EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 1, 2018, 51-68
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Some new Hermite-Hadamard-Fejér type inequalities
via k-fractional integrals concerning differentiable

generalized relative semi-(r;m,h1, h2)-preinvex mappings

Miftar Ramosaçaj1, Artion Kashuri1,∗, Rozana Liko1

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

Abstract. In this article, we first presented a new identity concerning differentiable mappings
defined on m-invex set via k-fractional integrals. By using the notion of generalized relative semi-
(r;m,h1, h2)-preinvexity and the obtained identity as an auxiliary result, some new estimates with
respect to Hermite-Hadamard-Fejér type inequalities via k-fractional integrals are established. It
is pointed out that some new special cases can be deduced from main results of the article.

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

Key Words and Phrases: Hermite-Hadamard inequality, Hermite-Hadamard-Fejér inequality,
Hölder’s inequality, Minkowski inequality, power mean inequality, k-fractional integrals, m-invex.

1. Introduction

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 on I 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: miftar.ramosaco@gmail.com (M. Ramosaçaj),
artionkashuri@gmail.com (A. Kashuri), rozanaliko86@gmail.com (R. Liko)

http://www.ejpam.com 51 c© 2018 EJPAM All rights reserved.



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 52

The most well-known inequalities related to the integral mean of a convex function
f are the Hermite-Hadamard inequalities or its weighted versions, the so-called Hermite-
Hadamard-Fejér inequalities.

Definition 1. [28] A function w : [a, b] ⊆ R −→ R is said to be symmetric with respect

to
a+ b

2
, if w(x) = w(a+ b− x) holds for all x ∈ [a, b].

Example 1. Assume that w1, w2 : [a, b] ⊆ R −→ R, w1(x) = c for c ∈ R, w2(x) =(
x− a+ b

2

)2

, then w1, w2 are symmetric functions with respect to
a+ b

2
.

In [12], Fejér established the following Hermite-Hadamard-Fejér inequality which is the
weighted generalization of the Hermite-Hadamard inequality (1).

Theorem 2. [12] Let f : [a, b] ⊆ R −→ R be a convex function. Then the inequality

f

(
a+ b

2

)∫ b

a
w(x)dx ≤

∫ b

a
f(x)w(x)dx ≤ f(a) + f(b)

2

∫ b

a
w(x)dx (2)

holds, where w : [a, b] −→ R is nonnegative, integrable and symmetric to
a+ b

2
.

In recent years, various generalizations, extensions and variants of such inequalities
have been obtained. For other recent results which generalize, improve and extend the
inequalities (1) and (2) through various classes of convex functions interested readers are
referred to (see [[1]-[31],[33],[36],[39]-[44],[48],[49]]).

Let us recall some special functions and evoke some basic definitions as follows.

Definition 2. 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)
. (3)

Definition 3. 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. (4)

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.



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 53

Definition 4. For k ∈ R+ and x ∈ C, the k-gamma function is defined by

Γk(x) = lim
n−→∞

n!knnk
x
k
−1

(x)n,k
. (5)

Its integral representation is given by

Γk(α) =

∫ ∞
0

tα−1e−
tk

k dt. (6)

One can note that
Γk(α+ k) = αΓk(α). (7)

For k = 1, (6) gives integral representation of gamma function.

Definition 5. [35] Let f ∈ L1[a, b]. Then k-fractional integrals of order α, k > 0 with
a ≥ 0 are defined as

Iα,ka+ f(x) =
1

kΓk(α)

∫ x

a
(x− t)

α
k
−1f(t)dt, x > a

and

Iα,kb− f(x) =
1

kΓk(α)

∫ b

x
(t− x)

α
k
−1f(t)dt, b > x. (8)

For k = 1, k-fractional integrals give Riemann-Liouville integrals.

Definition 6. [47] A set Mϕ ⊆ Rn is named as a relative convex (ϕ-convex) set, if and
only if, there exists a function ϕ : Rn −→ Rn such that,

tϕ(x) + (1− t)ϕ(y) ∈Mϕ, ∀ x, y ∈ Rn : ϕ(x), ϕ(y) ∈Mϕ, t ∈ [0, 1]. (9)

Definition 7. [47] A function f is named as a relative convex (ϕ-convex) function on a
relative convex (ϕ-convex) set Mϕ, if and only if, there exists a function ϕ : Rn −→ Rn
such that,

f(tϕ(x) + (1− t)ϕ(y)) ≤ tf(ϕ(x)) + (1− t)f(ϕ(y)), (10)

∀ x, y ∈ Rn : ϕ(x), ϕ(y) ∈Mϕ, t ∈ [0, 1].

Definition 8. [8] A nonnegative function f : I ⊆ R −→ [0,+∞) is said to be P -function,
if

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

Definition 9. [34] Let h : [0, 1] −→ R be a non-negative function and h 6= 0. The function
f on the invex set K is said to be h-preinvex with respect to η, if

f
(
x+ tη(y, x)

)
≤ h(1− t)f(x) + h(t)f(y) (11)

for each x, y ∈ K and t ∈ [0, 1] where f(·) > 0.



