SOME NOVEL ESTIMATIONS FOR DIFFERENT KINDS OF CONVEX FUNCTIONS VIA MODIFIED ATANGANA-BALEANU INTEGRAL OPERATORS

SOME NOVEL ESTIMATIONS FOR DIFFERENT KINDS OF
CONVEX FUNCTIONS VIA MODIFIED ATANGANA-BALEANU
INTEGRAL OPERATORS
BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR
Abs ac . F ac ional analysis has ecen ly been used qui e e ec i ely in he ield o
inequali y heo y, as in all subjec s o ma hema ics. Di e en and new ac ional ope a-
o s ob ained wi h he help o a ian s c ea ed in he ke nel s uc u es ha e con ibu ed
o he de elopmen o ac ional analysis and ha e also b ough new di ec ions o o he
ields. A angana-Baleanu ac ional in eg al ope a o has an impo an place among
ac ional in eg al ope a o s in e ms o i s non-singula and non-local p ope ies. In
his s udy, new in eg al inequali ies ha e been p o ed o di e en ypes o con ex
unc ion classes by using he Modi ied A angana-Baleanu ac ional in eg al ope a o
ob ained om A angana-Baleanu ac ional in eg al ope a o , which p oduces unc-
ional solu ions o many dynamic eal wo ld p oblems compa ed o o he ope a o s. In
he p oo s ages o he main indings, basic inequali ies such as H¨olde , Powe -mean,
Young and Jensen inequali ies ha e been used and many educed esul s ha e been
p o ided.
1. In oduc ion and P elimina ies
Con ex unc ions a e special ypes o unc ions ha play a cen al ole in ma he-
ma ical analysis and op imiza ion heo y. This s uc u e o con exi y p o ides a g ea
ad an age, especially in he analysis o solu ion se s and minimum alues. Di e en
ypes o con ex unc ions-such as s ic ly con ex, s ongly con ex, and piecewise con ex
unc ions-a e selec ed acco ding o he na u e o a ious p oblems and di ec ly a ec he
pe o mance o he solu ion me hods. Fo example, s ic ly con ex unc ions gua an ee
a singula minimum, while s ongly con ex unc ions enable op imiza ion algo i hms o
con e ge as e . These ypes play a c i ical ole in bo h heo e ical analysis and algo-
i hm design in disciplines such as nonlinea p og amming, machine lea ning, economics,
con ol heo y, and s a is ics. In addi ion, he geome ic simplici y p o ided by con ex-
i y makes he beha io o he solu ion space mo e p edic able and compu able, e en in
high-dimensional p oblems. Fo his eason, con ex unc ions a e among he indispens-
able ools no only in heo e ical ma hema ical s udies bu also in modeling complex
sys ems in he eal wo ld.
We will s a wi h some basic concep s ha will be used o p o e main indings as ollows.
2010 Ma hema ics Subjec Classi ica ion. 26A33, 26D10, 26D15.
Key wo ds and ph ases. Quasi-con ex unc ion, H¨olde inequali y, powe mean inequali y, Young
inequali y, modi ied A angana-Baleanu (AB) ac ional in eg al ope a o s.
1
2 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR
De ini ion 1.1. [2] The unc ion ℵ: [ , s]→Ris called a quasi-con ex i he ollowing
inequali y
ℵ(ρu + (1 −ρ) )≤max{ℵ(u),ℵ( )}
holds o all u, ∈[ , s]and ρ∈[0,1].
De ini ion 1.2. The unc ion ℵ: [ , s]→Ris η-con ex i
ℵ(ρu + (1 −ρ) )≤ ℵ( ) + ρη(ℵ(u),ℵ( ))
o all u, ∈[ , s],ρ∈[0,1], and η:R×R→Rwi h η(ℵ(u),ℵ( )) ≤ ℵ(u)− ℵ( ) o
ℵ(u)≥ ℵ( ). Simila ly, ℵis η-conca e i
ℵ(ρu + (1 −ρ) )≥ ℵ( ) + ρη(ℵ(u),ℵ( )).
