EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 4, 2017, 809-834
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Some new Hermite-Hadamard type conformable
fractional integral inequalities for twice differentiable

MT(r;g,m,ϕ)-preinvex functions

Akli Fundo1, Artion Kashuri2,∗, Miftar Ramosaço2, Rozana Liko2

1 Department of Mathematics, Polytechnic University of Tirana, Albania
2 Department of Mathematics, Faculty of Technical Science, University ”Ismail Qemali”,
Albania

Abstract. In the present paper, the notion of MT(r;g,m,ϕ)-preinvex function is introduced and
some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula
involving MT(r;g,m,ϕ)-preinvex functions are given. Moreover, some generalizations of Hermite-
Hadamard type inequalities for MT(r;g,m,ϕ)-preinvex functions that are twice differentiable via
conformable fractional integrals are established. At the end, some applications to special means
are given.
2010 Mathematics Subject Classifications: 26A51, 26A33, 26D07, 26D10, 26D15
Key Words and Phrases: Hermite-Hadamard type inequality, MT-convex function, Hölder’s
inequality, Minkowski inequality, power mean inequality, Riemann-Liouville fractional integral,
m-invex, P -function.

1. Introduction and Preliminaries

The following notations are used throughout this paper. We use I to denote an in-
terval on the real line R = (−∞,+∞) and I◦ to denote the interior of I. For any subset
K ⊆ Rn,K◦ is used to denote the interior of K. Rn is used to denote a n-dimensional
vector space. The set of integrable functions on the interval [a, b] is denoted by L1[a, b].

The following inequality, named Hermite-Hadamard inequality, is one of the most famous
inequalities in the literature for convex functions.

Theorem 1. Let f : I ⊆ R −→ R be a convex function and a, b ∈ I with a < b. Then the
following inequality holds:

f

(
a+ b

2

)
≤ 1

b− a

∫ b

a
f(x)dx ≤ f(a) + f(b)

2
. (1)

Definition 1.
∗Corresponding author.

Email addresses: aklifundo@yahoo.com (A. Fundo), artionkashuri@gmail.com (A. Kashuri),
miftar.ramosaco@gmail.com (M. Ramosaço), rozanaliko86@gmail.com (R. Liko)

http://www.ejpam.com 809 c© 2017 EJPAM All rights reserved.



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 810

In (see [32],[35]), Tunç and Yidirim defined the following so-called MT-convex function:

A function f : I ⊆ R −→ R is said to belong to the class of MT(I), if it is nonnegative
and for all x, y ∈ I and t ∈ (0, 1) satisfies the following inequality:

f(tx+ (1− t)y) ≤
√
t

2
√

1− t
f(x) +

√
1− t
2
√
t
f(y). (2)

In recent years, various generalizations, extensions and variants of such inequalities
have been obtained.
For other recent results concerning Hermite-Hadamard type inequalities through various
classes of convex functions, (see [12],[13],[18]-[27],[33],[34]).

Fractional calculus (see [31]), was introduced at the end of the nineteenth century by
Liouville and Riemann, the subject of which has become a rapidly growing area and has
found applications in diverse fields ranging from physical sciences and engineering to bio-
logical sciences and economics.

Definition 2. Let f ∈ L1[a, b]. The Riemann-Liouville integrals Jαa+f and Jαb−f of order
α > 0 with a ≥ 0 are defined by

Jαa+f(x) =
1

Γ(α)

∫ x

a
(x− t)α−1f(t)dt, x > a

and

Jαb−f(x) =
1

Γ(α)

∫ b

x
(t− x)α−1f(t)dt, b > x,

where Γ(α) =

∫ +∞

0
e−uuα−1du. Here J0

a+f(x) = J0
b−f(x) = f(x).

In the case of α = 1, the fractional integral reduces to the classical integral.

The following definitions will be used in the sequel.

Definition 3. The Euler beta function is defined for a, b > 0 as

β(a, b) =

∫ 1

0
ta−1(1− t)b−1dt =

Γ(a)Γ(b)

Γ(a+ b)
.

Definition 4. The incomplete beta function is defined for a, b > 0 as

βx(a, b) =

∫ x

0
ta−1(1− t)b−1dt, 0 < x ≤ 1.

For x = 1, the incomplete beta function coincides with the complete beta function.

Definition 5. Let g : [0, 1] −→ [0, 1] be a differentiable function. The new generalized
incomplete beta function is defined for a, b > 0 as

Bg(x)(a, b) =

∫ g(x)

g(0)
ta−1(1− t)b−1dt.



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 811

For g(x) = x, the new generalized incomplete beta function coincides with the incom-
plete beta function.

In the following, we give some definitions and properties of conformable fractional in-
tegrals which help to obtain main identity and results. Recently, some authors, started to
study on conformable fractional integrals (see [1],[2]). In (see [5]), Khalil et al. defined the
fractional integral of order 0 < α ≤ 1 only. In (see [6]), Abdeljawad gave the definition of
left and right conformable fractional integrals of any order α > 0.

Definition 6. Let α ∈ (n, n + 1] and set β = α − n, then the left conformable fractional
integral starting at a is defined by

(Iaαf) (t) =
1

n!

∫ t

a
(t− x)n(x− a)β−1f(x)dx.

Analogously, the right conformable fractional integral is defined by(
bIαf

)
(t) =

1

n!

∫ b

t
(x− t)n(b− x)β−1f(x)dx.

Notice that if α = n+ 1, then β = α−n = n+ 1−n = 1, where n = 0, 1, 2, . . . , and hence
(Iaαf) (t) =

(
Jan+1f

)
(t).

In (see [7]), Set et al. established a generalization of Hermite-Hadamard type inequality
for s-convex functions and gave some remarks to show the relationships with the classi-
cal and Riemann-Liouville fractional integrals inequality by using the given properties of
conformable fractional integrals.

Theorem 2. Let f : [a, b] −→ R be a function with 0 ≤ a < b, s ∈ (0, 1], and f ∈ L1[a, b].
If f is a convex function on [a, b], then the following inequalities for conformable fractional
integrals hold

Γ(α− n)

Γ(α+ 1)
f

(
a+ b

2

)
≤ 1

(b− a)α2s

[
(Iaαf) (b) +

(
bIαf

)
(a)
]

≤

[
β(n+ s+ 1, α− n) + β(n+ 1, α− n+ s)

n!

