Int. J. Nonlinear Anal. Appl. 10 (2019) No. 2, 275-299 [620036]

Int. J. Nonlinear Anal. Appl. 10 (2019) No. 2, 275-299
ISSN: 2008-6822 (electronic)
http://dx.doi.org/10.22075/ijnaa.2019.18455.2014
New inequalities for generalized m-convex functions
via generalized fractional integral operators and
their applications
Artion Kashuria,, Muhammad Aamir Alib, Mujahid Abbasc, H useyin Budakd
aDepartment of Mathematics, Faculty of Technical Science, University Ismail Qemali, 9400, Vlora, Albania
bSchool of Mathematical Sciences, Nanjing Normal University, 210023, China
cDepartment of Mathematics, Government College University, Lahore 54000, Pakistan and Department of Mathematics
and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa
dDepartment of Mathematics, Faculty of Science and Arts, D uzce University, D uzce, Turkey
Abstract
In the present work, we prove a parametrized identity for a dierentiable function via generalized
integral operators. By applying the established identity and the new so{called generalized m-convex
function, some generalized trapezium, Ostrowski and Simpson type integral inequalities have been
discovered. Various special cases have been studied as well. Some applications of the present results
to special means and new error estimates for the trapezium and midpoint quadrature formula have
been investigated. It is hoped that the methods and techniques of this paper could further stimulate
the research conducted in the eld of integral inequalities.
Keywords: Trapezium inequality, Ostrowski inequality, Simpson inequality, convexity, general
fractional integrals.
2010 MSC: Primary 26A51; Secondary 26A33, 26D07, 26D10, 26D15.
1. Introduction and preliminaries
The inequality below, known as Hermite{Hadamard inequality, is one of the most famous inequalities
in the literature relevant to convex functions.
Corresponding author
Email addresses: [anonimizat] (Artion Kashuri), [anonimizat] (Muhammad
Aamir Ali), [anonimizat] (Mujahid Abbas), [anonimizat] (H useyin Budak)
Received: August 2019 Revised: November 2019

276 Kashuri, A. Ali, Abbas, Budak
Theorem 1.1. Letf:IR!Rbe a convex function and u1;u22Iwithu1< u 2:Then the
following inequality holds:
fu1+u2
2
1
u2u1Zu2
u1f(x)dxf(u1) +f(u2)
2: (1.1)
The inequality (1.1) is also known as trapezium inequality.
The trapezium inequality has always been a source of great interest due to its wide ranging appli-
cations in the eld of mathematical analysis. In recent decades, several authors have intensively
investigates (1.1) motivated by the study of convex function. Interested readers are referred to [4]-
[8],[17, 22, 23, 25, 29, 31],[36]-[40],[47, 49, 52, 53, 55, 56],
The following result is known as Ostrowski's inequality, (see [32] and the references cited therein)
which provides an upper bound for the approximation of the integral average1
u2u1Zu2
u1f(t)dtfor
x2[u1;u2]:
Theorem 1.2. Letf:I!Rbe a mapping dierentiable on Iand letu1;u22Iwithu1< u 2:
Ifjf0(x)jMfor allx2[a;b];then
f(x)1
u2u1Zu2
u1f(t)dtM(u2u1)"
1
4+
xu1+u2
2
2
(u2u1)2#
;8×2[u1;u2]: (1.2)
For other recent results concerning Ostrowski type inequalities, see [1]-[3],[9]-[16],[20],[32]-[35],[41]-
[43],[45, 46, 50, 54, 57]. It must be noted that Ostrowski's inequality is essential in all elds of
mathematics, especially in approximation theory. Thus such types of inequalities have been studied
extensively by several researches and there has been a plethora of generalizations, extensions and
variants of these inequalities for various types of functions.
The following inequality is well known in the literature as Simpson's inequality.
Theorem 1.3. Letf: [u1;u2]!Rbe four time dierentiable on the interval (u1;u2)and having
the fourth derivative bounded on (u1;u2);that is
kf(4)k1= sup
x2(u1;u2)jf(4)j<+1:
Then, we have
Zu2
u1f(x)dxu2u1
3"
f(u1) +f(u2)
2+ 2fu1+u2
2#1
2880kf(4)k1(u2u1)5: (1.3)
The inequality (1.3) establishes an error bound for the classical Simpson quadrature formula, which
is one of the most commonly used quadrature formulae in practical applications. In recent years,
various generalizations, extensions and variants of such inequalities have been obtained. For other
recent results concerning Simpson type inequalities, see [30, 44, 51, 57].
In numerical analysis many quadrature rules have been established to approximate denite integrals.
Ostrowski's inequality provides bounds for many numerical quadrature rules, see [15, 16].

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 277
The aim of this paper is to establish trapezium, Ostrowski and Simpson type generalized integral
inequalities for generalized m-convex functions with respect to another function. Some applications
to special means and new error bounds for midpoint and trapezium quadrature formula are obtained.
Interestingly, special cases of some of our results constitute fractional integral inequalities. Hence, it
is important to recall some essential facts relevant to fractional integrals.
Initially, let us present some preliminaries and denitions which will be helpful for further study.
Denition 1.4. [37] Letf2L[u1;u2]:Thenk-fractional integrals of order ;k > 0withu10
are dened by
I;k
u+
1f(x) =1
kk()Zx
u1(xt)
k1f(t)dt; x>u 1
and
I;k
u
2f(x) =1
kk()Zu2
x(tx)
k1f(t)dt; u 2>x; (1.4)
where k(
)stands for the k-gamma function.
Fork= 1, thek-fractional integrals yield Riemann{Liouville integrals. For =k= 1, thek-
fractional integrals yield classical integrals.
LetKbe a nonempty closed set in Rn, andKbe the interior of K. We denote byh:;:iandk:kthe
inner product and norm on Rn;respectively. Let f; :K!Rbe continuous mappings.
Denition 1.5. [38] The function fon the-convex set Kis said to be -convex, if
f(u1+tei(u2u1))(1t)f(u1) +tf(u2);8u1;u22K; t2[0;1]:
The function fis said to be -concave ifis-convex. Note that, every convex function is
-convex but the converse does not hold in general.
Denition 1.6. [18] The special function
E(z) =1X
k=0zk
(1 +k); 2C; R ()>0; z2C;
where stands for the Gamma function, is called Mittag{Leer function.
This function plays an essential role in fractional calculus.
We are now in the position to introduce the notions of generalized m-convex set and generalized
m-convex function as follows:
Denition 1.7. A non empty set Kis said to be a generalized m-convex set for some xed m2
(0;1], if
mu1+tE(u2mu1)2K;8u1;u22K; t2[0;1]; (1.5)
where2C; R ()>0:
Remark 1.8. If we replace m= 1 and the function EwithE1, then the generalized m-convex set
reduces to the -convex set. The generalized m-convex sets are nonconvex.