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 54

Clearly, when putting h(t) = t in Definition 9, f becomes a preinvex function [38]. If
the mapping η(y, x) = y − x in Definition 9, then the non-negative function f reduces to
h-convex mappings [46].

Definition 10. [45] A non-negative function f : K ⊆ R −→ R is said to be a tgs-convex
function on K if the inequality

f
(
(1− t)x+ ty

)
≤ t(1− t)[f(x) + f(y)] (12)

grips for all x, y ∈ K and t ∈ (0, 1).

Definition 11. [32] A function f : I ⊆ R −→ R is said to MT-convex functions, if it is
non-negative and ∀ x, y ∈ I and t ∈ (0, 1) satisfies the subsequent inequality:

f(tx+ (1− t)y) ≤
√
t

2
√

1− t
f(x) +

√
1− t
2
√
t
f(y). (13)

Definition 12. [36] A function: I ⊆ R −→ R is said to be m-MT-convex, if f is positive
and for ∀ x, y∈I, and t ∈ (0, 1), among m ∈ [0, 1], satisfies the following inequality

f
(
tx+m(1− t)y

)
≤

√
t

2
√

1− t
f(x) +

m
√

1− t
2
√
t

f(y). (14)

Definition 13. [37] Let K ⊆ R be an open m-invex set respecting η : K×K×(0, 1] −→ R.
A function f : K −→ R and h1, h2 : [0, 1] −→ [0,+∞), if

f
(
mx+ tη(y, x,m)

)
≤ mh1(t)f(x) + h2(t)f(y) (15)

is valid for all x, y ∈ K and t ∈ [0, 1], together m ∈ (0, 1], is said to be generalized
(m,h1, h2)-preinvex functions with respect to η.

Motivated by the above literatures, the main objective of this article is to establish
some new estimates on Hermite-Hadamard-Fejér type inequalities via k-fractional integrals
associated with generalized relative semi-(r;m,h1, h2)-preinvex mappings. It is pointed
out that some new special cases will be deduced from main results of the article. Our
results also generalize Theorem 2.2 and Theorem 2.5 shown in [10].

2. Main results

The following definitions will be used in this section.

Definition 14. [9] A set K ⊆ Rn is named as m-invex with respect to the mapping
η : K ×K × (0, 1] −→ Rn for some fixed m ∈ (0, 1], if mx+ tη(y,mx) ∈ K grips for each
x, y ∈ K and any t ∈ [0, 1].

Remark 1. In Definition 14, under certain conditions, the mapping η(y,mx) could reduce
to η(y, x). For example when m = 1, then the m-invex set degenerates an invex set on K.



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 55

We next introduce a new class called generalized relative semi-(r;m,h1, h2)-preinvex
mappings.

Definition 15. Let K ⊆ R be an open nonempty m-invex set with respect to the mapping
η : K×K×(0, 1] −→ R. Suppose h1, h2 : [0, 1] −→ [0,+∞) and ϕ : I −→ K are continuous.
A function f : K −→ (0,+∞) is said to be generalized relative semi-(r;m,h1, h2)-preinvex,
if

f
(
mϕ(x) + tη(ϕ(y), ϕ(x),m)

)
≤Mr(h1(t), h2(t); f(x), f(y),m) (16)

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

Mr(h1(t), h2(t); f(x), f(y),m) :=


[
mh1(t)f

r(x) + h2(t)f
r(y)

] 1
r
, if r 6= 0;

[
f(x)

]mh1(t)[f(y)
]h2(t), if r = 0,

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

Remark 2. In Definition 15, if we choose r = 1 and ϕ(x) = x, then we get Definition 13.

Remark 3. For r = 1, let us discuss some special cases in Definition 15 as follows.

(I) If taking h1(t) = (1 − t)s, h2(t) = ts for s ∈ (0, 1], then we get generalized relative
semi-(m, s)− Breckner-preinvex mappings.
(II) If taking h1(t) = h2(t) = 1, then we get generalized relative semi-(m,P )-preinvex
mappings.
(III) If taking h1(t) = (1− t)−s, h2(t) = t−s for s ∈ (0, 1], then we get generalized relative
semi-(m, s)-Godunova-Levin-Dragomir-preinvex mappings.
(IV) If taking h1(t) = h(1− t), h2(t) = h(t), then we get generalized relative semi-(m,h)-
preinvex mappings.
(V) If taking h1(t) = h2(t) = t(1 − t), then we get generalized relative semi-(m, tgs)-
preinvex mappings.

(VI) If taking h1(t) =

√
1− t
2
√
t
, h2(t) =

√
t

2
√

1− t
, then we get generalized relative semi-m-

MT -preinvex mappings.