]
f(a) + f(b)

2s
,

with α ∈ (n, n+ 1], n ∈ N, n = 0, 1, 2, . . . , where Γ is Euler gamma function.

Also Set et al. established some results for some kind of inequalities via conformable
fractional integrals (see [8]-[11]).
Due to the wide application of fractional integrals, some authors extended to study frac-
tional Hermite-Hadamard type inequalities for functions of different classes (see [15]-[31]).

Now, let us evoke some definitions.



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 812

Definition 7. (see [4]) A nonnegative function f : I ⊆ R −→ [0,+∞) is said to be
P -function or P -convex, if

f(tx+ (1− t)y) ≤ f(x) + f(y), ∀x, y ∈ I, t ∈ [0, 1].

Definition 8. (see [14]) A set K ⊆ Rn is said to be invex with respect to the mapping
η : K ×K −→ Rn, if x+ tη(y, x) ∈ K for every x, y ∈ K and t ∈ [0, 1].

Notice that every convex set is invex with respect to the mapping η(y, x) = y− x, but
the converse is not necessarily true. For more details (see [14],[16]).

Definition 9. (see [17]) The function f defined on the invex set K ⊆ Rn is said to be
preinvex with respect η, if for every x, y ∈ K and t ∈ [0, 1], we have that

f (x+ tη(y, x)) ≤ (1− t)f(x) + tf(y).

The concept of preinvexity is more general than convexity since every convex function
is preinvex with respect to the mapping η(y, x) = y − x, but the converse is not true.

The Gauss-Jacobi type quadrature formula has the following∫ b

a
(x− a)p(b− x)qf(x)dx =

+∞∑
k=0

Bm,kf(γk) +R?m|f |, (3)

for certain Bm,k, γk and rest R?m|f | (see [28]).

Recently, Liu (see [29]) obtained several integral inequalities for the left-hand side of
(3) under the Definition 7 of P -function.
Also in (see [30]), Özdemir et al. established several integral inequalities concerning the
left-hand side of (3) via some kinds of convexity.

Motivated by these results, in Section 2, the notion of MT(r;g,m,ϕ)-preinvex function
is introduced and some new integral inequalities for the left-hand side of (3) involving
MT(r;g,m,ϕ)-preinvex functions are given. In Section 3, some generalizations of Hermite-
Hadamard type inequalities for MT(r;g,m,ϕ)-preinvex functions that are twice differentiable
via conformable fractional integrals are given. In Section 4, some applications to special
means are given. In Section 5, some conclusions and future research are given. These
general inequalities give us some new estimates for Hermite-Hadamard type conformable
fractional integral and fractional integral inequalities.

2. New integral inequalities for MT(r;g,m,ϕ)-preinvex functions

Definition 10. (see [3]) A set K ⊆ Rn is said to be m-invex with respect to the mapping
η : K ×K × (0, 1] −→ Rn for some fixed m ∈ (0, 1], if mx+ tη(y, x,m) ∈ K holds for each
x, y ∈ K and any t ∈ [0, 1].



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 813

Remark 1. In Definition 10, under certain conditions, the mapping η(y, x,m) could re-
duce to η(y, x). For example when m = 1, then the m-invex set degenerates an invex set
on K.

We next give new definition, to be referred as MT(r;g,m,ϕ)-preinvex function.

Definition 11. Let K ⊆ R be an open m-invex set with respect to η : K×K×(0, 1] −→ R,
g : [0, 1] −→ (0, 1) be a differentiable function and ϕ : I −→ K is a continuous function.
The function f : K −→ (0,∞) is said to be MT(r;g,m,ϕ)-preinvex function with respect to
η, if

f(mϕ(y) + g(t)η(ϕ(x), ϕ(y),m)) ≤Mr (f(ϕ(x)), f(ϕ(y)),m; g(t)) (4)

holds for any fixed m ∈ (0, 1] and for all x, y ∈ I, t ∈ [0, 1], where

Mr(f(ϕ(x)), f(ϕ(y)),m; g(t))

=



[
m
√
g(t)

2
√

1−g(t)
f r(ϕ(x)) +

m
√

1−g(t)
2
√
g(t)

f r(ϕ(y))

] 1
r

, if r 6= 0;

f(ϕ(x))
m
√
g(t)

2
√

1−g(t) f(ϕ(y))
m
√

1−g(t)
2
√
g(t) , if r = 0,

is the weighted power mean of order r for positive numbers f(ϕ(x)) and f(ϕ(y)).

Remark 2. In Definition 11, it is worthwhile to note that the class MT(r;g,m,ϕ)(I) is a
generalization of the class MT(I) given in Definition 1 for r = m = 1 with respect to
η(ϕ(x), ϕ(y),m) = ϕ(x)−mϕ(y), ϕ(x) = x, ∀x, y ∈ I, g(t) = t, ∀t ∈ (0, 1).

Let give below a nontrivial example for motivation of this new interesting class of
MT(r;g,m,ϕ)-preinvex functions.

Example 1. f1, f2 : (1,∞) −→ (0,∞), f1(x) = xp, f2(x) = (1 + x)p, p ∈
(
0, 1

1000

)
;

h : [1, 3/2] −→ (0,∞), h(x) = (1+x2)k, k ∈
(
0, 1

100

)
, are simple examples of the new class

of MT(1;t,m,x)-preinvex functions with respect to η(ϕ(x), ϕ(y),m) = ϕ(x)−mϕ(y), ϕ(x) =
x, g(t) = t, r = 1, for any fixed m ∈ (0, 1], but they are not convex.

In this section, in order to prove our main results regarding some new integral in-
equalities involving MT(r;g,m,ϕ)-preinvex functions, we need the following new interesting
Lemma:

Lemma 1. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ [0, 1] is a dif-
ferentiable function. Assume that f : K = [mϕ(a),mϕ(a) + η(ϕ(b), ϕ(a),m)] −→ R is
a continuous function on K◦ with respect to η : K × K × (0, 1] −→ R, for mϕ(a) <
mϕ(a) + η(ϕ(b), ϕ(a),m). Then for any fixed m ∈ (0, 1] and p, q > 0, we have∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 814

= η(ϕ(b), ϕ(a),m)p+q+1

×
∫ 1

0
gp(t)(1− g(t))qf(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)].