278 Kashuri, A. Ali, Abbas, Budak
Denition 1.9. The function fis said to be generalized m-convex for some xed m2(0;1], if
f(mu1+tE(u2mu1))(1t)f(mu1) +tf(u2);8u1;u22K; t2[0;1]: (1.6)
Remark 1.10. Clearly, a generalized m-convex function reduces to a -convex function if we set
m= 1 and replace EbyE1, which is a special case of the Mittag{Leer function. The generalized
m-convex functions are nonconvex.
Denition 1.11. [26, 27] Let g: [u1;u2]!Rbe an increasing and positive monotone function
on[u1;u2], having a continuous derivative on (u1;u2). The left{sided fractional integral of fwith
respect togon[u1;u2]of order>0is dened by:
I;g
u1+f(x) =1
()Zx
u1g0(u)f(u)
[g(x)g(u)]1du; x>u 1; (1.7)
provided that the integral exists. The right{sided fractional integral of fwith respect to gon[u1;u2]
of order>0is dened by:
I;g
u2f(x) =1
()Zu2
xg0(u)f(u)
[g(u)g(x)]1du; x<u 2; (1.8)
provided that the integral exists.
Jleli and Samet in [22], proved the Hadamard type inequality for Riemann-Liouville fractional integral
of a convex function fwith respect to another function g.
Also in [47], Sarikaya and Ertu gral dened a function : [0;+1)![0;+1) satisfying the following
conditions:
Z1
0(t)
tdt< +1; (1.9)
1
A(s)
(r)Afor1
2s
r2; (1.10)
(r)
r2B(s)
s2forsr; (1.11)
(r)
r2(s)
s2Cjrsj(r)
r2for1
2s
r2; (1.12)
whereA;B;C > 0 are independent of r;s > 0:If(r)ris increasing for some 0 and(r)
r
is decreasing for some 0;thensatises (1.9)-(1.12), see [48]. Therefore, the left{sided and
right{sided generalized integral operators are dened as follows:
u+
1If(x) =Zx
u1(xt)
xtf(t)dt; x>u 1; (1.13)
u
2If(x) =Zu2
x(tx)
txf(t)dt; x<u 2: (1.14)
The most important feature of generalized integrals is that they produce Riemann{Liouville fractional
integrals,k-Riemann{Liouville fractional integrals, Katugampola fractional integrals, conformable
fractional integrals, Hadamard fractional integrals, etc., see [21, 24, 47].
Recently, Farid in [19] generalised the above integral by introducing an increasing and positive
monotone function gon [u1;u2], having continuous derivative on ( u1;u2). The generalized fractional
integral operator dened by Farid may be given as follows.

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 279
Denition 1.12. The left and right{sided generalized fractional integral of a function fwith respect
to another function gmay be given as follows, respectively:
G;g
u1+f(x) =Zx
u1(g(x)g(u))
g(x)g(u)g0(u)f(u)du; x>u 1; (1.15)
G;g
u2f(x) =Zu2
x(g(u)g(x))
g(u)g(x)g0(u)f(u)du; x<u 2: (1.16)
This operator generalizes the various fractional integrals of a function fwith respect to another
functiong.
The following special cases are focussed in our study.
(i) If we take (u) =uthen the operator (1.15) and (1.16) reduces to Riemann{Liouville integral of
fwith respect to function g.
Ig
u1+f(x) =Zx
u1g0(u)f(u)du; x>u 1; (1.17)
Ig
u2f(x) =Zu2
xg0(u)f(u)du; x<u 2: (1.18)
Ifg(u) =u, then (1:17) and (1:18) will reduce to Riemann integral of f.
(ii) If we take (u) =u
()then the operator (1.15) and (1.16) reduces to Riemann{Liouville fractional
integral offwith respect to function g.
I;g
u1+f(x) =1
()Zx
u1[g(x)g(u)]1g0(u)f(u)du; x>u 1; (1.19)
I;g
u2f(x) =1
()Zu2
x[g(u)g(x)]1g0(u)f(u)du; x<u 2: (1.20)
Ifg(u) =u, then (1:19) and (1:20) will reduce to left and right{sided Riemann{Liouville fractional
integrals of frespectively.
(iii) If we take (u) =u
k
kk()then the operator (1.15) and (1.16) reduces to k-Riemann{Liouville
fractional integral of fwith respect to function g.
I;g
u1+;kf(x) =1
kk()Zx
u1[g(x)g(u)]
k1g0(u)f(u)du; x>u 1; (1.21)
I;g
u2;kf(x) =1
kk()Zu2
x[g(u)g(x)]1g0(u)f(u)du; x<u 2: (1.22)
Ifg(u) =u, then these operators in (1 :21) and (1:22) reduces to k-fractional integral operators given
in [37].
(iv) If we take g(u) =u(g(u2)u)1for2(0;1);then the operator given in (1.15) and (1.16)
reduces to conformable fractional integral operator of fwith respect to a function g.
K;g
u1f(x) =Zx
u1[g(u)]1g0(u)f(u)du; x>u 1: (1.23)