F ac ional analysis, as an ex ension o classical calculus, o e s a deep pe spec i e in
ma hema ical analysis by gene alizing he concep s o de i a i e and in eg al o non-
in ege deg ees. This ield has a ac ed he a en ion o a ious scien is s especially
since he 17 h cen u y, bu has s a ed o be esea ched mo e in ensi ely in ecen yea s
wi h he de eloping calcula ion me hods and inc easing applica ion a eas.
F ac ional de i a i es and in eg als, unlike classical de i a i es, include no only poin
changes bu also he e ec s o he pas beha io o he sys em. Thanks o his memo y
s uc u e, ac ional analysis p o ides a g ea ad an age in modeling he e ec s accumu-
la ed o e ime in ields such as physics, biology, economics and enginee ing. F ac ional
models o e mo e accu a e and lexible solu ions, especially in phenomena ha canno
be adequa ely explained by classical me hods such as anomalous di usion, iscoelas ici y,
elec ical ci cui s, con ol sys ems and biological issue modeling.
Ma hema ically, ac ional analysis adds a new dimension o he heo y o di e en ial
equa ions and de ines mo e gene al and comp ehensi e solu ion spaces beyond classi-
cal solu ions. In his espec , i p o ides a s ong heo e ical basis o mo e ealis ic
modeling, con ol and simula ion o sys ems.
As a esul , ac ional calculus is o c i ical impo ance in con empo a y science and
echnology, no only in e ms o heo e ical ma hema ics bu also in unde s anding and
managing complex sys ems.
Some ope a o s ha a e among he basic concep s o ac ional analysis and ha e become
amous o hei applica ions a e de ined as ollows.
De ini ion 1.3. [3,4] The AB- ac ional in eg al ope a o o ℵ ∈ L1[ , T ]and 0<ρ<1
is de ined by
AB
Iρ
ξℵ(ξ) = 1−ρ
M(ρ)ℵ(ξ) + ρ
M(ρ)Γ(ρ)Zξ
(ξ−η)ρ−1ℵ(ξ)dη.
De ini ion 1.4. [1] The modi ied le -sided ac ional in eg al ope a o o ℵ ∈ L1[ , T ]
and 0<ρ<1wi h espec o Ψis de ined by
MAB
Iρ
Ψ(ξ)ℵ(ξ) = 1−ρ
M(ρ)ℵ(ξ) + ρ
M(ρ)Γ(ρ)Zξ
(Ψ(ξ)−Ψ(η))ρ−1Ψ′(η)ℵ(η)dη.
SOME NOVEL INTEGRAL INEQUALITIES 3
De ini ion 1.5. [1] The modi ied igh -sided ac ional in eg al ope a o o ℵ ∈ L1[ , s]
and 0<ρ<1wi h espec o Ψis de ined by
MAB
sIρ
Ψ(ξ)ℵ(ξ) = 1−ρ
M(ρ)ℵ(ξ) + ρ
M(ρ)Γ(ρ)Zs
ξ
(Ψ(η)−Ψ(ξ))ρ−1Ψ′(η)ℵ(η)dη.
Rema k 1.1. I we conside Ψ(ξ) = ξin De ini ion 1.4, hen we ge he AB-ope a o
de ined in De ini ion 1.3.
2. Main Resul s
In he main indings pa , i s o all, an in eg al iden i y is eminded. Then, new
in eg al inequali ies a e ob ained by using his in eg al iden i y, some con ex unc ion
ypes and some basic inequali y de i a ion me hods. Since he esul s con ain modi ied
A angana-Baleanu (AB) ac ional in eg al ope a o s, new in eg al inequali ies, whose
p oo s a e gi en, in oduce new app oaches and add a new dimension o he li e a u e.
In [5], Rahman e al. es ablished he ollowing iden i y in ol ing modi ied A angana-
Baleanu (AB) ac ional in eg al ope a o s:
Lemma 2.1. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion on
( , s), whe e ℵ′∈L1[ , s]and < s. Fo modi ied A angana-Baleanu (AB) ac ional
in eg al ope a o s, he ollowing iden i y holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
=( − )ϱ+1
s− Z1
0
ρϱℵ′(ρ + (1 −ρ) )dρ −(s− )ϱ+1
s− Z1
0
ρϱℵ′(ρ + (1 −ρ)s)dρ,
whe e ϱ∈(0,1], ∈[ , s], and ρ∈[0,1].