Proof. It is easy to observe that∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

= η(ϕ(b), ϕ(a),m)

∫ 1

0
(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m)−mϕ(a))p

×(mϕ(a) + η(ϕ(b), ϕ(a),m)−mϕ(a)− g(t)η(ϕ(b), ϕ(a),m))q

×f(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

= η(ϕ(b), ϕ(a),m)p+q+1

×
∫ 1

0
gp(t)(1− g(t))qf(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)].

Theorem 3. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ (0, 1) is a
differentiable function. Assume that f : K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] −→ (0,∞)
is a continuous function on K◦ with respect to η : K × K × (0, 1] −→ R, for mϕ(a) <

mϕ(a) + η(ϕ(b), ϕ(a),m). Let k > 1 and 0 < r ≤ 1. If f
k
k−1 is a nonnegative MT(r;g,m,ϕ)-

preinvex function on an open m-invex set K for any fixed m ∈ (0, 1], then for any fixed
p, q > 0, we have∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

≤
(m

2

) k−1
rk |η(ϕ(b), ϕ(a),m)|p+q+1B

1
k (g(t); k, p, q)

×
[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f

rk
k−1 (ϕ(a)) +Ar(g(t); r)f

rk
k−1 (ϕ(b))

] k−1
rk
,

where

B(g(t); k, p, q) =

∫ 1

0
gkp(t)(1− g(t))kqd[g(t)];

A(g(t); r) =

∫ 1−g(0)

1−g(1)

(√
1− t
t

) 1
r

dt.



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 815

Proof. Let k > 1 and 0 < r ≤ 1. Since f
k
k−1 is a nonnegative MT(r;g,m,ϕ)-preinvex

function on K, combining with Lemma 1, Hölder inequality and Minkowski inequality for
all t ∈ [0, 1] and for any fixed m ∈ (0, 1], we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

≤ |η(ϕ(b), ϕ(a),m)|p+q+1

[∫ 1

0
gkp(t)(1− g(t))kqd[g(t)]

] 1
k

×

[∫ 1

0
f

k
k−1 (mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

] k−1
k

≤ |η(ϕ(b), ϕ(a),m)|p+q+1B
1
k (g(t); k, p, q)

×

[∫ 1

0

(
m
√
g(t)

2
√

1− g(t)
f r(ϕ(b))

k
k−1 +

m
√

1− g(t)

2
√
g(t)

f r(ϕ(a))
k
k−1

) 1
r

d[g(t)]

] k−1
k

≤
(m

2

) k−1
rk |η(ϕ(b), ϕ(a),m)|p+q+1B

1
k (g(t); k, p, q)

×

{∫ 1

0

( √
g(t)√

1− g(t)

) 1
r

f
k
k−1 (ϕ(b))d[g(t)]

r

+

∫ 1

0

(√
1− g(t)√
g(t)

) 1
r

f
k
k−1 (ϕ(a))d[g(t)]

r} k−1
rk

=
(m

2

) k−1
rk |η(ϕ(b), ϕ(a),m)|p+q+1B

1
k (g(t); k, p, q)

×
[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f

rk
k−1 (ϕ(a)) +Ar(g(t); r)f

rk
k−1 (ϕ(b))

] k−1
rk
.

Corollary 1. Under the same conditions as in Theorem 3 for r = 1 and g(t) = t, we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

≤
(mπ

4

) k−1
k |η(ϕ(b), ϕ(a),m)|p+q+1β

1
k (kp+ 1, kq + 1)

[
f

k
k−1 (ϕ(a)) + f

k
k−1 (ϕ(b))

] k−1
k
.



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 816

Theorem 4. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ (0, 1) is a
differentiable function. Assume that f : K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] −→ (0,∞)
is a continuous function on K◦ with respect to η : K × K × (0, 1] −→ R, for mϕ(a) <
mϕ(a) + η(ϕ(b), ϕ(a),m). Let l ≥ 1 and 0 < r ≤ 1. If f l is a nonnegative MT(r;g,m,ϕ)-
preinvex function on an open m-invex set K for any fixed m ∈ (0, 1], then for any fixed
p, q > 0, we have∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

≤
(m

2

) 1
rl |η(ϕ(b), ϕ(a),m)|p+q+1B

l−1
l (g(t); 1, p, q)

×

[
Br

(
g(t);

1

2r
, 2pr − 1, 2qr + 1

)
f rl(ϕ(a))

+Br

(
g(t);

1

2r
, 2pr + 1, 2qr − 1

)
f rl(ϕ(b))

] 1
rl

.

Proof. Let l ≥ 1 and 0 < r ≤ 1. Since f l is a nonnegative MT(r;g,m,ϕ)-preinvex function
on K, combining with Lemma 1, the well-known power mean inequality and Minkowski
inequality for all t ∈ [0, 1] and for any fixed m ∈ (0, 1], we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

= η(ϕ(b), ϕ(a),m)p+q+1

∫ 1

0

[
gp(t)(1− g(t))q

] l−1
l
[
gp(t)(1− g(t))q

] 1
l

×f(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

≤ |η(ϕ(b), ϕ(a),m)|p+q+1

[∫ 1

0
gp(t)(1− g(t))qd[g(t)]

] l−1
l

×

[∫ 1

0
gp(t)(1− g(t))qf l(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

] 1
l

≤ |η(ϕ(b), ϕ(a),m)|p+q+1B
l−1
l (g(t); 1, p, q)

×

[∫ 1

0
gp(t)(1− g(t))q

(
m
√
g(t)

2
√

1− g(t)
f r(ϕ(b))l +

m
√

1− g(t)

2
√
g(t)

f r(ϕ(a))l

) 1
r

d[g(t)]

] 1
l

≤
(m

2

) 1
rl |η(ϕ(b), ϕ(a),m)|p+q+1B

l−1
l (g(t); 1, p, q)



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 817

×

{(∫ 1

0
gp+

1
2r (t)(1− g(t))q−

1
2r f l(ϕ(b))d[g(t)]

)r

+

(∫ 1

0
gp−

1
2r (t)(1− g(t))q+

1
2r f l(ϕ(a))d[g(t)]

)r} 1
rl

=
(m

2

) 1
rl |η(ϕ(b), ϕ(a),m)|p+q+1B

l−1
l (g(t); 1, p, q)

×

[
Br

(
g(t);

1

2r
, 2pr − 1, 2qr + 1

)
f rl(ϕ(a))

+Br

(
g(t);

1

2r
, 2pr + 1, 2qr − 1

)
f rl(ϕ(b))

] 1
rl

.