It is worth to mention here that to the best of our knowledge all the special cases
discussed above are new in the literature.

For establishing our main results regarding some new Hermite-Hadamard-Fejér type
integral inequalities associated with generalized relative semi-(r;m,h1, h2)-preinvexity via
k-fractional integrals, we need the following crucial lemma.

Lemma 1. Suppose K = [mϕ(a),mϕ(a) + η(ϕ(b), ϕ(a),m)] ⊆ R be an open nonempty
m-invex subset with respect to η : K ×K × (0, 1] −→ R for some fixed m ∈ (0, 1], where



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 56

η(ϕ(b), ϕ(a),m) > 0. Also, let ϕ : I −→ K and g : K −→ R are continuous. Assume that
f : K −→ R be a differentiable mapping on K◦ such that f ′ ∈ L1(K). Then for α, k > 0,
the following equality holds for k-fractional integrals:

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
g(s)ds

)α
k [
f(mϕ(a)) + f(mϕ(a) + η(ϕ(b), ϕ(a),m))

]

−α
k

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ t

mϕ(a)
g(s)ds

)α
k
−1

g(t)f(t)dt

−α
k

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

)α
k
−1

g(t)f(t)dt

=

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ t

mϕ(a)
g(s)ds

)α
k

f ′(t)dt

−
∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

)α
k

f ′(t)dt. (17)

Proof. Let denote

If,g,η,ϕ(α, k,m, a, b) =

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ t

mϕ(a)
g(s)ds

)α
k

f ′(t)dt

−
∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

)α
k

f ′(t)dt. (18)

Integrating by parts, we get

If,g,η,ϕ(α, k,m, a, b) =

(∫ t

mϕ(a)
g(s)ds

)α
k

f(t)

∣∣∣∣∣
mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

−α
k

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ t

mϕ(a)
g(s)ds

)α
k
−1

g(t)f(t)dt

−

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

)α
k

f(t)

∣∣∣∣∣
mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

−α
k

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

)α
k
−1

g(t)f(t)dt



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 57

=

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
g(s)ds

)α
k [
f(mϕ(a)) + f(mϕ(a) + η(ϕ(b), ϕ(a),m))

]

−α
k

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ t

mϕ(a)
g(s)ds

)α
k
−1

g(t)f(t)dt

−α
k

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

)α
k
−1

g(t)f(t)dt.

This completes the proof of the lemma.

Remark 4. For η(ϕ(b), ϕ(a),m) = ϕ(b)−mϕ(a), where ϕ(x) = x, ∀x ∈ I and m = 1, we
get ([10], Lemma 2.1).

Using Lemma 1, we now state the following theorems for the corresponding version for
power of first derivative.

Theorem 3. Let α, k > 0 and 0 < r ≤ 1. Suppose K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] ⊆
R be an open nonempty m-invex subset with respect to η : K × K × (0, 1] −→ R for
some fixed m ∈ (0, 1], where η(ϕ(b), ϕ(a),m) > 0. Also, let h1, h2 : [0, 1] −→ [0,+∞),
ϕ : I −→ K and g : K −→ R are continuous. Assume that f : K −→ (0,+∞) be
a differentiable mapping on K◦ such that f ′ ∈ L1(K). If (f ′(x))q is generalized relative
semi-(r;m,h1, h2)-preinvex mapping, q > 1, p−1 + q−1 = 1 and ‖g‖∞ = sup |g(t)|, then
the following inequality for k-fractional integrals holds:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα
k + 1

)1/p
×
[
m(f ′(a))rqIr(h1(t); r) + (f ′(b))rqIr(h2(t); r)

] 1
rq
, (19)

where

I(hi(t); r) :=

∫ 1

0
h

1
r
i (t)dt, ∀ i = 1, 2.

Proof. Suppose that q > 1, p−1 + q−1 = 1 and 0 < r ≤ 1. From Lemma 1, generalized
relative semi-(r;m,h1, h2)-preinvexity of (f ′(x))q, Hölder inequality, Minkowski inequality,
properties of the modulus, the fact g(t) ≤ ‖g‖∞ and changing the variable t = mϕ(a) +
xη(ϕ(b), ϕ(a),m), we have

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ ∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ t

mϕ(a)
g(s)ds

∣∣∣∣∣
α
k

|f ′(t)|dt

+

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

∣∣∣∣∣
α
k

|f ′(t)|dt



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 58

≤

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ t

mϕ(a)
g(s)ds

∣∣∣∣∣
pα
k

dt

 1
p (∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(f ′(t))qdt

) 1
q

+

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

∣∣∣∣∣
pα
k

dt

 1
p

×

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(f ′(t))qdt

) 1
q

≤ ‖g‖
α
k∞ ×

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(f ′(t))qdt

) 1
q

×

{(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(t−mϕ(a))

pα
k dt

) 1
p

+

(∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)
(mϕ(a) + η(ϕ(b), ϕ(a),m)− t)

pα
k dt

) 1
p
}

=
2‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα
k + 1

)1/p ×
(∫ 1

0

(
f ′(mϕ(a) + tη(ϕ(b), ϕ(a),m))