280 Kashuri, A. Ali, Abbas, Budak
This operator (1 :23) generalizes conformable fractional integral operator which was given by Khalil
et al. in [28].
(v) If we take (u) =u
exp(Au), whereA=1
and2(0;1);then the operator given in (1.15)
and (1.16) reduces to fractional integral operator of fwith respect to function gwith exponential
kernel.
J;g
u1+f(x) =1
Zx
u1exp (A(g(x)g(u)))g0(u)f(u)du; x>u 1; (1.24)
J;g
u2f(x) =1
Zu2
xexp (A(g(x)g(u)))g0(u)f(u)du; x<u 2: (1.25)
Operators in (1 :24) and (1:25) generalizes fractional integral operator with exponential kernel which
was introduced by Kirane and Torebek in [29].
Motivated by the above literatures, the main objective of this paper is to discover in Section 2,
an interesting identity with parameter in order to study some new bounds regarding trapezium,
Ostrowski and Simpson type integral inequalities. By using the established identity as an auxiliary
result, some new estimates for trapezium, Ostrowski and Simpson type integral inequalities for
generalized m-convex functions via generalized integrals are obtained. It is pointed out that some new
fractional integral inequalities have been deduced from main results. In Section 3, some applications
to special means and new error estimates for the midpoint and trapezium quadrature formula are
given. In Section 4, a brie
y conlusion is given as well.
2. Main results
Throughout this study, let P= [mu1;u2];whereu1<u 2for some xed m2(0;1] and for all t2[0;1]:
For brevity, we dene
;g
m(x;t) =Zt
0(g(mu1+uE(xmu1))g(mu1))
g(mu1+uE(xmu1))g(mu1)(2.1)
g0(mu1+uE(xmu1))du< +1
and
;g
m(x;t) =Z1
t(g(mx+E(u2mx))g(mx+uE(u2mx)))
g(mx+E(u2mx))g(mx+uE(u2mx))(2.2)
g0(mx+uE(u2mx))du< +1;
wheregis an increasing and positive monotone function on P;having continuous derivative on
P= (mu1;u2):
For establishing some new results regarding general fractional integrals we need to prove the following
lemma.
Lemma 2.1. Letf:P!Rbe a dierentiable mapping on Pand2R:Iff02L(P)and
E(u2mu1)>0;then the following identity for generalized fractional integrals hold:
E(xmu1)f(mu1+tE(xmu1)) +E(u2mx)f(mx)
E(u2mu1)

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 281

E(u2mu1)"
E(xmu1)f(mu1+tE(xmu1))
;g
m(x;1)+E(u2mx)f(mx)
;g
m(x;0)#
+
E(u2mu1)"
E(xmu1)f(mu1)
;g
m(x;1)+E(u2mx)f(mx+E(u2mx))
;g
m(x;0)#
1
E(u2mu1)"G;g
(mu1+E(xmu1))f(mu1)
;g
m(x;1)+G;g
(mx)+f(mx+E(u2mx))
;g
m(x;0)#
=E2
(xmu1)
;g
m(x;1)E(u2mu1)Z1
0h
;g
m(x;t)i
f0(mu1+tE(xmu1))dt (2.3)
E2
(u2mx)
;g
m(x;0)E(u2mu1)Z1
0h
;g
m(x;t)i
f0(mx+tE(u2mx))dt:
We denote
Tf;;g
m;;g
m(;x;u 1;u2) =E2
(xmu1)
;g
m(x;1)E(u2mu1)(2.4)
Z1
0h
;g
m(x;t)i
f0(mu1+tE(xmu1))dt
E2
(u2mx)
;g
m(x;0)E(u2mu1)Z1
0h
;g
m(x;t)i
f0(mx+tE(u2mx))dt:
Proof . Integrating by parts equation (2.4) and changing the variables of integration, we have
Tf;;g
m;;g
m(;x;u 1;u2) =E2
(xmu1)
;g
m(x;1)E(u2mu1)
(Z1
0;g
m(x;t)f0(mu1+tE(xmu1))dtZ1
0f0(mu1+tE(xmu1))dt)
E2
(u2mx)
;g
m(x;0)E(u2mu1)
(Z1
0;g
m(x;t)f0(mx+tE(u2mx))dtZ1
0f0(mx+tE(u2mx))dt)
=E2
(xmu1)
;g
m(x;1)E(u2mu1)(
;g
m(x;t)f(mu1+tE(xmu1))
E(xmu1)1
01
E(xmu1)
Z1
0(g(mu1+tE(xmu1))g(mu1))
g(mu1+tE(xmu1))g(mu1)
g0(mu1+tE(xmu1))f(mu1+tE(xmu1))dt

E(xmu1)f(mu1+tE(xmu1))1
0)
E2
(u2mx)
;g
m(x;0)E(u2mu1)

282 Kashuri, A. Ali, Abbas, Budak
(
;g
m(x;t)f(mx+tE(u2mx))
E(u2mx)1
0
1
E(u2mx)Z1
0(g(mx+E(u2mx))g(mx+tE(u2mx)))
g(mx+E(u2mx))g(mx+tE(u2mx))
g0(mx+tE(u2mx))f(mx+tE(u2mx))dt

E(u2mx)f(mx+tE(u2mx))1
0)
=E(xmu1)f(mu1+tE(xmu1)) +E(u2mx)f(mx)
E(u2mu1)

E(u2mu1)"
E(xmu1)f(mu1+tE(xmu1))
;g
m(x;1)+E(u2mx)f(mx)
;g
m(x;0)#
+
E(u2mu1)"
E(xmu1)f(mu1)
;g
m(x;1)+E(u2mx)f(mx+E(u2mx))
;g
m(x;0)#
1
E(u2mu1)"G;g
(mu1+E(xmu1))f(mu1)
;g
m(x;1)+G;g
(mx)+f(mx+E(u2mx))
;g
m(x;0)#
:
The proof of Lemma 2.1 is completed. 
Remark 2.2. a Takingm= 1; = 0;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1andg(t) =(t) =tin Lemma 2.1, we get the following Ostrowski type
identity:
Tf(x;u 1;u2) =f(x)1
u2u1Zu2
u1f(t)dt:
bTakingm= 1; = 1;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2mu1) =u2mu1
andg(t) =(t) =tin Lemma 2.1, we get the following Hermite{Hadamard type identity:
Tf(x;u 1;u2) =(xu1)f(u1) + (u2x)f(u2)
u2u11
u2u1Zu2
u1f(t)dt:
cTakingm= 1; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2mu1) =
u2mu1andg(t) =(t) =tin Lemma 2.1, we get the following Simpson type identity:
Tf(;u1;u2) ="
f(u1) +f(u2)
2#
+ (1)fu1+u2
2
1
u2u1Zu2
u1f(t)dt:
Theorem 2.3. Letf:P!Rbe a dierentiable mapping on Pand2[0;1]:Ifjf0jqis generalized
m-convex on PandE(u2mu1)>0;then forq >1andp1+q1= 1;the following inequality
for generalized fractional integrals hold:
Tf;;g
m;;g
m(;x;u 1;u2)