Corollary 2. Under the same conditions as in Theorem 4 for r = 1 and g(t) = t, we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

≤
(m

2

) 1
l |η(ϕ(b), ϕ(a),m)|p+q+1β

l−1
l (p+ 1, q + 1)

×

[
β

(
p+

1

2
, q +

3

2

)
f l(ϕ(a)) + β

(
p+

3

2
, q +

1

2

)
f l(ϕ(b))

] 1
l

.

3. Some new Hermite-Hadamard type conformable fractional integral
inequalities for twice differentiable MT(r;g,m,ϕ)-preinvex functions

In this section, in order to prove our main results regarding some generalizations of
Hermite-Hadamard type inequalities for twice differentiable MT(r;g,m,ϕ)-preinvex functions
via conformable fractional integrals, we need the following new interesting integral identity:

Lemma 2. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ [0, 1] is a
differentiable function. Suppose K ⊆ R be an open m-invex subset with respect to η :
K ×K × (0, 1] −→ R for any fixed m ∈ (0, 1] and let mϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m).
Assume that f : K = [mϕ(a),mϕ(a) + η(ϕ(b), ϕ(a),m)] −→ R be a twice differentiable
function on K◦ and f ′′ ∈ L1[mϕ(a),mϕ(a) + η(ϕ(b), ϕ(a),m)]. Then for α > 0, we have

ηα+2(ϕ(x), ϕ(a),m)

η(ϕ(b), ϕ(a),m)

×

{
β(n+ 2, α− n)

η(ϕ(x), ϕ(a),m)



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 818

×
[
f ′(mϕ(a) + g(1)η(ϕ(x), ϕ(a),m))− f ′(mϕ(a) + g(0)η(ϕ(x), ϕ(a),m)

]
−
Bg(1)(n+ 2, α− n)f ′(mϕ(a) + g(1)η(ϕ(x), ϕ(a),m))

η(ϕ(x), ϕ(a),m)

+
1

η2(ϕ(x), ϕ(a),m)

×
[
gn+1(1)(1− g(1))α−n−1f(mϕ(a) + g(1)η(ϕ(x), ϕ(a),m))

−gn+1(0)(1− g(0))α−n−1f(mϕ(a) + g(0)η(ϕ(x), ϕ(a),m))
]

+
1

ηα+2(ϕ(x), ϕ(a),m)

×

[
(n+ 1)

∫ mϕ(a)+g(1)η(ϕ(x),ϕ(a),m)

mϕ(a)+g(0)η(ϕ(x),ϕ(a),m)
(t−mϕ(a))n

×(mϕ(a) + η(ϕ(x), ϕ(a),m)− t)α−n−1f(t)dt

−(α− n− 1)

∫ mϕ(a)+g(1)η(ϕ(x),ϕ(a),m)

mϕ(a)+g(0)η(ϕ(x),ϕ(a),m)
(t−mϕ(a))n+1

×(mϕ(a) + η(ϕ(x), ϕ(a),m)− t)α−n−2f(t)dt

]}

+
ηα+2(ϕ(x), ϕ(b),m)

η(ϕ(b), ϕ(a),m)

×

{
β(n+ 2, α− n)

η(ϕ(x), ϕ(b),m)

×
[
f ′(mϕ(b) + g(1)η(ϕ(x), ϕ(b),m))− f ′(mϕ(b) + g(0)η(ϕ(x), ϕ(b),m)

]
−
Bg(1)(n+ 2, α− n)f ′(mϕ(b) + g(1)η(ϕ(x), ϕ(b),m))

η(ϕ(x), ϕ(b),m)

+
1

η2(ϕ(x), ϕ(b),m)

×
[
gn+1(1)(1− g(1))α−n−1f(mϕ(b) + g(1)η(ϕ(x), ϕ(b),m))

−gn+1(0)(1− g(0))α−n−1f(mϕ(b) + g(0)η(ϕ(x), ϕ(b),m))
]

+
1

ηα+2(ϕ(x), ϕ(b),m)



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 819

×

[
(n+ 1)

∫ mϕ(b)+g(1)η(ϕ(x),ϕ(b),m)

mϕ(b)+g(0)η(ϕ(x),ϕ(b),m)
(t−mϕ(b))n

×(mϕ(b) + η(ϕ(x), ϕ(b),m)− t)α−n−1f(t)dt

−(α− n− 1)

∫ mϕ(b)+g(1)η(ϕ(x),ϕ(b),m)

mϕ(b)+g(0)η(ϕ(x),ϕ(b),m)
(t−mϕ(b))n+1

×(mϕ(b) + η(ϕ(x), ϕ(b),m)− t)α−n−2f(t)dt

]}

=
ηα+2(ϕ(x), ϕ(a),m)

η(ϕ(b), ϕ(a),m)

×
∫ 1

0
(β(n+ 2, α− n)−Bg(t)(n+ 2, α− n))f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))d[g(t)]

+
ηα+2(ϕ(x), ϕ(b),m)

η(ϕ(b), ϕ(a),m)

×
∫ 1

0
(β(n+ 2, α− n)−Bg(t)(n+ 2, α− n))f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))d[g(t)]. (5)

Proof. A simple proof of the equality can be done by performing two integration by
parts in the integrals from the right side, changing the variable and using Definition 5.
The details are left to the interested reader.

Throughout this paper we denote

If,g,η,ϕ(x;α, n,m, a, b) =
ηα+2(ϕ(x), ϕ(a),m)

η(ϕ(b), ϕ(a),m)

×
∫ 1

0
(β(n+ 2, α− n)−Bg(t)(n+ 2, α− n))f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))d[g(t)]

+
ηα+2(ϕ(x), ϕ(b),m)

η(ϕ(b), ϕ(a),m)

×
∫ 1

0
(β(n+ 2, α− n)−Bg(t)(n+ 2, α− n))f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))d[g(t)]. (6)

Using relation (6), the following results can be obtained for the corresponding version for
power of the second derivative.