)q
dt

) 1
q

≤ 2‖g‖
α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα
k + 1

)1/p ×
(∫ 1

0

[
mh1(t)(f

′(a))rq + h2(t)(f
′(b))rq

] 1
r
dt

) 1
q

≤ 2‖g‖
α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα
k + 1

)1/p
×

[(∫ 1

0
m

1
r (f ′(a))qh

1
r
1 (t)dt

)r
+

(∫ 1

0
(f ′(b))qh

1
r
2 (t)dt

)r ] 1
rq

=
2‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα
k + 1

)1/p
×
[
m(f ′(a))rqIr(h1(t); r) + (f ′(b))rqIr(h2(t); r)

] 1
rq
.

So, the proof of this theorem is complete.

Remark 5. For h1(t) = 1−t, h2(t) = t, r = m = 1, and η(ϕ(b), ϕ(a),m) = ϕ(b)−mϕ(a),
where ϕ(x) = x, ∀x ∈ I, we get ([10], Theorem 2.5).



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 59

We point out some special cases of Theorem 3.

Corollary 1. In Theorem 3 for p = q = 2, we have the following Hermite-Hadamard type
inequality for generalized relative semi-(r;m,h1, h2)-preinvex mappings via k-fractional
integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)√

2α
k + 1

×
[
m(f ′(a))2rIr(h1(t); r) + (f ′(b))2rIr(h2(t); r)

] 1
2r
. (20)

Corollary 2. In Theorem 3 for g(s) ≡ 1, we have the following Hermite-Hadamard type
inequality for generalized relative semi-(r;m,h1, h2)-preinvex mappings via k-fractional
integrals: ∣∣∣∣∣f(mϕ(a)) + f(mϕ(a) + η(ϕ(b), ϕ(a),m))

2
− Γk(α+ k)

2η
α
k (ϕ(b), ϕ(a),m)

×
[
Iα,k(mϕ(a))+f(mϕ(a) + η(ϕ(b), ϕ(a),m)) + Iα,k(mϕ(a)+η(ϕ(b),ϕ(a),m))−f(mϕ(a))

]∣∣∣∣∣
≤ η

α
k
+1(ϕ(b), ϕ(a),m)(pα

k + 1
)1/p ×

[
m(f ′(a))rqIr(h1(t); r) + (f ′(b))rqIr(h2(t); r)

] 1
rq
. (21)

Corollary 3. In Theorem 3 for h1(t) = h(1 − t), h2(t) = h(t) and f ′(x) ≤ K, ∀x ∈ I,
we get the following Hermite-Hadamard-Fejér type inequality for generalized relative semi-
(r;m,h)-preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2K(m+ 1)

1
rq ‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα

k + 1
)1/p I

1
q (h(t); r). (22)

Corollary 4. In Corollary 3 for h1(t) = (1−t)s, h2(t) = ts, we get the following Hermite-
Hadamard-Fejér type inequality for generalized relative semi-(r;m, s)-Breckner-preinvex
mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2K(m+ 1)

1
rq ‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα

k + 1
)1/p (

r

r + s

) 1
q

. (23)

Corollary 5. In Corollary 3 for h1(t) = (1− t)−s, h2(t) = t−s and 0 < s < r, we get the
following Hermite-Hadamard-Fejér type inequality for generalized relative semi-(r;m, s)-
Godunova-Levin-Dragomir-preinvex mappings via k-fractional integrals:



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 60

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2K(m+ 1)

1
rq ‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα

k + 1
)1/p (

r

r − s

) 1
q

. (24)

Corollary 6. In Theorem 3 for h1(t) = h2(t) = t(1−t) and f ′(x) ≤ K, ∀x ∈ I, we get the
following Hermite-Hadamard-Fejér type inequality for generalized relative semi-(m, tgs)-
preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2K(m+ 1)

1
rq ‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα

k + 1
)1/p β

1
q

(
1 +

1

r
, 1 +

1

r

)
. (25)

Corollary 7. In Theorem 3 for h1(t) =

√
1− t
2
√
t
, h2(t) =

√
t

2
√

1− t
and f ′(x) ≤ K, ∀x ∈ I,

we get the following Hermite-Hadamard-Fejér type inequality for generalized relative semi-
(r;m)-MT -preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2

1− 1
rqK(m+ 1)

1
rq ‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(pα

k + 1
)1/p β

1
q

(
1− 1

2r
, 1 +

1

2r

)
.