In his sec ion, we use he modi ied A angana-Baleanu (AB) ac ional in eg al op-
e a o s o ob ain some ac ional in eg al inequali ies o he quasi-con ex unc ion and
η-con ex unc ion, espec i ely, as ollows.
Theo em 2.1. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion on
( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|qis a quasi-con ex unc ion, he ollowing inequali y
holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤1
ϱp + 1
1
p( − )ϱ+1
s− max |ℵ′( )|q,|ℵ′( )|q
1
q
4 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR
+(s− )ϱ+1
s− max |ℵ′( )|q,|ℵ′( )|q
1
q,
whe e 1
p+1
q= 1, ∈[ , s],ϱ∈(0,1], and M(ϱ)>0.
P oo . Using Lemma 2.1, well known H¨olde inequali y and he quasi-con exi y o |ℵ′|q
on [ , s], we can w i e
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱp dρ
1
pZ1
0
|ℵ′(ρ + (1 −ρ) )|qdρ
1
q
+(s− )ϱ+1
s− Z1
0
ρϱp dρ
1
pZ1
0
|ℵ′(ρ + (1 −ρ)s)|qdρ
1
q
≤( − )ϱ+1
s− Z1
0
ρϱp dρ
1
p
max |ℵ′( )|q,|ℵ′( )|q
1
q
+(s− )ϱ+1
s− Z1
0
ρϱp dρ
1
p
max |ℵ′( )|q,|ℵ′(s)|q
1
q
=( − )ϱ+1
s− 1
ϱp + 1
1
pmax |ℵ′( )|q,|ℵ′( )|q
1
q
+(s− )ϱ+1
s− 1
ϱp + 1
1
pmax |ℵ′( )|q,|ℵ′(s)|q
1
q
=1
ϱp + 1
1
p( − )ϱ+1
s− max |ℵ′( )|q,|ℵ′( )|q
1
q
+(s− )ϱ+1
s− max |ℵ′( )|q,|ℵ′( )|q
1
q
whe e i is easily seen ha R1
0ρϱp dρ =1
ϱp+1 . This comple es he p oo . □
Theo em 2.2. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion on
( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|qis a quasi-con ex unc ion, he ollowing inequali y
holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤1
ϱ+ 1( − )ϱ+1
s− max |ℵ′( )|q,|ℵ′( )|q
1
q+(s− )ϱ+1
s− max |ℵ′( )|q,|ℵ′(s)|q
1
q,
SOME NOVEL INTEGRAL INEQUALITIES 5
whe e q≥1, ∈[ , s],ϱ∈(0,1], and M(ϱ)>0.
P oo . Using Lemma 2.1, well known powe mean inequali y and he quasi-con exi y o
|ℵ′|qon [ , s], we can w i e
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱ|ℵ′(ρ + (1 −ρ) )|qdρ
1
q
+(s− )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱ|ℵ′(ρ + (1 −ρ)s)|qdρ
1
q
≤( − )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱmax |ℵ′( )|q,|ℵ′( )|qdρ
1
q
+(s− )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱmax |ℵ′( )|q,|ℵ′(s)|qdρ
1
q
=1
ϱ+ 11−1
q1
ϱ+ 1
1
q
×( − )ϱ+1
s− max |ℵ′( )|q,|ℵ′( )|q
1
q+(s− )ϱ+1
s− max |ℵ′( )|q,|ℵ′(s)|q
1
q.
So he p oo is comple ed. □
Theo em 2.3. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion on
( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|qis a quasi-con ex unc ion, he ollowing inequali y
holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− 1
p(ϱp + 1) +1
qmax |ℵ′( )|q,|ℵ′( )|q
+(s− )ϱ+1
s− 1
p(ϱp + 1) +1
qmax |ℵ′( )|q,|ℵ′(s)|q,
whe e 1
p+1
q= 1, ∈[ , s],ϱ∈(0,1], and M(ϱ)>0.