Theorem 5. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ (0, 1) is a
differentiable function. Suppose K ⊆ R be an open m-invex subset with respect to η : K ×
K×(0, 1] −→ R for any fixed m ∈ (0, 1] and let mϕ(a) < mϕ(a)+η(ϕ(b), ϕ(a),m). Assume
that f : K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] −→ (0,∞) be a twice differentiable function



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 820

on K◦. If f ′′q is a nonnegative MT(r;g,m,ϕ)-preinvex function on K, q > 1, p−1 + q−1 = 1,
then for α > 0 and 0 < r ≤ 1, we have

|If,g,η,ϕ(x;α, n,m, a, b)| ≤
(m

2

) 1
rq δ

1
p (g(t); p, α, n)

|η(ϕ(b), ϕ(a),m)|

×

{
|η(ϕ(x), ϕ(a),m)|α+2

[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f ′′(ϕ(a))rq

+Br
g(1)

(
1 +

1

2r
, 1− 1

2r

)
f ′′(ϕ(x))rq

] 1
rq

+|η(ϕ(x), ϕ(b),m)|α+2

[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f ′′(ϕ(b))rq

+Br
g(1)

(
1 +

1

2r
, 1− 1

2r

)
f ′′(ϕ(x))rq

] 1
rq
}
, (7)

where δ(g(t); p, α, n) =

∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]p
d[g(t)].

Proof. Suppose that q > 1 and 0 < r ≤ 1. Using relation (6), Hölder inequality, the
fact that f ′′q is a nonnegative MT(r;g,m,ϕ)-preinvex function on an open m-invex set K◦,
combining with Minkowski inequality for all t ∈ [0, 1] and for any fixed m ∈ (0, 1] and
taking the modulus, we have

|If,g,η,ϕ(x;α, n,m, a, b)| ≤ |η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×
∫ 1

0

(
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

)
f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))d[g(t)]

+
|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×
∫ 1

0

(
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

)
f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))d[g(t)]

≤ |η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]p
d[g(t)]

) 1
p

×
(∫ 1

0
f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))qd[g(t)]

) 1
q

+
|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]p
d[g(t)]

) 1
p



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 821

×
(∫ 1

0
f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))qd[g(t)]

) 1
q

≤ |η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]p
d[g(t)]

) 1
p

×

∫ 1

0

[
m
√
g(t)

2
√

1− g(t)
f ′′(ϕ(x))rq +

m
√

1− g(t)

2
√
g(t)

f ′′(ϕ(a))rq

] 1
r

d[g(t)]


1
q

+
|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]p
d[g(t)]

) 1
p

×

∫ 1

0

[
m
√
g(t)

2
√

1− g(t)
f ′′(ϕ(x))rq +

m
√

1− g(t)

2
√
g(t)

f ′′(ϕ(b))rq

] 1
r

d[g(t)]


1
q

≤
(m

2

) 1
rq
δ

1
p (g(t); p, α, n)

|η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×

[∫ 1

0

(√
1− g(t)

g(t)

) 1
r

f ′′(ϕ(a))qd[g(t)]

r

+

∫ 1

0

(√
g(t)

1− g(t)

) 1
r

f ′′(ϕ(x))qd[g(t)]

r ] 1
rq

+
(m

2

) 1
rq
δ

1
p (g(t); p, α, n)

|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×

[∫ 1

0

(√
1− g(t)

g(t)

) 1
r

f ′′(ϕ(b))qd[g(t)]

r

+

∫ 1

0

(√
g(t)

1− g(t)

) 1
r

f ′′(ϕ(x))qd[g(t)]

r ] 1
rq

=
(m

2

) 1
rq δ

1
p (g(t); p, α, n)

|η(ϕ(b), ϕ(a),m)|

×

{
|η(ϕ(x), ϕ(a),m)|α+2

[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f ′′(ϕ(a))rq

+Br
g(1)

(
1 +

1

2r
, 1− 1

2r

)
f ′′(ϕ(x))rq

] 1
rq



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 822

+|η(ϕ(x), ϕ(b),m)|α+2

[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f ′′(ϕ(b))rq

+Br
g(1)

(
1 +

1

2r
, 1− 1

2r

)
f ′′(ϕ(x))rq

] 1
rq
}
.

Corollary 3. Under the same conditions as in Theorem 5, if we choose α ∈ (n, n + 1]
where n = 0, 1, 2, . . . and g(t) = t, we get the following inequality for conformable fractional
integrals: ∣∣∣∣∣−ηα+1(ϕ(x), ϕ(a),m)f ′(mϕ(a))− ηα+1(ϕ(x), ϕ(b),m)f ′(mϕ(b))

η(ϕ(b), ϕ(a),m)

−(n+ 2− α)(n+ 1)!

η(ϕ(b), ϕ(a),m)

×

[(
(mϕ(a)+η(ϕ(x),ϕ(a),m))Iαf

)
(mϕ(a)) +

(
(mϕ(b)+η(ϕ(x),ϕ(b),m))Iαf

)
(mϕ(b))

]∣∣∣∣∣
≤
(m

2

) 1
rq

(
π

2r sin
(
π
2r

)) 1
q δ

1
p (p, α, n)

|η(ϕ(b), ϕ(a),m)|

×

{
|η(ϕ(x), ϕ(a),m)|α+2

[
f ′′(ϕ(a))rq + f ′′(ϕ(x))rq

] 1
rq

+|η(ϕ(x), ϕ(b),m)|α+2
[
f ′′(ϕ(b))rq + f ′′(ϕ(x))rq

] 1
rq

}
,

where δ(p, α, n) =

∫ 1

0

[
β(n+ 2, α− n)− βt(n+ 2, α− n)

]p
dt.

Corollary 4. Under the same conditions as in Corollary 3, if we choose α = n+ 1 where
n = 0, 1, 2, . . . , r = 1 and f ′′ ≤ K, we get the following inequality for fractional integrals:∣∣∣∣∣−ηα+1(ϕ(x), ϕ(a),m)f ′(mϕ(a))− ηα+1(ϕ(x), ϕ(b),m)f ′(mϕ(b))

(α+ 1)η(ϕ(b), ϕ(a),m)

+
ηα(ϕ(x), ϕ(a),m)f(mϕ(a) + η(ϕ(x), ϕ(a),m)) + ηα(ϕ(x), ϕ(b),m)f(mϕ(b) + η(ϕ(x), ϕ(b),m))

η(ϕ(b), ϕ(a),m)

− Γ(α+ 1)

η(ϕ(b), ϕ(a),m)



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 823

×
[
Jα(mϕ(a)+η(ϕ(x),ϕ(a),m))−f(mϕ(a)) + Jα(mϕ(b)+η(ϕ(x),ϕ(b),m))−f(mϕ(b))

]∣∣∣∣∣
≤ K

(mπ
2

) 1
q

Γ(p+ 1)Γ
(

1
α+1

)
Γ
(
p+ 1 + 1

α+1

)


1
p

×

[
|η(ϕ(x), ϕ(a),m)|α+2 + |η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

]
.