(26)

Theorem 4. Let α, k > 0 and 0 < r ≤ 1. Suppose K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] ⊆
R be an open nonempty m-invex subset with respect to η : K × K × (0, 1] −→ R for
some fixed m ∈ (0, 1], where η(ϕ(b), ϕ(a),m) > 0. Also, let h1, h2 : [0, 1] −→ [0,+∞),
ϕ : I −→ K and g : K −→ R are continuous. Assume that f : K −→ (0,+∞) be
a differentiable mapping on K◦ such that f ′ ∈ L1(K). If (f ′(x))q is generalized relative
semi-(r;m,h1, h2)-preinvex mapping, q ≥ 1 and ‖g‖∞ = sup |g(t)|, then the following
inequality for k-fractional integrals holds:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ ‖g‖αk∞η αk+1(ϕ(b), ϕ(a),m)(

α
k + 1

)1− 1
q

×

{[
m(f ′(a))rqIr(h1(t);α, k, r) + (f ′(b))rqIr(h2(t);α, k, r)

] 1
rq

+
[
m(f ′(a))rqI

r
(h1(t);α, k, r) + (f ′(b))rqI

r
(h2(t);α, k, r)

] 1
rq

}
, (27)

where

I(hi(t);α, k, r) :=

∫ 1

0
t
α
k h

1
r
i (t)dt, I(hi(t);α, k, r) :=

∫ 1

0
(1− t)

α
k h

1
r
i (t)dt, ∀ i = 1, 2.



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 61

Proof. Suppose that q ≥ 1 and 0 < r ≤ 1. From Lemma 1, generalized relative semi-
(r;m,h1, h2)-preinvexity of (f ′(x))q, the well-known power mean inequality, Minkowski
inequality, properties of the modulus, the fact g(t) ≤ ‖g‖∞ and changing the variable
t = mϕ(a) + xη(ϕ(b), ϕ(a),m), we have

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ ∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ t

mϕ(a)
g(s)ds

∣∣∣∣∣
α
k

|f ′(t)|dt

+

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

∣∣∣∣∣
α
k

|f ′(t)|dt

≤

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ t

mϕ(a)
g(s)ds

∣∣∣∣∣
α
k

dt

1− 1
q

×

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ t

mϕ(a)
g(s)ds

∣∣∣∣∣
α
k

(f ′(t))qdt

 1
q

+

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

∣∣∣∣∣
α
k

dt

1− 1
q

×

∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

∣∣∣∣∣
∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

t
g(s)ds

∣∣∣∣∣
α
k

(f ′(t))qdt

 1
q

≤ ‖g‖
α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[∫ 1

0
t
α
k
(
f ′(mϕ(a) + tη(ϕ(b), ϕ(a),m))

)q
dt

] 1
q

+

[∫ 1

0
(1− t)

α
k
(
f ′(mϕ(a) + tη(ϕ(b), ϕ(a),m))

)q
dt

] 1
q
}

≤ ‖g‖
α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[∫ 1

0
t
α
k

[
mh1(t)(f

′(a))rq + h2(t)(f
′(b))rq

] 1
r
dt

] 1
q



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 62

+

[∫ 1

0
(1− t)

α
k

[
mh1(t)(f

′(a))rq + h2(t)(f
′(b))rq

] 1
r
dt

] 1
q
}

≤ ‖g‖
α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[(∫ 1

0
m

1
r (f ′(a))qt

α
k h

1
r
1 (t)dt

)r
+

(∫ 1

0
(f ′(b))qt

α
k h

1
r
2 (t)dt

)r ] 1
rq

+

[(∫ 1

0
m

1
r (f ′(a))q(1− t)

α
k h

1
r
1 (t)dt

)r
+

(∫ 1

0
(f ′(b))q(1− t)

α
k h

1
r
2 (t)dt

)r ] 1
rq
}

=
‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[
m(f ′(a))rqIr(h1(t);α, k, r) + (f ′(b))rqIr(h2(t);α, k, r)

] 1
rq

+
[
m(f ′(a))rqI

r
(h1(t);α, k, r) + (f ′(b))rqI

r
(h2(t);α, k, r)

] 1
rq

}
.

So, the proof of this theorem is complete.

Remark 6. For h1(t) = 1 − t, h2(t) = t, r = m = q = 1, and η(ϕ(b), ϕ(a),m) =
ϕ(b)−mϕ(a), where ϕ(x) = x, ∀x ∈ I, we get ([10], Theorem 2.2).

We point out some special cases of Theorem 4.