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 283
E2
(xmu1)
qp
2;g
m(x;1)E(u2mu1)pq
B;g
m(x;;p)qp
jf0(mu1)jq+jf0(x)jq (2.5)
+E2
(u2mx)
qp
2;g
m(x;0)E(u2mu1)pq
B;g
m(x;;p)qp
jf0(mx)jq+jf0(u2)jq;
where
B;g
m(x;;p) =Z1
0;g
m(x;t)p
dt; B;g
m(x;;p) =Z1
0;g
m(x;t)p
dt: (2.6)
Proof . From Lemma 2.1, generalized m-convexity ofjf0jq;H older's inequality and properties of the
modulus, we have
Tf;;g
m;;g
m(;x;u 1;u2)
E2
(xmu1)
;g
m(x;1)E(u2mu1)Z1
0;g
m(x;t)f0(mu1+tE(xmu1))dt
+E2
(u2mx)
;g
m(x;0)E(u2mu1)Z1
0;g
m(x;t)f0(mx+tE(u2mx))dt
E2
(xmu1)
;g
m(x;1)E(u2mu1)
Z1
0;g
m(x;t)p
dt1
pZ1
0f0(mu1+tE(xmu1))q
dt1
q
+E2
(u2mx)
;g
m(x;0)E(u2mu1)
Z1
0;g
m(x;t)p
dt1
pZ1
0f0(mx+tE(u2mx))q
dt1
q
E2
(xmu1)
;g
m(x;1)E(u2mu1)pq
B;g
m(x;;p)Z1
0h
(1t)jf0(mu1)jq+tjf0(x)jqi
dt1
q
+E2
(u2mx)
;g
m(x;0)E(u2mu1)pq
B;g
m(x;;p)Z1
0h
(1t)jf0(mx)jq+tjf0(u2)jqi
dt1
q
=E2
(xmu1)
qp
2;g
m(x;1)E(u2mu1)pq
B;g
m(x;;p)qp
jf0(mu1)jq+jf0(x)jq
+E2
(u2mx)
qp
2;g
m(x;0)E(u2mu1)pq
B;g
m(x;;p)qp
jf0(mx)jq+jf0(u2)jq:
The proof of Theorem 2.3 is completed. 
We point out some special cases of Theorem 2.3.

284 Kashuri, A. Ali, Abbas, Budak
Corollary 2.4. Takingp=q= 2 in Theorem 2.3, we get
Tf;;g
m;;g
m(;x;u 1;u2)
E2
(xmu1)p
2;g
m(x;1)E(u2mu1)q
B;g
m(x;;2)p
jf0(mu1)j2+jf0(x)j2 (2.7)
+E2
(u2mx)p
2;g
m(x;0)E(u2mu1)q
B;g
m(x;;2)p
jf0(mx)j2+jf0(u2)j2:
Corollary 2.5. Takingjf0jKin Theorem 2.3, we have
Tf;;g
m;;g
m(;x;u 1;u2)K
E(u2mu1)(2.8)
"
E2
(xmu1)
;g
m(x;1)pq
B;g
m(x;;p) +E2
(u2mx)
;g
m(x;0)pq
B;g
m(x;;p)#
:
Corollary 2.6. Takingm= 1; = 0;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1andg(t) =(t) =tin Theorem 2.3, we get the following Ostrowski type inequality:
Tf(x;u 1;u2)1
qp
2ppp+ 1(u2u1)(2.9)

(xu1)2qp
jf0(u1)jq+jf0(x)jq+ (u2x)2qp
jf0(x)jq+jf0(u2)jq
:
Corollary 2.7. Takingx=u1+u2
2in Corollary 2.6, we get the following midpoint type inequality:
Tf(u1;u2)(u2u1)
4qp
2ppp+ 1(2.10)
(
qs
jf0(u1)jq+f0u1+u2
2q
+qsf0u1+u2
2q
+jf0(u2)jq)
:
Corollary 2.8. Takingm= 1; = 1;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1andg(t) =(t) =tin Theorem 2.3, we get the following trapezium type inequality:
Tf(x;u 1;u2)1
qp
2ppp+ 1(u2u1)(2.11)

(xu1)2qp
jf0(u1)jq+jf0(x)jq+ (u2x)2qp
jf0(x)jq+jf0(u2)jq
:
Corollary 2.9. Takingm= 1; =1
3;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1andg(t) =(t) =tin Theorem 2.3, we get the following Simpson type inequality:
Tf1
3;u1;u21
qp
2(u2u1)ps
2p+1+ 1
3p+1(p+ 1)(2.12)

(xu1)2qp
jf0(u1)jq+jf0(x)jq+ (u2x)2qp
jf0(x)jq+jf0(u2)jq
:

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 285
Corollary 2.10. Taking= 0 and(t) =tin Theorem 2.3, we get
Tf;g
m;g
m(0;x;u 1;u2)1
qp
2E(u2mu1)(2.13)
(
qp
E(xmu1)pp
Bg
1(x;p)qp
jf0(mu1)jq+jf0(x)jq
+qp
E(u2mx)pp
Bg
2(x;p)qp
jf0(mx)jq+jf0(u2)jq)
;
where
Bg
1(x;p) =Zmu1+E(xmu1)
mu1
g(t)g(mu1)pdt (2.14)
and
Bg
2(x;p) =Zmx+E(u2mx)
mx
g(mx+E(u2mx))g(t)pdt: (2.15)
Corollary 2.11. Taking= 0 and(t) =t
()in Theorem 2.3, we have
Tf;g
m;g
m(0;x;u 1;u2)1
qp
2E(u2mu1)(2.16)
(
qp
E(xmu1)pp
Bg
3(x;p;)qp
jf0(mu1)jq+jf0(x)jq
+qp
E(u2mx)pp
Bg
4(x;p;)qp
jf0(mx)jq+jf0(u2)jq)
;
where
Bg
3(x;p;) =Zmu1+E(xmu1)
mu1
g(t)g(mu1)pdt (2.17)
and
Bg
4(x;p;) =Zmx+E(u2mx)
mx
g(mx+E(u2mx))g(t)pdt: (2.18)
Corollary 2.12. Taking= 0 and(t) =t
k
kk()in Theorem 2.3, we obtain
Tf;g
m;g
m(0;x;u 1;u2)1
qp
2E(u2mu1)(2.19)
(
qp
E(xmu1)pp
Bg
5(x;p;;k )qp
jf0(mu1)jq+jf0(x)jq
+qp
E(u2mx)pp
Bg
6(x;p;;k )qp
jf0(mx)jq+jf0(u2)jq)
;

