EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 3, 2017, 495-505
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Hermite-Hadamard type fractional integral inequalities
for generalized (r; s,m, ϕ)-preinvex functions

Artion Kashuri1,∗, Rozana Liko1

1 Department of Mathematics, Faculty of Technical Science, University ”Ismail Qemali”,
Vlora, Albania

Abstract. In the present paper, a new class of generalized (r; s,m, ϕ)-preinvex functions is intro-
duced and some new integral inequalities for the left hand side of Gauss-Jacobi type quadrature
formula involving generalized (r; s,m, ϕ)-preinvex functions are given. Moreover, some general-
izations of Hermite-Hadamard type inequalities for generalized (r; s,m, ϕ)-preinvex functions via
Riemann-Liouville fractional integrals are established. These results not only extend the results
appeared in the literature (see [1], [2]), but also provide new estimates on these types.

2010 Mathematics Subject Classifications: Primary: 26A51. Secondary: 26A33, 26D07,
26D10, 26D15.

Key Words and Phrases: Hermite-Hadamard type inequality, Hölder’s inequality, Minkowski’s
inequality, Cauchy’s inequality, power mean inequality, Riemann-Liouville fractional integral, s-
convex function in the second sense, 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 sub-
set K ⊆ Rn,K◦ is used to denote the interior of K. Rn is used to denote a generic
n-dimensional vector space. The nonnegative real numbers are denoted by R◦ = [0,+∞).
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 on an interval I of real numbers
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)

∗Corresponding author.

Email addresses: artionkashuri@gmail.com (A. Kashuri), rozanaliko86@gmail.com (R. Liko)

http://www.ejpam.com 495 c© 2017 EJPAM All rights reserved.



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 496

Fractional calculus (see [14]) and the references cited therein, 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 biological sciences and economics.

Definition 1. 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.

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 [13], [14])
and the references cited therein.
Now, let us recall some definitions of various convex functions.

Definition 2. (see [4]) A nonnegative function f : I ⊆ R −→ R◦ 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 3. (see [5]) A function f : R◦ −→ R is said to be s-convex in the second
sense, if

f(λx+ (1− λ)y) ≤ λsf(x) + (1− λ)sf(y) (2)

for all x, y ∈ R◦, λ ∈ [0, 1] and s ∈ (0, 1].

It is clear that a 1-convex function must be convex on R◦ as usual. The s-convex
functions in the second sense have been investigated in (see [5]).

Definition 4. (see [6]) 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 please see (see [6], [7]) and the
references therein.

Definition 5. (see [8]) 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).



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 497

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 [9]).
Recently, Liu (see [10]) obtained several integral inequalities for the left hand side of (3)
under the Definition 2 of P -function.
Also in (see [11]), 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 generalized (r; s,m, ϕ)-preinvex
function is introduced and some new integral inequalities for the left hand side of (3) in-
volving generalized (r; s,m, ϕ)-preinvex functions are given. In Section 3, some generaliza-
tions of Hermite-Hadamard type inequalities for generalized (r; s,m, ϕ)-preinvex functions
via fractional integrals are given. These general inequalities give us some new estimates
for the left hand side of Gauss-Jacobi type quadrature formula and Hermite-Hadamard
type fractional integral inequalities.

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

Definition 6. (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].

Remark 1. In Definition 6, under certain conditions, the mapping η(y, x,m) could reduce
to η(y, x). For example when m = 1, then the m-invex set degenerates an invex set on K.

Definition 7. (see [12]) A positive function f on the invex set K is said to be logarith-
mically preinvex, if

f(u+ tη(v, u)) ≤ f1−t(u)f t(v)

for all u, v ∈ K and t ∈ [0, 1].

Definition 8. (see [12]) The function f on the invex set K is said to be r-preinvex with
respect to η, if

f(u+ tη(v, u)) ≤Mr(f(u), f(v); t)

holds for all u, v ∈ K and t ∈ [0, 1], where

Mr(x, y; t) =

{ [
(1− t)xr + tyr

] 1
r , if r 6= 0;

x1−tyt, if r = 0,

is the weighted power mean of order r for positive numbers x and y.



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 498

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

Definition 9. Let K ⊆ Rn be an open m-invex set with respect to η : K×K×(0, 1] −→ Rn,
and ϕ : I −→ K is a continuous increasing function. The function f : K −→ (0,∞) is
said to be generalized (r; s,m, ϕ)-preinvex with respect to η, if

f (mϕ(x) + tη(ϕ(y), ϕ(x),m)) ≤Mr(f(ϕ(x)), f(ϕ(y)),m, s; t) (4)

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

Mr(f(ϕ(x)), f(ϕ(y)),m, s; t) =


[
m(1− t)sf r(ϕ(x)) + tsf r(ϕ(y))

] 1
r
, if r 6= 0;

f(ϕ(x))m(1−t)sf(ϕ(y))t
s
, 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 9, it is worthwhile to note that the class of generalized (r; s,m, ϕ)-
preinvex function is a generalization of the class of s-convex in the second sense function
given in Definition 3. Also, for r = 1 and ϕ(x) = x, ∀x ∈ I, we get the notion of
generalized (s,m)-preinvex function (see [3]).

Example 1. Let f(x) = |x|, ϕ(x) = x, r = s = 1 and

η(y, x,m) =


y −mx, if x ≥ 0, y ≥ 0;
y −mx, if x ≤ 0, y ≤ 0;
mx− y, if x ≥ 0, y ≤ 0;
mx− y, if x ≤ 0, y ≥ 0.

Then f(x) is a generalized (1; 1,m, x)-preinvex function of with respect to η : R × R ×
(0, 1] −→ R and any fixed m ∈ (0, 1]. However, it is obvious that f(x) = |x| is not a
convex function on R.

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

Lemma 1. Let ϕ : I −→ K be a continuous increasing function. Assume that f : K =
[mϕ(a),mϕ(a) + η(ϕ(b), ϕ(a),m)] −→ R is a continuous function on the interval of real
numbers 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

= η(ϕ(b), ϕ(a),m)p+q+1

∫ 1

0
tp(1− t)qf(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt.



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 499

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) + tη(ϕ(b), ϕ(a),m)−mϕ(a))p

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

×f(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt

= η(ϕ(b), ϕ(a),m)p+q+1

∫ 1

0
tp(1− t)qf(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt.

The following definition will be used in the sequel.

Definition 10. The Euler Beta function is defined for x, y > 0 as

β(x, y) =

∫ 1

0
tx−1(1− t)y−1dt =

Γ(x)Γ(y)

Γ(x+ y)
.

Theorem 2. Let ϕ : I −→ K be a continuous increasing function. Assume that f : K =
[mϕ(a),mϕ(a) + η(ϕ(b), ϕ(a),m)] −→ (0,∞) is a continuous function on the interval of
real numbers K◦ with mϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m). Let k > 1 and 0 < r ≤ 1. If

f
k
k−1 is a generalized (r; s,m, ϕ)-preinvex function on an open m-invex set K with respect

to η : K ×K × (0, 1] −→ R for any fixed s,m ∈ (0, 1], then for any fixed p, q > 0,∫ 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

(
r

s+ r

) k−1
k

β
1
k (kp+ 1, kq + 1)

×
[
mf

rk
k−1 (ϕ(a)) + f

rk
k−1 (ϕ(b))

] k−1
rk
. (5)

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

function on K, combining with Lemma 1, Hölder inequality and Minkowski inequality for
all t ∈ [0, 1] and for any fixed s,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
tkp(1− t)kqdt

] 1
k



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 500

×

[∫ 1

0
f

k
k−1 (mϕ(a) + tη(ϕ(b), ϕ(a),m))dt

] k−1
k

≤ |η(ϕ(b), ϕ(a),m)|p+q+1β
1
k (kp+ 1, kq + 1)

×

[∫ 1

0

(
m(1− t)sf r(ϕ(a))

k
k−1 + tsf r(ϕ(b))

k
k−1

) 1
r
dt

] k−1
k

≤ |η(ϕ(b), ϕ(a),m)|p+q+1β
1
k (kp+ 1, kq + 1)

×

[(∫ 1

0
m

1
r (1− t)

s
r f

k
k−1 (ϕ(a))dt

)r
+

(∫ 1

0
t
s
r f

k
k−1 (ϕ(b))dt

)r ] k−1
rk

= |η(ϕ(b), ϕ(a),m)|p+q+1

(
r

s+ r

) k−1
k

β
1
k (kp+ 1, kq + 1)

×
[
mf

rk
k−1 (ϕ(a)) + f

rk
k−1 (ϕ(b))

] k−1
rk
.

Corollary 1. Under the same conditions as in Theorem 2 for r = 1, we get (see [1],
Theorem 2.2).

Theorem 3. Let ϕ : I −→ K be a continuous increasing function. Assume that f : K =
[mϕ(a),mϕ(a) + η(ϕ(b), ϕ(a),m)] −→ (0,∞) is a continuous function on the interval of
real numbers K◦ with mϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m). Let l ≥ 1 and 0 < r ≤ 1. If
f l is a generalized (r; s,m, ϕ)-preinvex function on an open m-invex set K with respect to
η : K ×K × (0, 1] −→ R for any fixed s,m ∈ (0, 1], then for any fixed p, q > 0,∫ 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+1β
l−1
l (p+ 1, q + 1)

×
[
mf rl(ϕ(a))βr

(
p+ 1, q +

s

r
+ 1
)

+ f rl(ϕ(b))βr
(
p+

s

r
+ 1, q + 1

) ] 1
rl
. (6)

Proof. Let l ≥ 1 and 0 < r ≤ 1. Since f l is a generalized (r; s,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 s,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



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 501

×
∫ 1

0

[
tp(1− t)q

] l−1
l
[
tp(1− t)q

] 1
l
f(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt

≤ |η(ϕ(b), ϕ(a),m)|p+q+1

[∫ 1

0
tp(1− t)qdt

] l−1
l

×

[∫ 1

0
tp(1− t)qf l(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt

] 1
l

≤ |η(ϕ(b), ϕ(a),m)|p+q+1β
l−1
l (p+ 1, q + 1)

×

[∫ 1

0
tp(1− t)q

(
m(1− t)sf r(ϕ(a))l + tsf r(ϕ(b))l

) 1
r
dt

] 1
l

≤ |η(ϕ(b), ϕ(a),m)|p+q+1β
l−1
l (p+ 1, q + 1)

×

[(∫ 1

0
m

1
r tp(1− t)q+

s
r f l(ϕ(a))dt

)r
+

(∫ 1

0
tp+

s
r (1− t)qf l(ϕ(b))dt

)r ] 1
rl

= |η(ϕ(b), ϕ(a),m)|p+q+1β
l−1
l (p+ 1, q + 1)

×
[
mf rl(ϕ(a))βr

(
p+ 1, q +

s

r
+ 1
)

+ f rl(ϕ(b))βr
(
p+

s

r
+ 1, q + 1

) ] 1
rl
.

Corollary 2. Under the same conditions as in Theorem 3 for r = 1, we get (see [1],
Theorem 2.3).

3. Hermite-Hadamard type fractional integral inequalities for
generalized (r; s,m, ϕ)-preinvex functions

In this section, we prove our main results regarding some generalizations of Hermite-
Hadamard type inequalities for generalized (r; s,m, ϕ)-preinvex functions via fractional
integrals.

Theorem 4. Let ϕ : I −→ K be a continuous increasing function. Suppose K ⊆ R be
an open m-invex subset with respect to η : K × K × (0, 1] −→ R for any fixed s,m ∈
(0, 1] with 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 generalized (r; s,m, ϕ)-preinvex function on an open m-
invex set K◦. Then for α > 0 and 0 < r ≤ 1, we have

Γ(α)

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

≤

[
mf r(ϕ(a))βr

(
α,
s

r
+ 1
)

+ f r(ϕ(b))

(
r

αr + s

)r ] 1
r

. (7)



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 502

Proof. Let 0 < r ≤ 1. Since f is a generalized (r; s,m, ϕ)-preinvex function on an open
m-invex set K◦, combining with Minkowski inequality for all t ∈ [0, 1] and for any fixed
s,m ∈ (0, 1], we get

Γ(α)

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

=

∫ 1

0
tα−1f(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt

≤
∫ 1

0
tα−1

[
m(1− t)sf r(ϕ(a)) + tsf r(ϕ(b))

] 1
r
dt

≤

{[∫ 1

0
tα−1+

s
r f(ϕ(b))dt

]r
+

[∫ 1

0
m

1
r tα−1(1− t)

s
r f(ϕ(a))dt

]r} 1
r

=

[
mf r(ϕ(a))βr

(
α,
s

r
+ 1
)

+ f r(ϕ(b))

(
r

αr + s

)r ] 1
r

.

Corollary 3. Under the same conditions as in Theorem 4 for m = s = 1, ϕ(x) = x and
η(ϕ(b), ϕ(a),m) = η(b, a), we get (see [2], Theorem 3.1).

Theorem 5. Let ϕ : I −→ K be a continuous increasing function. Suppose K ⊆ R be
an open m-invex subset with respect to η : K × K × (0, 1] −→ R for any fixed s,m ∈
(0, 1] with mϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m). Assume that f, h : K = [mϕ(a),mϕ(a) +
η(ϕ(b), ϕ(a),m)] −→ (0,∞) are respectively generalized (r; s,m, ϕ)-preinvex function and
generalized (l; s,m, ϕ)-preinvex function on an open m-invex set K◦. Then for α > 0, r > 1
and r−1 + l−1 = 1, we have

Γ(α)

ηα(ϕ(b), ϕ(a),m)
Jα(mϕ(a)+η(ϕ(b),ϕ(a),m))−f(mϕ(a))h(mϕ(a))

≤ 1

2

{[
mf r(ϕ(a))β

r
2

(
2(α− 1)

r
+ 1,

2s

r
+ 1

)
+ f r(ϕ(b))

(
r

2(α− 1 + s) + r

) r
2

] 2
r

(8)

+

[
mhl(ϕ(a))β

l
2

(
2(α− 1)

l
+ 1,

2s

l
+ 1

)
+ hl(ϕ(b))

(
l

2(α− 1 + s) + l

) l
2

] 2
l
}
.

Proof. Let r > 1 and r−1 + l−1 = 1. Since f and h are respectively generalized
(r; s,m, ϕ)-preinvex function and generalized (l; s,m, ϕ)-preinvex function on an open m-
invex set K◦, combining with Cauchy and Minkowski inequalities for all t ∈ [0, 1] and for
any fixed s,m ∈ (0, 1], we get

Γ(α)

ηα(ϕ(b), ϕ(a),m)
Jα(mϕ(a)+η(ϕ(b),ϕ(a),m))−f(mϕ(a))h(mϕ(a))



A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 10 (3) (2017), 495-505 503

=

∫ 1

0
t(α−1)(

1
r
+ 1
l )f(mϕ(a) + tη(ϕ(b), ϕ(a),m))

×h(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt

≤
∫ 1

0
t(α−1)(

1
r
+ 1
l )
[
m(1− t)sf r(ϕ(a)) + tsf r(ϕ(b))

] 1
r

×
[
m(1− t)shl(ϕ(a)) + tshl(ϕ(b))

] 1
l
dt

≤ 1

2

{∫ 1

0

[
tα−1+sf r(ϕ(b)) +mtα−1(1− t)sf r(ϕ(a))

] 2
r
dt

+

∫ 1

0

[
tα−1+shl(ϕ(b)) +mtα−1(1− t)shl(ϕ(a))

] 2
l
dt

}

≤ 1

2

[{(∫ 1

0
t
2(α−1+s)

r f2(ϕ(b))dt

) r
2

+

(∫ 1

0
m

2
r t

2(α−1)
r (1− t)

2s
r f2(ϕ(a))dt

) r
2

} 2
r

+

{(∫ 1

0
t
2(α−1+s)

l h2(ϕ(b))dt

) l
2

+

(∫ 1

0
m

2
l t

2(α−1)
l (1− t)

2s
l h2(ϕ(a))dt

) l
2

} 2
l
]

=
1

2

{[
mf r(ϕ(a))β

r
2

(
2(α− 1)

r
+ 1,

2s

r
+ 1

)
+ f r(ϕ(b))

(
r

2(α− 1 + s) + r

) r
2

] 2
r

+

[
mhl(ϕ(a))β

l
2

(
2(α− 1)

l
+ 1,

2s

l
+ 1

)
+ hl(ϕ(b))

(
l

2(α− 1 + s) + l

) l
2

] 2
l
}
.

Corollary 4. Under the same conditions as in Theorem 5 for m = s = 1, ϕ(x) = x and
η(ϕ(b), ϕ(a),m) = η(b, a), we get (see [2], Theorem 3.3).

Theorem 6. Let ϕ : I −→ K be a continuous increasing function. Suppose K ⊆ R be
an open m-invex subset with respect to η : K × K × (0, 1] −→ R for any fixed s,m ∈
(0, 1] with mϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m). Assume that f, h : K = [mϕ(a),mϕ(a) +
η(ϕ(b), ϕ(a),m)] −→ (0,∞) are respectively generalized (r; s,m, ϕ)-preinvex function and
generalized (l; s,m, ϕ)-preinvex function on an open m-invex set K◦. Then for α > 0, r > 1
and r−1 + l−1 = 1, we have

Γ(α)

ηα(ϕ(b), ϕ(a),m)
Jα(mϕ(a)+η(ϕ(b),ϕ(a),m))−f(mϕ(a))h(mϕ(a))

≤

{
f r(ϕ(b))

s+ α
+mf r(ϕ(a))β(α, s+ 1)

} 1
r

+

{
hl(ϕ(b))

s+ α
+mhl(ϕ(a))β(α, s+ 1)

} 1
l

. (9)



REFERENCES 504

Proof. Let r > 1 and r−1 + l−1 = 1. Since f and h are respectively generalized
(r; s,m, ϕ)-preinvex function and generalized (l; s,m, ϕ)-preinvex function on an open m-
invex set K◦, combining with Hölder inequality for all t ∈ [0, 1] and for any fixed s,m ∈
(0, 1], we get

Γ(α)

ηα(ϕ(b), ϕ(a),m)
Jα(mϕ(a)+η(ϕ(b),ϕ(a),m))−f(mϕ(a))h(mϕ(a))

=

∫ 1

0
t(α−1)(

1
r
+ 1
l )f(mϕ(a) + tη(ϕ(b), ϕ(a),m))

×h(mϕ(a) + tη(ϕ(b), ϕ(a),m))dt

≤

{∫ 1

0

[
tα−1+sf r(ϕ(b)) +mtα−1(1− t)sf r(ϕ(a))

] 1
r

×
[
tα−1+shl(ϕ(b)) +mtα−1(1− t)shl(ϕ(a))

] 1
l
dt

}

≤

{∫ 1

0

[
tα−1+sf r(ϕ(b)) +mtα−1(1− t)sf r(ϕ(a))

]
dt

} 1
r

+

{∫ 1

0

[
tα−1+shl(ϕ(b)) +mtα−1(1− t)shl(ϕ(a))

]
dt

} 1
l

=

{
f r(ϕ(b))

s+ α
+mf r(ϕ(a))β(α, s+ 1)

} 1
r

+

{
hl(ϕ(b))

s+ α
+mhl(ϕ(a))β(α, s+ 1)

} 1
l

.

Corollary 5. Under the same conditions as in Theorem 6 for m = s = 1, ϕ(x) = x and
η(ϕ(b), ϕ(a),m) = η(b, a), we get (see [2], Theorem 3.9).

Remark 3. For different choices of positive values r, l =
1

2
,
1

3
, 2, etc., for any fixed s,m ∈

(0, 1] and a particular choices of a continuous increasing function ϕ(x) = ex for all x ∈
R, xn for all x > 0 and for all n ∈ N, etc., by Theorem 4, Theorem 5 and Theorem 6 we
can get some special kinds of Hermite-Hadamard type fractional integral inequalities.

References

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[2] A. Akkurt, H. Yildirim, On some fractional integral inequalities of Hermite-Hadamard
type for r-preinvex functions, Khayyam J. Math., 2, 2 (2016), 119-126.



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