Theorem 6. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ (0, 1) is a
differentiable function. Suppose K ⊆ R be an open m-invex subset with respect to η :
K ×K × (0, 1] −→ R for any fixed m ∈ (0, 1] and let mϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m).
Assume that f : K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] −→ (0,∞) be a twice differentiable
function on K◦. If f ′′q is a nonnegative MT(r;g,m,ϕ)-preinvex function on K, q ≥ 1, then
for α > 0 and 0 < r ≤ 1, we have

|If,g,η,ϕ(x;α, n,m, a, b)| ≤
(m

2

) 1
rq H

1− 1
q

|η(ϕ(b), ϕ(a),m)|

×

{
|η(ϕ(x), ϕ(a),m)|α+2

×

[(
β(n+ 2, α− n)Bg(1)

(
1− 1

2r
, 1 +

1

2r

)
−D(g(t);α, n, r)

)r
f ′′(ϕ(a))rq

+

(
β(n+ 2, α− n)Bg(1)

(
1 +

1

2r
, 1− 1

2r

)
− C(g(t);α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq

+|η(ϕ(x), ϕ(b),m)|α+2

×

[(
β(n+ 2, α− n)Bg(1)

(
1− 1

2r
, 1 +

1

2r

)
−D(g(t);α, n, r)

)r
f ′′(ϕ(b))rq

+

(
β(n+ 2, α− n)Bg(1)

(
1 +

1

2r
, 1− 1

2r

)
− C(g(t);α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq
}
, (8)

where

H =

∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
d[g(t)]

= β(n+ 2, α− n)(g(1)− g(0))− g(1)Bg(1)(n+ 2, α− n) +Bg(1)(n+ 3, α− n);

C(g(t);α, n, r) =

∫ 1

0
Bg(t)(n+ 2, α− n)

(√
g(t)

1− g(t)

) 1
r

d[g(t)];



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 824

D(g(t);α, n, r) =

∫ 1

0
Bg(t)(n+ 2, α− n)

(√
1− g(t)

g(t)

) 1
r

d[g(t)].

Proof. Suppose that q ≥ 1 and 0 < r ≤ 1. Using relation (6), the well-known power
mean inequality, the fact that f ′′q is a nonnegative MT(r;g,m,ϕ)-preinvex function on an
open m-invex set K◦, combining with Minkowski inequality for all t ∈ [0, 1] and for any
fixed m ∈ (0, 1] and taking the modulus, we have

|If,g,η,ϕ(x;α, n,m, a, b)| ≤ |η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×
∫ 1

0

(
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

)
f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))d[g(t)]

+
|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×
∫ 1

0

(
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

)
f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))d[g(t)]

≤ |η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
d[g(t)]

)1− 1
q

×

[∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]

×f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))qd[g(t)]

] 1
q

+
|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
d[g(t)]

)1− 1
q

×

[∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]

×f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))qd[g(t)]

] 1
q

≤ |η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
d[g(t)]

)1− 1
q

×

[∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 825

×

[
m
√
g(t)

2
√

1− g(t)
f ′′(ϕ(x))rq +

m
√

1− g(t)

2
√
g(t)

f ′′(ϕ(a))rq

] 1
r

d[g(t)]

] 1
q

+
|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
d[g(t)]

)1− 1
q

×

[∫ 1

0

[
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]

×

[
m
√
g(t)

2
√

1− g(t)
f ′′(ϕ(x))rq +

m
√

1− g(t)

2
√
g(t)

f ′′(ϕ(b))rq

] 1
r

d[g(t)]

] 1
q

≤
(m

2

) 1
rq
H

1− 1
q
|η(ϕ(x), ϕ(a),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×

[∫ 1

0

(√
1− g(t)

g(t)

) 1
r [
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
f ′′(ϕ(a))qd[g(t)]

r

+

∫ 1

0

(√
g(t)

1− g(t)

) 1
r [
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
f ′′(ϕ(x))qd[g(t)]

r ] 1
rq

+
(m

2

) 1
rq
H

1− 1
q
|η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

×

[∫ 1

0

(√
1− g(t)

g(t)

) 1
r [
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
f ′′(ϕ(b))qd[g(t)]

r

+

∫ 1

0

(√
g(t)

1− g(t)

) 1
r [
β(n+ 2, α− n)−Bg(t)(n+ 2, α− n)

]
f ′′(ϕ(x))qd[g(t)]

r ] 1
rq

=
(m

2

) 1
rq H

1− 1
q

|η(ϕ(b), ϕ(a),m)|

×

{
|η(ϕ(x), ϕ(a),m)|α+2

×

[(
β(n+ 2, α− n)Bg(1)

(
1− 1

2r
, 1 +

1

2r

)
−D(g(t);α, n, r)

)r
f ′′(ϕ(a))rq

+

(
β(n+ 2, α− n)Bg(1)

(
1 +

1

2r
, 1− 1

2r

)
− C(g(t);α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 826

+|η(ϕ(x), ϕ(b),m)|α+2

×

[(
β(n+ 2, α− n)Bg(1)

(
1− 1

2r
, 1 +

1

2r

)
−D(g(t);α, n, r)

)r
f ′′(ϕ(b))rq

+

(
β(n+ 2, α− n)Bg(1)

(
1 +

1

2r
, 1− 1

2r

)
− C(g(t);α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq
}
.

Corollary 5. Under the same conditions as in Theorem 6, if we choose α ∈ (n, n + 1]
where n = 0, 1, 2, . . . and g(t) = t, we get the following inequality for conformable fractional
integrals: ∣∣∣∣∣−ηα+1(ϕ(x), ϕ(a),m)f ′(mϕ(a))− ηα+1(ϕ(x), ϕ(b),m)f ′(mϕ(b))

η(ϕ(b), ϕ(a),m)

−(n+ 2− α)(n+ 1)!