286 Kashuri, A. Ali, Abbas, Budak
where
Bg
5(x;p;;k ) =Zmu1+E(xmu1)
mu1
g(t)g(mu1)p
kdt (2.20)
and
Bg
6(x;p;;k ) =Zmx+E(u2mx)
mx
g(mx+E(u2mx))g(t)p
kdt: (2.21)
Corollary 2.13. Taking= 0;8u2[0;t]; g(x;t) =t(g(mu1+E(xmu1))t)1and8u2
[t;1]; g(x;t) =t(g(mx+E(u2mx))t)1in Theorem 2.3, we get
Tf;g
m;g
m(0;x;u 1;u2)Eq+1
q
(xmu1)
qp
2
g(mu1+E(xmu1))g(mu1)
E(u2mu1)(2.22)
pp
Bg
7(x;p)qp
jf0(mu1)jq+jf0(x)jq
+Eq+1
q
(u2mx)
qp
2
g(mx+E(u2mx))g(mx)
E(u2mu1)
pp
Bg
8(x;p;)qp
jf0(mx)jq+jf0(u2)jq;
where
Bg
7(x;p) =Zmu1+E(xmu1)
mu1
g(t)g(mu1)pdt (2.23)
and
Bg
8(x;p;) =Zmx+E(u2mx)
mxh
g(mx+E(u2mx))g(t)ip
dt: (2.24)
Corollary 2.14. Taking= 0 and(t) =t
exp(At), whereA=1
, in Theorem 2.3, we have
Tf;g
m;g
m(0;x;u 1;u2)Eq+1
q
(xmu1)
qp
2E(u2mu1)pp
Bg
9(x;p;A)qp
jf0(mu1)jq+jf0(x)jq (2.25)
+Eq+1
q
(u2mx)
qp
2E(u2mu1)pp
Bg
10(x;p;A)qp
jf0(mx)jq+jf0(u2)jq;
where
Bg
9(x;p;A) =Zmu1+E(xmu1)
mu1n
1exp
A(g(mu1)g(t))op
dt (2.26)
and
Bg
10(x;p;A) =Zmx+E(u2mx)
mxn
1exp
A(g(t)g(mx+E(u2mx)))op
dt: (2.27)
Theorem 2.15. Letf:P!Rbe a dierentiable mapping on Pand2[0;1]:Ifjf0jqis
generalized m-convex on PandE(u2mu1)>0;then forq1;the following inequality for
generalized fractional integrals hold:
Tf;;g
m;;g
m(;x;u 1;u2)E2
(xmu1)
;g
m(x;1)E(u2mu1)h
B;g
m(x;;1)i11
q(2.28)

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 287
qrh
B;g
m(x;;1)E;g
m(x;)i
jf0(mu1)jq+E;g
m(x;)jf0(x)jq
+E2
(u2mx)
;g
m(x;0)E(u2mu1)h
B;g
m(x;;1)i11
q
qrh
B;g
m(x;;1)G;g
m(x;)i
jf0(mx)jq+G;g
m(x;)jf0(u2)jq;
where
E;g
m(x;) =Z1
0t;g
m(x;t)dt; G;g
m(x;) =Z1
0t;g
m(x;t)dt; (2.29)
andB;g
m(x;;1); B;g
m(x;;1)are dened as in Theorem 2.3.
Proof . From Lemma 2.1, generalized m-convexity ofjf0jq;the well{known power mean inequality
and properties of the modulus, we have
Tf;;g
m;;g
m(;x;u 1;u2)
E2
(xmu1)
;g
m(x;1)E(u2mu1)Z1
0;g
m(x;t)f0(mu1+tE(xmu1))dt
+E2
(u2mx)
;g
m(x;0)E(u2mu1)Z1
0;g
m(x;t)f0(mx+tE(u2mx))dt
E2
(xmu1)
;g
m(x;1)E(u2mu1)
Z1
0;g
m(x;t)dt11
qZ1
0;g
m(x;t)f0(mu1+tE(xmu1))q
dt1
q
+E2
(u2mx)
;g
m(x;0)E(u2mu1)
Z1
0;g
m(x;t)dt11
qZ1
0;g
m(x;t)f0(mx+tE(u2mx))q
dt1
q
E2
(xmu1)
;g
m(x;1)E(u2mu1)pq
B;g
m(x;;p)
Z1
0;g
m(x;t)h
(1t)jf0(mu1)jq+tjf0(x)jqi
dt1
q
+E2
(u2mx)
;g
m(x;0)E(u2mu1)pq
B;g
m(x;;p)
Z1
0;g
m(x;t)h
(1t)jf0(mx)jq+tjf0(u2)jqi
dt1
q
=E2
(xmu1)
;g
m(x;1)E(u2mu1)h
B;g
m(x;;1)i11
q

288 Kashuri, A. Ali, Abbas, Budak
qrh
B;g
m(x;;1)E;g
m(x;)i
jf0(mu1)jq+E;g
m(x;)jf0(x)jq
+E2
(u2mx)
;g
m(x;0)E(u2mu1)h
B;g
m(x;;1)i11
q
qrh
B;g
m(x;;1)G;g
m(x;)i
jf0(mx)jq+G;g
m(x;)jf0(u2)jq:
The proof of Theorem 2.15 is completed. 
We point out some special cases of Theorem 2.15.
Corollary 2.16. Takingq= 1 in Theorem 2.15, we get
Tf;;g
m;;g
m(;x;u 1;u2)E2
(xmu1)
;g
m(x;1)E(u2mu1)(2.30)
h
B;g
m(x;;1)E;g
m(x;)
jf0(mu1)j+E;g
m(x;)jf0(x)ji
+E2
(u2mx)
;g
m(x;0)E(u2mu1)h
B;g
m(x;;1)G;g
m(x;)
jf0(mx)j+G;g
m(x;)jf0(u2)ji
:
Corollary 2.17. Takingjf0jKin Theorem 2.15, we have
Tf;;g
m;;g
m(;x;u 1;u2)K
E(u2mu1)(2.31)
"
E2
(xmu1)
;g
m(x;1)B;g
m(x;;1) +E2
(u2mx)
;g
m(x;0)B;g
m(x;;1)#
:
Corollary 2.18. Takingm= 1; = 0;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1andg(t) =(t) =tin Theorem 2.15, we get the following Ostrowski type inequality:
Tf(x;u 1;u2)1
2qp
3(u2u1)(2.32)

(xu1)2qp
jf0(u1)jq+ 2jf0(x)jq+ (u2x)2qp
2jf0(x)jq+jf0(u2)jq
:
Corollary 2.19. Takingx=u1+u2
2in Corollary 2.18 we get the following midpoint type inequality:
Tf(u1;u2)(u2u1)
8qp
3(2.33)
(
qs
jf0(u1)jq+ 2f0u1+u2
2q
+qs
2f0u1+u2
2q
+jf0(u2)jq)
:
Corollary 2.20. Takingm= 1; = 1;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1andg(t) =(t) =tin Theorem 2.15, we get the following trapezium type inequality:
Tf(x;u 1;u2)1
2qp
3(u2u1)(2.34)