P oo . F om Lemma 2.1, apply he Young inequali y ab ≤ap
p+bq
q, we ob ain
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]

6 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− 1
pZ1
0
ρϱp dρ +1
qZ1
0
|ℵ′(ρ + (1 −ρ) )|qdρ
+(s− )ϱ+1
s− 1
pZ1
0
ρϱp dρ +1
qZ1
0
|ℵ′(ρ + (1 −ρ)s)|qdρ
≤( − )ϱ+1
s− 1
p(ϱp + 1) +1
qmax |ℵ′( )|q,|ℵ′( )|q
+(s− )ϱ+1
s− 1
p(ϱp + 1) +1
qmax |ℵ′( )|q,|ℵ′(s)|q.
So, he p oo is comple ed. □
Theo em 2.4. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion
on ( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|is η-con ex wi h espec o η, he ollowing
inequali y holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− |ℵ′( )|
ϱ+ 1 +η(|ℵ′( )|,|ℵ′( )|)
ϱ+ 2 
+(s− )ϱ+1
s− |ℵ′(s)|
ϱ+ 1 +η(|ℵ′( )|,|ℵ′(s)|)
ϱ+ 2 ,
whe e ∈[ , s],ϱ∈(0,1], and M(ϱ)>0.
P oo . By using he iden i y ha is gi en in Lemma 2.1, we ob ain
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱ|ℵ′(ρ + (1 −ρ) )|dρ +(s− )ϱ+1
s− Z1
0
ρϱ|ℵ′(ρ + (1 −ρ)s)|dρ.
Since |ℵ′|is η-con ex, we ge
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱ[|ℵ′( )|+ρη(|ℵ′( )|,|ℵ′( )|)] dρ
SOME NOVEL INTEGRAL INEQUALITIES 7
+(s− )ϱ+1
s− Z1
0
ρϱ[|ℵ′(s)|+ρη(|ℵ′( )|,|ℵ′(s)|)] dρ
=( − )ϱ+1
s− |ℵ′( )|
ϱ+ 1 +η(|ℵ′( )|,|ℵ′( )|)
ϱ+ 2 
+(s− )ϱ+1
s− |ℵ′(s)|
ϱ+ 1 +η(|ℵ′( )|,|ℵ′(s)|)
ϱ+ 2 
and he p oo is comple ed. □
Theo em 2.5. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion on
( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|qis η-con ex wi h espec o η, whe e q≥1, he
ollowing inequali y holds o modi ied AB- ac ional in eg al ope a o s:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− 1
ϱp + 1
1
p|ℵ′( )|q+η(|ℵ′( )|q,|ℵ′( )|q)
2
1
q
+(s− )ϱ+1
s− 1
ϱp + 1
1
p|ℵ′(s)|q+η(|ℵ′( )|q,|ℵ′(s)|q)
2
1
q
,
whe e 1
p+1
q= 1, ∈[ , s],ϱ∈(0,1], and M(ϱ)>0.
P oo . Using Lemma 2.1 and applying he H¨olde inequali y, we ha e
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱp dρ
1
pZ1
0
|ℵ′(ρ + (1 −ρ) )|qdρ
1
q
+(s− )ϱ+1
s− Z1
0
ρϱp dρ
1
pZ1
0
|ℵ′(ρ + (1 −ρ)s)|qdρ
1
q
.
Since |ℵ′|qis η-con ex, we ob ain
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱp dρ
1
pZ1
0
[|ℵ′( )|q+ρη(|ℵ′( )|q,|ℵ′( )|q)] dρ
1
q
8 BARIS¸ C¸ELIK, ERHAN SET, AND AHMET OCAK AKDEMIR
+(s− )ϱ+1
s− Z1
0
ρϱp dρ
1
pZ1
0
[|ℵ′(s)|q+ρη(|ℵ′( )|q,|ℵ′(s)|q)] dρ
1
q
=( − )ϱ+1
s− 1
ϱp + 1
1
p|ℵ′( )|q+η(|ℵ′( )|q,|ℵ′( )|q)
2
1
q
+(s− )ϱ+1
s− 1
ϱp + 1
1
p|ℵ′(s)|q+η(|ℵ′( )|q,|ℵ′(s)|q)
2
1
q
.