Corollary 8. In Theorem 4 for q = 1, we have the following Hermite-Hadamard type
inequality for generalized relative semi-(r;m,h1, h2)-preinvex mappings via k-fractional
integrals: ∣∣If,g,η,ϕ(α, k,m, a, b)

∣∣ ≤ ‖g‖αk∞η αk+1(ϕ(b), ϕ(a),m)

×

{[
m(f ′(a))rIr(h1(t);α, k, r) + (f ′(b))rIr(h2(t);α, k, r)

] 1
r

+
[
m(f ′(a))rI

r
(h1(t);α, k, r) + (f ′(b))rI

r
(h2(t);α, k, r)

] 1
r

}
. (28)



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 63

Corollary 9. In Theorem 4 for g(s) ≡ 1, we have the following Hermite-Hadamard type
inequality for generalized relative semi-(r;m,h1, h2)-preinvex mappings via k-fractional
integrals: ∣∣∣∣∣f(mϕ(a)) + f(mϕ(a) + η(ϕ(b), ϕ(a),m))

2
− Γk(α+ k)

2η
α
k (ϕ(b), ϕ(a),m)

×
[
Iα,k(mϕ(a))+f(mϕ(a) + η(ϕ(b), ϕ(a),m)) + Iα,k(mϕ(a)+η(ϕ(b),ϕ(a),m))−f(mϕ(a))

]∣∣∣∣∣
≤ η

α
k
+1(ϕ(b), ϕ(a),m)

2
(
α
k + 1

)1− 1
q

×

{[
m(f ′(a))rqIr(h1(t);α, k, r) + (f ′(b))rqIr(h2(t);α, k, r)

] 1
rq

+
[
m(f ′(a))rqI

r
(h1(t);α, k, r) + (f ′(b))rqI

r
(h2(t);α, k, r)

] 1
rq

}
. (29)

Corollary 10. In Theorem 4 for h1(t) = h(1 − t), h2(t) = h(t) and f ′(x) ≤ K, ∀x ∈ I,
we get the following Hermite-Hadamard-Fejér type inequality for generalized relative semi-
(r;m,h)-preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ K‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[
mIr(h(1−t);α, k, r)+Ir(h(t);α, k, r)

] 1
rq

+
[
mIr(h(t);α, k, r)+Ir(h(1−t);α, k, r)

] 1
rq

}
.

(30)

Corollary 11. In Corollary 10 for h1(t) = (1 − t)s, h2(t) = ts, we get the following
Hermite-Hadamard-Fejér type inequality for generalized relative semi-(r;m, s)-Breckner-
preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ K‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[
mβr

(s
r

+ 1,
α

k
+ 1
)

+

(
1

s
r + α

k + 1

)r ] 1
rq

+

[
m

(
1

s
r + α

k + 1

)r
+βr

(s
r

+ 1,
α

k
+ 1
)] 1

rq
}
.

(31)



M. Ramosaçaj, A. Kashuri, R. Liko / Eur. J. Pure Appl. Math, 11 (1) (2018), 51-68 64

Corollary 12. In Corollary 10 for h1(t) = (1−t)−s, h2(t) = t−s and 0 < s < r, we get the
following Hermite-Hadamard-Fejér type inequality for generalized relative semi-(r;m, s)-
Godunova-Levin-Dragomir-preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ K‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[
mβr

(
1− s

r
,
α

k
+ 1
)

+

(
1

α
k −

s
r + 1

)r ] 1
rq

+

[
m

(
1

α
k −

s
r + 1

)r
+βr

(
1− s

r
,
α

k
+ 1
)] 1

rq
}
.

(32)

Corollary 13. In Theorem 4 for h1(t) = h2(t) = t(1 − t) and f ′(x) ≤ K, ∀x ∈ I, we
get the following Hermite-Hadamard-Fejér type inequality for generalized relative semi-
(m, tgs)-preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ 2K(m+ 1)

1
rq ‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(

α
k + 1

)1− 1
q

β
1
q

(
1 +

1

r
,
α

k
+

1

r
+ 1

)
.

(33)

Corollary 14. In Theorem 4 for h1(t) =

√
1− t
2
√
t
, h2(t) =

√
t

2
√

1− t
and f ′(x) ≤ K, ∀x ∈ I,

we get the following Hermite-Hadamard-Fejér type inequality for generalized relative semi-
(r;m)-MT -preinvex mappings via k-fractional integrals:

∣∣If,g,η,ϕ(α, k,m, a, b)
∣∣ ≤ (1

2

) 1
rq K‖g‖

α
k∞η

α
k
+1(ϕ(b), ϕ(a),m)(
α
k + 1

)1− 1
q

×

{[
mβr

(
α

k
− 1

2r
+ 1, 1 +

1

2r

)
+ βr

(
α

k
+

1

2r
+ 1, 1− 1

2r

)] 1
rq

+

[
mβr

(
α

k
+

1

2r
+ 1, 1− 1

2r

)
+ βr

(
α

k
− 1

2r
+ 1, 1 +

1

2r

)] 1
rq
}
. (34)

Remark 7. For k = 1, by our Theorems 3 and 4, we can get some new special Hermite-
Hadamard-Fejér type inequalities associated with generalized relative semi-(r;m,h1, h2)-
preinvex mappings via fractional integrals of order α > 0.

Remark 8. Also, applying our Theorems 3 and 4, we can deduce some new inequalities
using special means associated with generalized relative semi-(r;m,h1, h2)-preinvex map-
pings.