(xu1)2qp
2jf0(u1)jq+jf0(x)jq+ (u2x)2qp
jf0(x)jq+ 2jf0(u2)jq
:

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 289
Corollary 2.21. Takingm= 1; =1
3;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1andg(t) =(t) =tin Theorem 2.15, we get the following Simpson type inequality:
Tf1
3;u1;u21
2qp
243(u2u1)(2.35)

(xu1)2qp
185jf0(u1)jq+ 58jf0(x)jq+ (u2x)2qp
195jf0(x)jq+ 48jf0(u2)jq
:
Corollary 2.22. Taking= 0 and(t) =tin Theorem 2.15, we get
Tf;g
m;g
m(0;x;u 1;u2)1
Eq+1
q
(xmu1)E(u2mu1)(2.36)
h
Bg
1(x; 1)i11
qqrh
Bg
1(x; 1)E(xmu1)Cg
1(x)i
jf0(mu1)jq+Cg
1(x)jf0(x)jq
+1
Eq+1
q
(u2mx)E(u2mu1)h
Bg
2(x; 1)i11
q
qrh
Bg
2(x; 1)E(u2mx)Eg
1(x)i
jf0(mx)jq+Eg
1(x)jf0(u2)jq;
where
Cg
1(x) =Zmu1+E(xmu1)
mu1(tmu1)(g(t)g(mu1))dt; (2.37)
Eg
1(x) =Zmx+E(u2mx)
mx(tmx)(g(mx+E(u2mx))g(t))dt; (2.38)
andBg
1(x; 1); Bg
2(x; 1)are dened as in Corollary 2.10 for value p= 1:
Corollary 2.23. Taking= 0 and(t) =t
()in Theorem 2.15, we have
Tf;g
m;g
m(0;x;u 1;u2)1
Eq+1
q
(xmu1)E(u2mu1)(2.39)
h
Bg
3(x; 1;)i11
qqrh
Bg
3(x; 1;)E(xmu1)Cg
1(x;)i
jf0(mu1)jq+Cg
1(x;)jf0(x)jq
+1
Eq+1
q
(u2mx)E(u2mu1)h
Bg
4(x; 1;)i11
q
qrh
Bg
4(x; 1;)E(u2mx)Eg
1(x;)i
jf0(mx)jq+Eg
1(x;)jf0(u2)jq;
where
Cg
1(x;) =Zmu1+E(xmu1)
mu1(tmu1)
g(t)g(mu1)dt; (2.40)
Eg
1(x;) =Zmx+E(u2mx)
mx(tmx)
g(mx+E(u2mx))g(t)dt; (2.41)
andBg
3(x; 1;); Bg
4(x; 1;)are dened as in Corollary 2.11 for value p= 1:

290 Kashuri, A. Ali, Abbas, Budak
Corollary 2.24. Taking= 0 and(t) =t
k
kk()in Theorem 2.15, we obtain
Tf;g
m;g
m(0;x;u 1;u2)1
Eq+1
q
(xmu1)E(u2mu1)h
Bg
5(x; 1;;k )i11
q(2.42)
qrh
Bg
5(x; 1;;k )E(xmu1)Cg
1(x;;k )i
jf0(mu1)jq+Cg
1(x;;k )jf0(x)jq
+1
Eq+1
q
(u2mx)E(u2mu1)h
Bg
6(x; 1;;k )i11
q
qrh
Bg
6(x; 1;;k )E(u2mx)Eg
1(x;;k )i
jf0(mx)jq+Eg
1(x;;k )jf0(u2)jq;
where
Cg
1(x;;k ) =Zmu1+E(xmu1)
mu1(tmu1)
g(t)g(mu1)
kdt; (2.43)
Eg
1(x;;k ) =Zmx+E(u2mx)
mx(tmx)
g(mx+E(u2mx))g(t)
kdt; (2.44)
andBg
5(x; 1;;k ); Bg
6(x; 1;;k )are dened as in Corollary 2.12 for value p= 1:
Corollary 2.25. Taking= 0;8u2[0;t]; g(x;t) =t(g(mu1+E(xmu1))t)1and8u2
[t;1]; g(x;t) =t(g(mx+E(u2mx))t)1in Theorem 2.15, we get
Tf;g
m;g
m(0;x;u 1;u2)1
Eq+1
q
(xmu1)E(u2mu1)(2.45)
h
Bg
7(x; 1;)i11
qqrh
Bg
7(x; 1;)E(xmu1)Cg
1(x)i
jf0(mu1)jq+Cg
1(x)jf0(x)jq
+1
Eq+1
q
(u2mx)E(u2mu1)h
Bg
8(x; 1;)i11
q
qrh
Bg
8(x; 1;)E(u2mx)Lg
2(x;)i
jf0(mx)jq+Lg
2(x;)jf0(u2)jq;
where
Lg
2(x;) =Zmx+E(u2mx)
mx(tmx)
g(mx+E(u2mx))g(t)
dt; (2.46)
andBg
7(x; 1;); Bg
8(x; 1;)are dened as in Corollary 2.13 for value p= 1 andCg
1(x)is dened as
in Corollary 2.22.
Corollary 2.26. Taking= 0 and(t) =t
exp(At), whereA=1
in Theorem 2.15, we have
Tf;g
m;g
m(0;x;u 1;u2)1
(1)Eq+1
q
(xmu1)E(u2mu1)(2.47)
(h
Bg
9(x; 1;A)i11
qqp
Lg
3(x;A)jf0(mu1)jq+Lg
4(x;A)jf0(x)jq