So, he p oo is comple ed. □
Theo em 2.6. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion on
( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|qis η-con ex wi h espec o η, whe e q≥1, he
ollowing inequali y holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− 1
ϱ+ 11−1
q|ℵ′( )|q
ϱ+ 1 +η(|ℵ′( )|q,|ℵ′( )|q)
ϱ+ 2 
1
q
+(s− )ϱ+1
s− 1
ϱ+ 11−1
q|ℵ′(s)|q
ϱ+ 1 +η(|ℵ′( )|q,|ℵ′(s)|q)
ϱ+ 2 
1
q
,
whe e ∈[ , s],ϱ∈(0,1], and M(ϱ)>0.
P oo . F om Lemma 2.1, applying he powe -mean inequali y, we ge
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱ|ℵ′(ρ + (1 −ρ) )|qdρ
1
q
+(s− )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱ|ℵ′(ρ + (1 −ρ)s)|qdρ
1
q
.
Since |ℵ′|qis η-con ex, we ha e
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱ[|ℵ′( )|q+ρη(|ℵ′( )|q,|ℵ′( )|q)] dρ
1
q
SOME NOVEL INTEGRAL INEQUALITIES 9
+(s− )ϱ+1
s− Z1
0
ρϱdρ1−1
qZ1
0
ρϱ[|ℵ′(s)|q+ρη(|ℵ′( )|q,|ℵ′(s)|q)] dρ
1
q
=( − )ϱ+1
s− 1
ϱ+ 11−1
q|ℵ′( )|q
ϱ+ 1 +η(|ℵ′( )|q,|ℵ′( )|q)
ϱ+ 2 
1
q
+(s− )ϱ+1
s− 1
ϱ+ 11−1
q|ℵ′(s)|q
ϱ+ 1 +η(|ℵ′( )|q,|ℵ′(s)|q)
ϱ+ 2 
1
q
.
So, he p oo is comple ed. □
Theo em 2.7. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion on
( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|qis η-con ex wi h espec o η, whe e q≥1, he
ollowing inequali y holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− 1
p(ϱp + 1) +1
q|ℵ′( )|q+η(|ℵ′( )|q,|ℵ′( )|q)
2
+(s− )ϱ+1
s− 1
p(ϱp + 1) +1
q|ℵ′(s)|q+η(|ℵ′( )|q,|ℵ′(s)|q)
2,
whe e 1
p+1
q= 1, ∈[ , s],ϱ∈(0,1], and M(ϱ)>0.
P oo . F om Lemma 2.1, apply he Young inequali y ab ≤ap
p+bq
q, we ge
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]
−M(ϱ)Γ(ϱ)
s− hMAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1( )) + MAB
ΨIϱ
Ψ−1( )(ℵ ◦ Ψ)(Ψ−1(s))i
≤( − )ϱ+1
s− 1
pZ1
0
ρϱp dρ +1
qZ1
0
|ℵ′(ρ + (1 −ρ) )|qdρ
+(s− )ϱ+1
s− 1
pZ1
0
ρϱp dρ +1
qZ1
0
|ℵ′(ρ + (1 −ρ)s)|qdρ.
Since |ℵ′|qis η-con ex and by a simple compu a ion, we ha e he desi ed esul . □
Theo em 2.8. Assume ha Ψ:[ , s]→Ris a s ic ly inc easing and posi i e unc ion
wi h a con inuous de i a i e on [ , s]. Le ℵ: [ , s]→Rbe a di e en iable unc ion
on ( , s),ℵ′∈L1[ , s], and < s. I |ℵ′|is η-conca e wi h espec o η, he ollowing
inequali y holds:
( − )ϱ+ (s− )ϱ
s− ℵ( ) + 1−ϱ
s− Γ(ϱ)[ℵ( ) + ℵ(s)]