REFERENCES 65

References

[1] A. Akkurt, H. Yildirim, On Hermite-Hadamard-Fejér type inequalities for convex
functions via fractional integrals, Math. Morav., 21, 1 (2017), 105-123.

[2] F. Chen, A note on Hermite-Hadamard inequalities for products of convex functions
via Riemann-Liouville fractional integrals, Ital. J. Pure Appl. Math., 33, (2014),
299-306.

[3] Y. M. Chu, M. A. Khan, T. Ali, S. S. Dragomir, Inequalities for α-fractional differ-
entiable functions, J. Inequal. Appl., (2017) 2017:93, DOI10.1186/s13660-017-1371-6,
12 pages.

[4] Y. M. Chu, G. D. Wang, X. H. Zhang, Schur convexity and Hadamard’s inequality,
Math. Inequal. Appl., 13, 4 (2010), 725-731.

[5] Y. M. Chu, M. A. Khan, T. U. Khan, T. Ali, Generalizations of Hermite-Hadamard
type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9, 5 (2016), 4305-
4316.

[6] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional
integration, Ann. Funct. Anal., 1, 1 (2010), 51-58.

[7] S. S. Dragomir, Generalization, Refinement and Reverses of the Right Fejér Inequality
for Convex Functions, J. Math., Punjab Univ., 49, 3 (2017), 1-13.

[8] S. S. Dragomir, J. Pečarić, L. E. Persson, Some inequalities of Hadamard type, Soo-
chow J. Math., 21, (1995), 335-341.

[9] T. S. Du, J. G. Liao, Y. J. Li, Properties and integral inequalities of Hadamard-
Simpson type for the generalized (s,m)-preinvex functions, J. Nonlinear Sci. Appl.,
9, (2016), 3112-3126.

[10] G. Farid, A. U. Rehman, Generalizations of some integral inequalities for fractional
integrals, Ann. Math. Sil., 31, (2017), pp. 14.

[11] G. Farid, A. Javed, A. U. Rehman, On Hadamard inequalities for n-times differen-
tiable functions which are relative convex via Caputo k-fractional derivatives, Non-
linear Anal. Forum, To appear.

[12] L. Fejér, Uber die fourierreihen, ii, Math. Naturwise. Anz Ungar. Akad., Wiss 24,
(1906), 369-390.

[13] A. Fundo, A. Kashuri, M. Ramosaçaj, R. Liko, Some new Hermite-Hadamard
type conformable fractional integral inequalities for twice differentiable MT(r;g,m,ϕ)-
preinvex functions, Eur. J. Pure Appl. Math., 10, 4 (2017), 809-834.



REFERENCES 66

[14] I. Işcan, M. Kunt, Hermite-Hadamard-Fejér Type Inequalities for Harmonically
Quasi-convex Functions via Fractional Integrals, Kyungpook Math. J., 56, (2016),
845-859.

[15] A. Kashuri, R. Liko, On Hermite-Hadamard type inequalities for generalized (s,m, ϕ)-
preinvex functions via k-fractional integrals, Adv. Inequal. Appl., 6, (2017), 1-12.

[16] A. Kashuri, R. Liko, Generalizations of Hermite-Hadamard and Ostrowski type in-
equalities for MTm-preinvex functions, Proyecciones, 36, 1 (2017), 45-80.

[17] A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for gener-
alized (r; s,m, ϕ)-preinvex functions, Eur. J. Pure Appl. Math., 10, 3 (2017), 495-505.

[18] A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for twice
differentiable generalized (s,m, ϕ)-preinvex functions, Konuralp J. Math., 5, 2 (2017),
228-238.

[19] A. Kashuri, R. Liko, Hermite-Hadamard type inequalities for generalized (s,m, ϕ)-
preinvex functions via k-fractional integrals, Tbil. Math. J., 10, 4 (2017), 73-82.

[20] A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for
MT(m,ϕ)-preinvex functions, Stud. Univ. Babeş-Bolyai, Math., 62, 4 (2017), 439-450.

[21] A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for twice
differentiable generalized beta-preinvex functions, J. Fract. Calc. Appl., 9, 1 (2018),
241-252.

[22] A. Kashuri, R. Liko, M. Adil Khan, Y. M. Chu, Some new Ostrowski type frac-
tional integral inequalities for generalized (r; s,m, ϕ)-preinvex functions via Caputo
k-fractional derivatives, J. Fract. Calc. Appl., 9, 2 (2018), 163-177.

[23] M. A. Khan, T. Ali, S. S. Dragomir, M. Z. Sarikaya, Hermite-Hadamard type inequal-
ities for conformable fractional integrals, Revista de la Real Academia de Ciencias
Exactas, Fsicas y Naturales. Serie A. Matemticas, (2017), DOI:10.1007/s13398-017-
0408-5.