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 291
+1
(1)Eq+1
q
(u2mx)E(u2mu1)
h
Bg
10(x; 1;A)i11
qqp
Lg
5(x;A)jf0(mx)jq+Lg
6(x;A)jf0(u2)jq)
;
where
Lg
3(x;A) =Zmu1+E(xmu1)
mu1(mu1+E(xmu1)t) (2.48)
n
1exp
A(g(mu1)g(t))o
dt;
Lg
4(x;A) =Zmu1+E(xmu1)
mu1(tmu1)n
1exp
A(g(mu1)g(t))o
dt; (2.49)
Lg
5(x;A) =Zmx+E(u2mx)
mx(mx+E(u2mx)t) (2.50)
n
1exp
A(g(t)g(mx+E(u2mx)))o
dt;
Lg
6(x;A) =Zmx+E(u2mx)
mx(tmx)n
1exp
A(g(t)g(mx+E(u2mx)))o
dt; (2.51)
andBg
9(x; 1;A); Bg
10(x; 1;A)are dened as in Corollary 2.14 for value p= 1:
Remark 2.27. Applying our Theorems 2.3 and 2.15 for special values of parameter 2[0;1];
for appropriate choices of function g(t) =t;g(t) = lnt;8t > 0;; g(t) =et;etc., where (t) =
t;t
();t
k
kk();g(t) =t(g(u2)t)1for2(0;1);(t) =t
exph
1


ti
for2(0;1);such
thatjf0jqto be convex, we can deduce some new general fractional integral inequalities. We omit
their proofs and the details are left to the interested readers.
3. Applications
Consider the following special means for dierent real numbers u1;u2andu1u26= 0;as follows:
1. the arithmetic mean:
A:=A(u1;u2) =u1+u2
2;
2. the harmonic mean:
H:=H(u1;u2) =2
1
u1+1
u2;
3. the logarithmic mean:
L:=L(u1;u2) =u2u1
lnju2jlnju1j;
4. the generalized log{mean:
Lr:=Lr(u1;u2) ="
ur+1
2ur+1
1
(r+ 1)(u2u1)#1
r
;r2Znf1;0g:

292 Kashuri, A. Ali, Abbas, Budak
It is well known that Lris monotonic nondecreasing over r2ZwithL1:=L:In particular, we
have the following inequality HLA:Now, using the theory results in Section 2, we give some
applications to special means for dierent real numbers.
Proposition 3.1. Letu1;u22Rnf0g;whereu1<u 2:Then forr2Nandr2;whereq >1and
p1+q1= 1;the following inequality hold:
Ar(u1;u2)Lr
r(u1;u2)r(u2u1)
4ppp+ 1(3.1)
(
qs
A
ju1jq(r1);u1+u2
2q(r1)
+qs
Au1+u2
2q(r1)
;ju2jq(r1))
:
Proof . Takingm= 1; = 0; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =trandg(t) =(t) =t;in Theorem 2.3, one can obtain the result
immediately. 
Proposition 3.2. Letu1;u22Rnf0g;whereu1<u 2:Then forr2Nandr2;whereq >1and
p1+q1= 1;the following inequality hold:
A(ur
1;ur
2)Lr
r(u1;u2)r(u2u1)
4ppp+ 1(3.2)
(
qs
A
ju1jq(r1);u1+u2
2q(r1)
+qs
Au1+u2
2q(r1)
;ju2jq(r1))
:
Proof . Takingm= 1; = 1; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =trandg(t) =(t) =t;in Theorem 2.3, one can obtain the result
immediately. 
Proposition 3.3. Letu1;u22Rnf0g;whereu1< u 2:Then forq > 1andp1+q1= 1;the
following inequality hold: 1
A(u1;u2)1
L(u1;u2)(u2u1)
4ppp+ 1(3.3)
(
1
qs
H
ju1j2q;u1+u2
22q+1
qs
Hu1+u2
22q
;ju2j2q)
:
Proof . Takingm= 1; = 0; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =1
tandg(t) =(t) =t;in Theorem 2.3, one can obtain the result
immediately. 
Proposition 3.4. Letu1;u22Rnf0g;whereu1< u 2:Then forq > 1andp1+q1= 1;the
following inequality hold: 1
H(u1;u2)1
L(u1;u2)(u2u1)
4ppp+ 1(3.4)

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 293
(
1
qs
H
ju1j2q;u1+u2
22q+1
qs
Hu1+u2
22q
;ju2j2q)
:
Proof . Takingm= 1; = 1; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =1
tandg(t) =(t) =t;in Theorem 2.3, one can obtain the result
immediately. 
Proposition 3.5. Letu1;u22Rnf0g;whereu1<u 2:Then forr2Nandr2;whereq1;the
following inequality hold:
Ar(u1;u2)Lr
r(u1;u2)qr
2
3r(u2u1)
8(3.5)
(
qs
A
ju1jq(r1);2u1+u2
2q(r1)
+qs
A
2u1+u2
2q(r1)
;ju2jq(r1))
:
Proof . Takingm= 1; = 0; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =trandg(t) =(t) =t;in Theorem 2.15, one can obtain the result
immediately. 
Proposition 3.6. Letu1;u22Rnf0g;whereu1<u 2:Then forr2Nandr2;whereq1;the
following inequality hold:
A(ur
1;ur
2)Lr
r(u1;u2)qr
2
3r(u2u1)
8(3.6)
(
qs
A
2ju1jq(r1);u1+u2
2q(r1)
+qs
Au1+u2
2q(r1)
;2ju2jq(r1))
:
Proof . Takingm= 1; = 1; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =trandg(t) =(t) =t;in Theorem 2.15, one can obtain the result
immediately. 
Proposition 3.7. Letu1;u22Rnf0g;whereu1<u 2:Then forq1;the following inequality hold:
1
A(u1;u2)1
L(u1;u2)qr
4
3(u2u1)
8(3.7)
(
1
qs
H
2ju1j2q;u1+u2
22q+1
qs
Hu1+u2
22q
;2ju2j2q)
:
Proof . Takingm= 1; = 0; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =1
tandg(t) =(t) =t;in Theorem 2.15, one can obtain the result
immediately. 

294 Kashuri, A. Ali, Abbas, Budak
Proposition 3.8. Letu1;u22Rnf0g;whereu1<u 2andE(u2mu1)>0:Then forq1;the
following inequality hold: 1
H(u1;u2)1
L(u1;u2)qr
4
3(u2u1)
8(3.8)
(
1
qs
H
ju1j2q;2u1+u2
22q+1
qs
H
2u1+u2
22q
;ju2j2q)
:
Proof . Takingm= 1; = 1; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =u2mx;E(u2
mu1) =u2mu1; f(t) =1
tandg(t) =(t) =t;in Theorem 2.15, one can obtain the result
immediately. 
Remark 3.9. Applying our Theorems 2.3 and 2.15 for special values of parameter 2[0;1];for
appropriate choices of function g(t) =t;g(t) = lnt;8t > 0;; g(t) =et;etc., where (t) =
t;t
();t
k
kk();g(t) =t(g(u2)t)1for2(0;1);(t) =t
exph
1