η(ϕ(b), ϕ(a),m)

×

[(
(mϕ(a)+η(ϕ(x),ϕ(a),m))Iαf

)
(mϕ(a)) +

(
(mϕ(b)+η(ϕ(x),ϕ(b),m))Iαf

)
(mϕ(b))

]∣∣∣∣∣
≤
(m

2

) 1
rq β

1− 1
q (n+ 3, α− n)

|η(ϕ(b), ϕ(a),m)|

×

{
|η(ϕ(x), ϕ(a),m)|α+2

×

[(
π

2r sin
(
π
2r

)β(n+ 2, α− n)−D(t;α, n, r)

)r
f ′′(ϕ(a))rq

+

(
π

2r sin
(
π
2r

)β(n+ 2, α− n)− C(t;α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq

+|η(ϕ(x), ϕ(b),m)|α+2

×

[(
π

2r sin
(
π
2r

)β(n+ 2, α− n)−D(t;α, n, r)

)r
f ′′(ϕ(b))rq

+

(
π

2r sin
(
π
2r

)β(n+ 2, α− n)− C(t;α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq
}
.



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 827

Corollary 6. Under the same conditions as in Corollary 5, if we choose α = n+ 1 where
n = 0, 1, 2, . . . , r = 1 and f ′′ ≤ K, we get the following inequality for fractional integrals:∣∣∣∣∣−ηα+1(ϕ(x), ϕ(a),m)f ′(mϕ(a))− ηα+1(ϕ(x), ϕ(b),m)f ′(mϕ(b))

(α+ 1)η(ϕ(b), ϕ(a),m)

+
ηα(ϕ(x), ϕ(a),m)f(mϕ(a) + η(ϕ(x), ϕ(a),m)) + ηα(ϕ(x), ϕ(b),m)f(mϕ(b) + η(ϕ(x), ϕ(b),m))

η(ϕ(b), ϕ(a),m)

− Γ(α+ 1)

η(ϕ(b), ϕ(a),m)

×
[
Jα(mϕ(a)+η(ϕ(x),ϕ(a),m))−f(mϕ(a)) + Jα(mϕ(b)+η(ϕ(x),ϕ(b),m))−f(mϕ(b))

]∣∣∣∣∣
≤
(m

2

) 1
q K

(α+ 2)
1− 1

q

[
π

α+ 1
− (C(t;α, n, 1) +D(t;α, n, 1))

] 1
q

×

[
|η(ϕ(x), ϕ(a),m)|α+2 + |η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

]
.

Remark 3. If we choose α = (n, n + 1] where n = 0, 1, 2, . . . , for different choices of

positive value r =
1

2
,
1

3
etc., for any fixed m ∈ (0, 1], for a particular choices of a differen-

tiable function g(t), for example: e−(t+1), sin
(
π(t+1)

3

)
, cos

(
π(t+1)

3

)
, etc., and a particular

choices of a continuous function ϕ(x) = ex for all x ∈ R, xn for all x > 0 and for all n ∈ N,
etc., by Theorem 5 and Theorem 6 we can get some special kinds of Hermite-Hadamard
type conformable fractional integral inequalities. In particular for α = n+1, n = 0, 1, 2, . . . ,
we get some special kinds of Hermite-Hadamard type fractional integral inequalities.

4. Applications to special means

In the following we give certain generalizations of some notions for a positive valued
function of a positive variable.

Definition 12. (see [36]) A function M : R2
+ −→ R+, is called a Mean function if it has

the following properties:

(i) Homogeneity: M(ax, ay) = aM(x, y), for all a > 0,

(ii) Symmetry: M(x, y) = M(y, x),

(iii) Reflexivity: M(x, x) = x,

(iv) Monotonicity: If x ≤ x′ and y ≤ y′, then M(x, y) ≤M(x′, y′),



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 828

(v) Internality: min{x, y} ≤M(x, y) ≤ max{x, y}.

We consider some means for arbitrary positive real numbers α, β (α 6= β).

(i) The arithmetic mean:

A := A(α, β) =
α+ β

2

(ii) The geometric mean:
G := G(α, β) =

√
αβ

(iii) The harmonic mean:

H := H(α, β) =
2

1
α + 1

β

(iv) The power mean:

Pr := Pr(α, β) =

(
αr + βr

2

) 1
r

, r ≥ 1.

(v) The identric mean:

I := I(α, β) =

{
1
e

(
ββ

αα

)
, α 6= β;

α, α = β.

(vi) The logarithmic mean:

L := L(α, β) =
β − α

ln(β)− ln(α)
.

(vii) The generalized log-mean:

Lp := Lp(α, β) =

[
βp+1 − αp+1

(p+ 1)(β − α)

] 1
p

; p ∈ R \ {−1, 0}.

(viii) The weighted p-power mean:

Mp

(
α1, α2, · · · , αn
u1, u2, · · · , un

)
=

(
n∑
i=1

αiu
p
i

) 1
p

where 0 ≤ αi ≤ 1, ui > 0 (i = 1, 2, . . . , n) with
∑n

i=1 αi = 1.

It is well known that Lp is monotonic nondecreasing over p ∈ R with L−1 := L and L0 := I.
In particular, we have the following inequality H ≤ G ≤ L ≤ I ≤ A. Now, let a and b
be positive real numbers such that a < b. Consider the function M := M(ϕ(x), ϕ(y)) :
[ϕ(x), ϕ(x) + η(ϕ(y), ϕ(x))] × [ϕ(x), ϕ(x) + η(ϕ(y), ϕ(x))] −→ R+, which is one of the
above mentioned means, ϕ : I −→ K be a continuous function and g : [0, 1] −→ (0, 1) is



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 829

a differentiable function. Therefore one can obtain various inequalities using the results
of Section 3 for these means as follows: Replace η(ϕ(x), ϕ(y),m) with η(ϕ(x), ϕ(y)) and
setting η(ϕ(x), ϕ(y)) = M(ϕ(x), ϕ(y)), ∀x, y ∈ I, for value m = 1 in (7) and (8), one can
obtain the following interesting inequalities involving means:

|If,g,M(·,·),ϕ(x;α, n, 1, a, b)| =

∣∣∣∣∣Mα+2(ϕ(a), ϕ(x))

M(ϕ(a), ϕ(b))

×

{
β(n+ 2, α− n)

M(ϕ(a), ϕ(x))

×
[
f ′(ϕ(a) + g(1)M(ϕ(a), ϕ(x)))− f ′(ϕ(a) + g(0)M(ϕ(a), ϕ(x))

]
−
Bg(1)(n+ 2, α− n)f ′(ϕ(a) + g(1)M(ϕ(a), ϕ(x)))

M(ϕ(a), ϕ(x))

+
1

M2(ϕ(a), ϕ(x))

×
[
gn+1(1)(1− g(1))α−n−1f(ϕ(a) + g(1)M(ϕ(a), ϕ(x)))

−gn+1(0)(1− g(0))α−n−1f(ϕ(a) + g(0)M(ϕ(a), ϕ(x)))
]

+
1

Mα+2(ϕ(a), ϕ(x))

×

[
(n+ 1)

∫ ϕ(a)+g(1)M(ϕ(a),ϕ(x))

ϕ(a)+g(0)M(ϕ(a),ϕ(x))
(t− ϕ(a))n

×(ϕ(a) +M(ϕ(a), ϕ(x))− t)α−n−1f(t)dt

−(α− n− 1)

∫ ϕ(a)+g(1)M(ϕ(a),ϕ(x))

ϕ(a)+g(0)M(ϕ(a),ϕ(x))
(t− ϕ(a))n+1

×(ϕ(a) +M(ϕ(a), ϕ(x))− t)α−n−2f(t)dt

]}

+
Mα+2(ϕ(b), ϕ(x))

M(ϕ(a), ϕ(b))

×

{
β(n+ 2, α− n)

M(ϕ(b), ϕ(x))

×
[
f ′(ϕ(b) + g(1)M(ϕ(b), ϕ(x)))− f ′(ϕ(b) + g(0)M(ϕ(b), ϕ(x))

]
−
Bg(1)(n+ 2, α− n)f ′(ϕ(b) + g(1)M(ϕ(b), ϕ(x)))

M(ϕ(b), ϕ(x))



A. Fundo, A. Kashuri, M. Ramosaço, R. Liko / Eur. J. Pure Appl. Math, 10 (4) (2017), 809-834 830

+
1

M2(ϕ(b), ϕ(x))

×
[
gn+1(1)(1− g(1))α−n−1f(ϕ(b) + g(1)M(ϕ(b), ϕ(x)))

−gn+1(0)(1− g(0))α−n−1f(ϕ(b) + g(0)M(ϕ(b), ϕ(x)))
]

+
1

Mα+2(ϕ(b), ϕ(x))

×

[
(n+ 1)

∫ ϕ(b)+g(1)M(ϕ(b),ϕ(x))

ϕ(b)+g(0)M(ϕ(b),ϕ(x))
(t− ϕ(b))n

×(ϕ(b) +M(ϕ(b), ϕ(x))− t)α−n−1f(t)dt

−(α− n− 1)

∫ ϕ(b)+g(1)M(ϕ(b),ϕ(x))

ϕ(b)+g(0)M(ϕ(b),ϕ(x))
(t− ϕ(b))n+1

×(ϕ(b) +M(ϕ(b), ϕ(x))− t)α−n−2f(t)dt

]}∣∣∣∣∣
≤
(

1

2

) 1
rq δ

1
p (g(t); p, α, n)

M(ϕ(a), ϕ(b))

×

{
Mα+2(ϕ(a), ϕ(x))

[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f ′′(ϕ(a))rq

+Br
g(1)

(
1 +

1

2r
, 1− 1

2r

)
f ′′(ϕ(x))rq

] 1
rq

+Mα+2(ϕ(b), ϕ(x))

[
Br
g(1)

(
1− 1

2r
, 1 +

1

2r

)
f ′′(ϕ(b))rq

+Br
g(1)

(
1 +

1

2r
, 1− 1

2r

)
f ′′(ϕ(x))rq

] 1
rq
}
, (9)

|If,g,M(·,·),ϕ(x;α, n, 1, a, b)| ≤
(

1

2

) 1
rq H

1− 1
q

M(ϕ(a), ϕ(b))

×

{
Mα+2(ϕ(a), ϕ(x))

×

[(
β(n+ 2, α− n)Bg(1)

(
1− 1

2r
, 1 +

1

2r

)
−D(g(t);α, n, r)

)r
f ′′(ϕ(a))rq



REFERENCES 831

+

(
β(n+ 2, α− n)Bg(1)

(
1 +

1

2r
, 1− 1

2r

)
− C(g(t);α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq

+Mα+2(ϕ(b), ϕ(x))

×

[(
β(n+ 2, α− n)Bg(1)

(
1− 1

2r
, 1 +

1

2r

)
−D(g(t);α, n, r)

)r
f ′′(ϕ(b))rq

+

(
β(n+ 2, α− n)Bg(1)

(
1 +

1

2r
, 1− 1

2r

)
− C(g(t);α, n, r)

)r
f ′′(ϕ(x))rq

] 1
rq
}
. (10)

Letting M(ϕ(x), ϕ(y)) = A,G,H, Pr, I, L, Lp,Mp, ∀x, y ∈ I in (9) and (10), we get the
inequalities involving means for a particular choices of a nonnegative twice differentiable
MT(r;g,1,ϕ)-preinvex function f. The details are left to the interested reader.

5. Conclusions

In this paper, we proved some new integral inequalities for the left-hand side of Gauss-
Jacobi type quadrature formula involving MT(r;g,m,ϕ)-preinvex functions. Also, we es-
tablished some new Hermite-Hadamard type integral inequalities for MT(r;g,m,ϕ)-preinvex
functions via conformable fractional integrals. These general inequalities give us some
new estimates for Hermite-Hadamard type conformable fractional integral and fractional
integral inequalities.

Motivated by this new interesting class of MT(r;g,m,ϕ)-preinvex functions we can indeed
see to be vital for fellow researchers and scientists working in the same domain.

We conclude that our methods considered here may be a stimulant for further investiga-
tions concerning Hermite-Hadamard type integral inequalities for various kinds of preinvex
functions involving classical integrals, Riemann-Liouville fractional integrals, k-fractional
integrals, local fractional integrals, fractional integral operators, q-calculus, (p, q)-calculus,
time scale calculus and conformable fractional integrals.

Acknowledgements

We would like to thank the anonymous referees for the useful comments and sugges-
tions.

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