Capu o ac ional modeling o he mos a wi h hyb id bounda y alue condi ions.
Bound. Value P obl. 2020,2020, 64. [C ossRe ]
3.
Thaip ayoon, C.; Sudsu ad, W.; Alzabu , J.; E emad, S.; Rezapou , S. On he quali a i e analysis o he ac ional bounda y alue
p oblem desc ibing he mos a con ol model ia ψ-Hil e ac ional ope a o . Ad . Di e . Equ. 2021,2021, 201. [C ossRe ]
4.
Cesa one, F.; Capu o, M.; Came i, C. Memo y o malism in he passi e di usion ac oss a biological memb ane. J. Memb . Sci.
2004,250, 79–84. [C ossRe ]
5.
Bonyah, E.; Chukwu, C.W.; Juga, M.L.; Fa mawa i. Modeling ac ional-o de dynamics o Syphilis ia Mi ag-Le le law. AIMS
Ma h 2021,6, 8367–8389. [C ossRe ]
6.
Mohammad, H.; Kuma , S.; Rezapou , S.; E emad, S. A heo e ical s udy o he Capu o–Fab izio ac ional modeling o hea ing
loss due o Mumps i us wi h op imal con ol. Chaos Soli ons F ac als 2021,144, 110668. [C ossRe ]
7.
E emad, S.; A ci, I.; Kuma , P.; Baleanu, D.; Rezapou , S. Some no el ma hema ical analysis on he ac al– ac ional model o he
AH1N1/09 i us and i s gene alized Capu o- ype e sion. Chaos Soli ons F ac als 2022,162, 112511. [C ossRe ]
8.
Ia aldano, G.; Capu o, M.; Ma ino, S. Expe imen al and heo e ical memo y di usion o wa e in he sand. Hyd ol. Ea h Sys . Sci.
2006,10, 93–100. [C ossRe ]
9.
Juma ie, G. New s ochas ic ac ional models o Mal husiang ow h, he Poissonian bi h p ocess and op imal managemen o
popula ions. Ma h. Compu . Model. 2006,44, 231–254. [C ossRe ]
10.
Capu o, M. Modeling Social and Economic Cycles. In Al e na i e Public Economics; Fo e, F., Na a a, P., Mudambi, R., Eds.; Elga :
Chel enham, UK, 2014.
11.
E emad, S.; Iqbal, I.; Samei, M.E.; Rezapou , S.; Alzabu , J.; Sudsu ad, W.; Goksel, I. Some inequali ies on mul i- unc ions o
applying in he ac ional Capu o–Hadama d je k inclusion sys em. J. Inequal. Appl. 2022,2022, 84. [C ossRe ]
12.
A angana, A.; A az, S.I. Nonlinea equa ions wi h global di e en ial and in eg al ope a o s: Exis ence, uniqueness wi h applica ion
o epidemiology. Res. Phys. 2021,20, 103593. [C ossRe ]
13.
Akdemi , A.O.; Ka aoglan, A.; Ragusa, M.A.; Se , E. F ac ional in eg al inequali ies ia A angana-Baleanu ope a o s o con ex
and conca e unc ions. J. Func . Spaces 2021,2021, 1055434. [C ossRe ]
14.
Rizwan, R.; Zada, A.; Wang, X. S abili y analysis o nonlinea implici ac ional Lange in equa ion wi h nonins an aneous
impulses. Ad . Di e . Equ. 2019,2019, 85. [C ossRe ]
15.
Belmo , S.; Ja ad, F.; Abdeljawad, T.; Kinic, G. A s udy o bounda y alue p oblem o gene alized ac ional di e en ial inclusion
ia endpoin heo y o weak con ac ions. Ad . Di e . Equ. 2020,2020, 348. [C ossRe ]
16.
Mohammad, H.; Rezapou , S.; E emad, S.; Baleanu, D. Two sequen ial ac ional hyb id di e en ial inclusions. Ad . Di e . Equ.
2020,2020, 385. [C ossRe ]
17.
Rezapou , S.; E emad, S.; Tellab, B.; Aga wal, P.; Gui ao, J.L.G. Nume ical solu ions caused by DGJIM and ADM me hods o
mul i- e m ac ional BVP in ol ing he gene alized ψ-RL-ope a o s. Symme y 2021,13, 532. [C ossRe ]
18.
Phuong, N.D.; Saka , F.M.; E emad, S.; Rezapou , S. A no el ac ional s uc u e o a mul i-o de quan um mul i-in eg o-di e en ial
p oblem. Ad . Di e . Equ. 2020,2020, 633. [C ossRe ]
19.
Rezapou , S.; Souid, M.S.; Bouazza, Z.; Hussain, A.; E emad, S. On he ac ional a iable o de he mos a model: Exis ence heo y
on cones ia piece-wise cons an unc ions. J. Func . Spaces 2022,2022, 8053620. [C ossRe ]
20.
Kun , M.; Iscan, I.; Yazici, N.; Gozu ok, U. On new inequali ies o He mi e-Hadama d-Feje ype o ha monically con ex unc ions
ia ac ional in eg als. Sp inge Plus 2016,5, 635. [C ossRe ]
21.