ti
for2(0;1);
such thatjf0jqto be convex, we can deduce some new general fractional integral inequalities using
above special means (and other special means). We omit their proofs and the details are left to the
interested readers.
Next, we provide some new error estimates for the midpoint and trapezium quadrature formula. Let
Qbe the partition of the points u1=x0< x 1< ::: < x k=u2of the interval [ u1;u2]:Let consider
the following quadrature formula:
Zu2
u1f(x)dx=M(f;Q) +E(f;Q);Zu2
u1f(x)dx=T(f;Q) +E(f;Q)
where
M(f;Q) =k1X
i=0fxi+xi+1
2
(xi+1xi); T(f;Q) =k1X
i=0f(xi) +f(xi+1)
2(xi+1xi)
are the midpoint and trapezium version and E(f;Q); E(f;Q) are denote their associated approxi-
mation errors.
Proposition 3.10. Letf: [u1;u2]!Rbe a dierentiable function on (u1;u2);whereu1<u 2:If
jf0jqis convex on [u1;u2]forq>1andp1+q1= 1;then the following inequality holds:
E(f;Q)1
4qp
2ppp+ 1k1X
i=0(xi+1xi)2(3.9)
(
qs
jf0(xi)jq+f0xi+xi+1
2q
+qsf0xi+xi+1
2q
+jf0(xi+1)jq)
:

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 295
Proof . Applying Theorem 2.3 for m= 1; = 0; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =
u2mx;E(u2mu1) =u2mu1andg(t) =(t) =ton the subintervals [ xi;xi+1] (i= 0;:::;k1)
of the partition Q, we have
fxi+xi+1
2
1
xi+1xiZxi+1
xif(x)dx(xi+1xi)
4qp
2ppp+ 1(3.10)
(
qs
jf0(xi)jq+f0xi+xi+1
2q
+qsf0xi+xi+1
2q
+jf0(xi+1)jq)
:
Hence from (3.10), we get
E(f;Q)=Zu2
u1f(x)dxM(f;Q)
k1X
i=0(Zxi+1
xif(x)dxfxi+xi+1
2
(xi+1xi))
k1X
i=0(Zxi+1
xif(x)dxfxi+xi+1
2
(xi+1xi))
1
4qp
2ppp+ 1k1X
i=0(xi+1xi)2
(
qs
jf0(xi)jq+f0xi+xi+1
2q
+qsf0xi+xi+1
2q
+jf0(xi+1)jq)
:
The proof of Proposition 3.10 is completed. 
Proposition 3.11. Letf: [u1;u2]!Rbe a dierentiable function on (u1;u2);whereu1<u 2:If
jf0jqis convex on [u1;u2]forq1;then the following inequality holds:
E(f;Q)1
8qp
3k1X
i=0(xi+1xi)2(3.11)
(
qs
jf0(xi)jq+ 2f0xi+xi+1
2q
+qs
2f0xi+xi+1
2q
+jf0(xi+1)jq)
:
Proof . The proof is analogous as to that of Proposition 3.10 taking m= 1; = 0; x=u1+u2
2;E(x
mu1) =xmu1;E(u2mx) =u2mx;E(u2mu1) =u2mu1andg(t) =(t) =tusing
Theorem 2.15. 
Proposition 3.12. Letf: [u1;u2]!Rbe a dierentiable function on (u1;u2);whereu1<u 2:If
jf0jqis convex on [u1;u2]forq>1andp1+q1= 1;then the following inequality holds:
E(f;Q)1
4qp
2ppp+ 1k1X
i=0(xi+1xi)2(3.12)
(
qs
jf0(xi)jq+f0xi+xi+1
2q
+qsf0xi+xi+1
2q
+jf0(xi+1)jq)
:

296 Kashuri, A. Ali, Abbas, Budak
Proof . Applying Theorem 2.3 for m= 1; = 1; x=u1+u2
2;E(xmu1) =xmu1;E(u2mx) =
u2mx;E(u2mu1) =u2mu1andg(t) =(t) =ton the subintervals [ xi;xi+1] (i= 0;:::;k1)
of the partition Q, we have
f(xi) +f(xi+1)
21
xi+1xiZxi+1
xif(x)dx(xi+1xi)
4qp
2ppp+ 1(3.13)
(
qs
jf0(xi)jq+f0xi+xi+1
2q
+qsf0xi+xi+1
2q
+jf0(xi+1)jq)
:
Hence from (3.13), we get
E(f;Q)=Zu2
u1f(x)dxT(f;Q)
k1X
i=0(Zxi+1
xif(x)dxf(xi) +f(xi+1)
2(xi+1xi))
k1X
i=0(Zxi+1
xif(x)dxf(xi) +f(xi+1)
2(xi+1xi))
1
4qp
2ppp+ 1k1X
i=0(xi+1xi)2
(
qs
jf0(xi)jq+f0xi+xi+1
2q
+qsf0xi+xi+1
2q
+jf0(xi+1)jq)
:
The proof of Proposition 3.12 is completed. 
Proposition 3.13. Letf: [u1;u2]!Rbe a dierentiable function on (u1;u2);whereu1<u 2:If
jf0jqis convex on [u1;u2]forq1;then the following inequality holds:
E(f;Q)1
8qp
3k1X
i=0(xi+1xi)2(3.14)
(
qs
2jf0(xi)jq+f0xi+xi+1
2q
+qsf0xi+xi+1
2q
+ 2jf0(xi+1)jq)
:
Proof . The proof is analogous as to that of Proposition 3.12 taking m= 1; = 1; x=u1+u2
2;E(x
mu1) =xmu1;E(u2mx) =u2mx;E(u2mu1) =u2mu1andg(t) =(t) =tusing
Theorem 2.15. 
Remark 3.14. Applying our Theorems 2.3 and 2.15, where m= 1;for special values of parameter
2[0;1];for appropriate choices of function g(t) =t;g(t) = lnt;8t >0;; g(t) =et;etc., where
(t) =t;t
();t
k
kk();g(t) =t(g(u2)t)1for2(0;1);(t) =t
exph
1


ti
for2
(0;1);such thatjf0jqto be convex, we can deduce some new bounds for the midpoint and trapezium
quadrature formula using above ideas and techniques. We omit their proofs and the details are left
to the interested readers.

New inequalities for generalized m-convex functions. . . 10 (2019) No. 2, 275-299 297
4. Conclusion
The new class of functions called generalized m-convex can be applied to obtain several results in
convex analysis, related optimization theory and may stimulate further research in dierent areas of
pure and applied sciences.
Acknowledgements
We thank anonymous referee for his/her valuable suggestions for improved our manuscript.
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