[24] M. Adil Khan, Y.-M. Chu, A. Kashuri, R. Liko, G. Ali, New Hermite-Hadamard
inequalities for conformable fractional integrals, J. Funct. Spaces, In press.

[25] M. Adil Khan, Y.-M. Chu, T. U. Khan, J. Khan, Some new inequalities of Hermite-
Hadamard type for s-convex functions with applications, Open Math., 15, (2017),
1414-1430.

[26] M. A. Khan, Y. Khurshid, T. Ali, N. Rehman, Inequalities for three times differen-
tiable functions, J. Math., Punjab Univ., 48, 2 (2016), 35-48.

[27] M. A. Khan, Y. Khurshid, T. Ali, Hermite-Hadamard inequality for fractional inte-
grals via η-convex functions, Acta Math. Univ. Comenianae, 79, 1 (2017), 153-164.



REFERENCES 67

[28] M. Kunt, I. Işcan, Hermite-Hadamard-Fejér type inequalities for p-convex functions,
Arab J. Math. Sci., 23, (2017), 215-230.

[29] M. Kunt, I. Işcan, N. Yazici, U. Gözütok, On new inequalities of Hermite-Hadamard-
Fejér type for harmonically convex functions via fractional integrals, SpringerPlus,
(2016) 5:635, 1-19.

[30] K. L. Tseng, S. R. Hwang, S. S. Dragomir, Fejér-type inequalities (II), Math. Slovaca,
67, 1 (2017), 109-120.

[31] M. A. Latif, S. S. Dragomir, E. Momoniat, Some Fejér type inequalities for
harmonically-convex functions with applications to special means, Int. J. Anal. Appl.,
13, 1 (2017), 1-14.

[32] W. Liu, W. Wen, J. Park, Ostrowski type fractional integral inequalities for MT-
convex functions, Miskolc Math. Notes, 16, 1 (2015), 249-256.

[33] W. Liu, W. Wen, J. Park, Hermite-Hadamard type inequalities for MT-convex func-
tions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl., 9, (2016),
766-777.

[34] M. Mat loka, Inequalities for h-preinvex functions, Appl. Math. Comput., 234, (2014),
52-57.

[35] S. Mubeen, G. M. Habibullah, k-Fractional integrals and applications, Int. J. Con-
temp. Math. Sci., 7, (2012), 89-94.

[36] O. Omotoyinbo, A. Mogbodemu, Some new Hermite-Hadamard integral inequalities
for convex functions, Int. J. Sci. Innovation Tech., 1, 1 (2014), 1-12.

[37] C. Peng, C. Zhou, T. S. Du, Riemann-Liouville fractional Simpson’s inequalities
through generalized (m,h1, h2)-preinvexity, Ital. J. Pure Appl. Math., 38, (2017),
345-367.

[38] R. Pini, Invexity and generalized convexity, Optimization, 22, (1991), 513-525.

[39] E. Set, A. Gözpinar, A study on Hermite-Hadamard type inequalities for s-convex
functions via conformable fractional integrals, Submitted.

[40] E. Set, A. Gözpnar, J. Choi, Hermite-Hadamard type inequalities for twice differ-
entiable m-convex functions via conformable fractional integrals, Far East J. Math.
Sci., 101, 4 (2017), 873-891.

[41] E. Set, S. S. Karataş, M. Adil Khan, Hermite-Hadamard type inequalities obtained
via fractional integral for differentiable m-convex and (α,m)-convex functions, Inter-
national Journal of Analysis, 2016, Article ID 4765691, 8 pages.



REFERENCES 68

[42] E. Set, I. Mumcu, Hermite-Hadamard-Fejér type inequalies for conformable fractional
integrals, Submitted.

[43] E. Set, M. Z. Sarikaya, A. Gözpinar, Some Hermite-Hadamard type inequalities for
convex functions via conformable fractional integrals and related inequalities, Creat.
Math. Inform., Accepted paper.

[44] H. N. Shi, Two Schur-convex functions related to Hadamard-type integral inequalities,
Publ. Math. Debrecen, 78, 2 (2011), 393-403.

[45] M. Tunç, E. Göv, Ü. Şanal, On tgs-convex function and their inequalities, Facta Univ.
Ser. Math. Inform., 30, 5 (2015), 679-691.

[46] S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326, 1 (2007), 303-311.

[47] E. A. Youness, E-convex sets, E-convex functions, and E-convex programming, J.
Optim. Theory Appl., 102, (1999), 439-450.

[48] X. M. Zhang, Y. M. Chu, X. H. Zhang, The Hermite-Hadamard type inequality
of GA-convex functions and its applications, J. Inequal. Appl., (2010), Article ID
507560, 11 pages.

[49] Y. C. Zhang, T. S. Du, J. Pan, On new inequalities of Fejér-Hermite-Hadamard type
for differentiable (α,m)-preinvex mappings, ScienceAsia, 43, (2017), 258-266.