Sa ikaya, M.Z.; Se , E.; Yaldiz, H.; Basak, N. He mi e-Hadama d’s inequali ies o ac ional in eg als and ela ed ac ional
inequali ies. Ma h. Compu . Model. 2013,57, 2403–2407. [C ossRe ]
22.
Sa ikaya, M.Z.; Yildi im, H. On He mi e-Hadama d ype inequali ies o Riemann-Liou ille ac ional in eg als. Miskolc Ma h.
No es 2017,17, 1049–1059. [C ossRe ]
23. Beckenbach, E.; Bellman, R. An In oduc ion o Inequali ies; Random House Inc.: New Yo k, NY, USA, 1961.
24.
El-Hamid, H.A.A.; Rezk, H.M.; Ahmed, A.M.; Al Neme , G.; Zaka ya, M.; El Sai y, H.A. Dynamic inequali ies in quo ien s wi h
gene al ke nels and measu es. J. Func . Spaces 2020,2020, 5417084. [C ossRe ]
25.
Sake , S.H.; Rezk, H.M.; Abohela, I.; Baleanu, D. Re inemen mul idimensional dynamic inequali ies wi h gene al ke nels and
measu es. J. Inequal. Appl. 2019,2019, 306. [C ossRe ]
26.
AlNeme , G.; Zaka ya, M.; El-Hamid, H.A.A.; Kenawy, M.R.; Rezk, H.M. Dynamic Ha dy- ype inequali ies wi h non-conjuga e
pa ame e s. Alex. Eng. J. 2020,59, 4523–4532. [C ossRe ]
27.
Zaka ya, M.; Al anji, M.; AlNeme , G.; El-Hamid, H.A.A.; Cesa ano, C.; Rezk, H.M. F ac ional e e se Coposn’s inequali ies ia
con o mable calculus on ime scales. Symme y 2021,13, 542. [C ossRe ]
28.
Hadama d, J. E ude su les p op ié és des onc ions en iè es e en pa iculie d’une onc ion considé ée pa Riemann. J. Ma h.
Pu es Appl.1893,58, 171–216.
29.
Fa id, G.; Peca ic, J.; Nonlaopon, K. Inequali ies o ac ional Riemann–Liou ille in eg als o ce ain class o con ex unc ions.
Ad . Con . Disc . Models 2022,8, 766. [C ossRe ]
Symme y 2023,15, 479 18 o 18
30. Habibullah, G.M.; Mubeen, S. k-F ac ional in eg als and Applica ion. In . J. Con emp. Ma h. Sci. 2012,7, 89–94.
31.
Fa id, G.; Nazee , W.; Saleem, M.; Mehmood, S.; Kang, S. Bounds o Riemann-Liou ille ac ional in eg als in gene al o m ia
con ex unc ions and hei applica ions. Ma hema ics 2018,6, 248. [C ossRe ]
32.
Kwun, Y.C.; Fa id, G.; Nazee , W.; Ullah, S.; Kang, S.M. Gene alized Riemann-liou ille
k
- ac ional in eg als associa ed wi h
Os owski ype inequali ies and e o bounds o Hadama d inequali ies. IEEE Access 2018,6, 64946–64953. [C ossRe ]
33.
Yang, X.; Fa id, G.; Nazee , W.; Yussou , M.; Chu, Y.-M.; Dong, C. F ac ional gene alized Hadama d and Feje -Hadama d
inequali ies o m-con ex unc ions. AIMS Ma h. 2020,5, 6325–6340. [C ossRe ]
34.
Iscan, I. He mi e-Hadama d ype inequali ies o ha monically con ex unc ions. Hace . J. Ma h. S a .
2014
,43, 935–942. [C ossRe ]
35.
Tu han, S.; Kun , M.; Iscan, I. He mi e-Hadama d ype inequali ies o
MΦA
-con ex unc ions. In . J. Ma h. Model. Comp.
2020
,10,
57–75.
36. Diaz, R.; Pa iguan, E. On hype geome ic unc ions and Pochhamme k-symbol. Di ulg. Ma h. 2007,15, 179–192.
37.
Kilbas, A.A.; S i as a a, H.M.; T ujillo, J.J. Theo y and Applica ions o F ac ional Di e en ial Equa ions; No h-Holland: New Yo k, NY,
USA, 2006.
38.
D agomi , S.S.; Aga wal, R.P. Two inequali ies o di e en iable mappings and applica ions o special means o eal numbe s and
o apezoidal o mula. Appl. Ma h. Le . 1998,11, 91–95. [C ossRe ]
39.
Iscan, I.; Wu, S. He mi e-Hadama d ype inequali ies o ha monically con ex unc ions ia ac ional in eg als. Appl. Ma h.
Compu . 2014,238, 237–244.
40.
Toplu, T.; Se , E.; Iscan, I.; Maden, S. He mi e-Hadama d- ype inequali ies o p-con ex unc ions ia Ka ugampola ac ional
in eg als. Fac a Uni . Se . Ma h. In o m. 2019,34, 149–164. [C ossRe ]
41.
P udniko , A.P.; B ychko , I.A.; Ma iche , O.I. In eg als and Se ies: Special Func ions; CRC P ess: Boca Ra on, FL, USA, 1986;
Volume 2.
Disclaime /Publishe ’s No e:
The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .