1 Introduction
The Hermite-Hadamard (HH) inequality, which was independently found by Hermite and Hadamard (see, also [1], and [2, p. 137]), is particularly important in convex functions theory:
(1)
ϝ
π
1
+
π
2
2
≤
1
π
2
−
π
1
∫
π
1
π
2
ϝ
(
x
)
d
x
≤
ϝ
(
π
1
)
+
ϝ
(
π
2
)
2
,
where
ϝ
is a convex function on
[
π
1
,
π
2
]
in this case. The aforementioned inequality is true in reverse order for concave mappings.
In [3], Hudzik and Maligranda defined s-convex functions in the second sense as follows: a mapping
ϝ
:
R
+
→
R
, where
R
+
=
[
0
,
∞
)
is called s-convex in the second sense if
ϝ
(
t
x
+
(
1
−
t
)
y
)
≤
t
s
ϝ
(
x
)
+
(
1
−
t
)
s
ϝ
(
y
)
for all
x
,
y
∈
R
+
and
t
∈
[
0
,
1
]
and
s
∈
(
0
,
1
]
. Dragomir and Fitzpatrick [4] then used this newly discovered class of functions to prove the HH inequality on
[
π
1
,
π
2
]
as follows:
(2)
2
s
−
1
ϝ
π
1
+
π
2
2
≤
1
π
2
−
π
1
∫
π
1
π
2
ϝ
(
x
)
d
x
≤
ϝ
(
π
1
)
+
ϝ
(
π
2
)
s
+
1
.
On the other hand, several works in the field of
q
-analysis are being carried out, beginning with Euler, to achieve mastery in the mathematics that underpins quantum computing. The link between physics and mathematics is referred to as
q
-calculus. It has a wide range of applications in different areas of pure and applied mathematics [5,6]. Euler is thought to be the inventor of this significant branch of mathematics. In Newton’s work on infinite series, he used the
q
parameter. Later, Jackson [7,8] presented the
q
-calculus that knew without limits calculus in a logical approach. Al-Salam [9] presented the
q
-analogue of the
q
-fractional integral and the
q
-Riemann-Liouville fractional in 1966. Since then, the amount of study in this area has steadily expanded. In particular, in 2013, Tariboon and Ntouyas introduced
D
q
π
1
-difference operator and
q
π
1
-integral in [10]. In 2020, Bermudo et al. introduced the notion of
D
q
π
2
derivative and
q
π
2
-integral in [11]. Sadjang generalized to quantum calculus and introduced the notions of post-quantum calculus or shortly
(
p
,
q
)
-calculus in [12]. Soontharanon and Sitthiwirattham [13] introduced the notions of fractional
(
p
,
q
)
-calculus later on. In [14], Tunç and Göv gave the post-quantum variant of
D
q
π
1
-difference operator and
q
π
1
-integral. Recently, in 2021, Vivas-Cortez et al. introduced the notions of
D
p
,
q
π
2
derivative and
(
p
,
q
)
π
2
-integral in [15].
Many integral inequalities have been studied using quantum integrals for various types of functions. For example, in [16,17,18, 19,11,20, 21,22,23], the authors used
D
q
π
1
,
D
q
π
2
-derivatives and
q
π
1
,
q
π
2
-integrals to prove HH integral inequalities and their left-right estimates for convex and coordinated convex functions. In [24], Noor et al. presented a generalized version of quantum HH integral inequalities. For generalized quasi-convex functions, Nwaeze and Tameru proved certain parameterized quantum integral inequalities in [25]. Khan et al. proved quantum HH inequality using the green function in [26]. Budak et al. [27], Ali et al. [28,29] and Vivas-Cortez et al. [30] developed new quantum Simpson’s and quantum Newton’s type inequalities for convex and coordinated convex functions. For quantum Ostrowski’s inequalities for convex and co-ordinated convex functions, readers refer to [31,32,33]. Kunt et al. [34] generalized the results of [18] and proved Hermite-Hadamard type inequalities and their left estimates using
D
p
,
q
π
1
-difference operator and
(
p
,
q
)
π
1
-integral. Recently, Latif et al. [35] found the right estimates of Hermite-Hadamard type inequalities proved by Kunt et al. [34].
Inspired by these ongoing studies, in the context of
(
p
,
q
)
-calculus, we prove several new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in the second sense.
The following is the structure of this article: Section 2 provides a brief overview of the fundamentals of
q
-calculus as well as other related studies in this field. In Section 3, we go over some basic
(
p
,
q
)
-calculus notions and inequalities. In Section 4, we show the relationship between the results presented here and related results in the literature by proving post-quantum HH inequalities for s-convex functions in the second sense. Post-quantum Ostrowski type inequalities for s-convex functions in the second are presented in Section 5. In Section 6, we present some applications to special means of real numbers for newly established inequalities. Section 7 concludes with some recommendations for future research.
2 Preliminaries of
q
-calculus and some inequalities
In this section, we revisit several previously regarded ideas. In addition, throughout the paper,
s
∈
(
0
,
1
]
, and we use the following notations (see, [6]):
[
n
]
q
=
1
−
q
n
1
−
q
=
1
+
q
+
q
2
+
⋯
+
q
n
−
1
,
q
∈
(
0
,
1
)
.
In [8], Jackson gave the
q
-Jackson integral from 0 to
π
2
for
0
<
q
<
1
as follows:
(3)
∫
0
π
2
ϝ
(
x
)
d
q
x
=
(
1
−
q
)
π
2
∑
n
=
0
∞
q
n
ϝ
(
π
2
q
n
)
provided the sum converge absolutely.
Definition 1
[10] For a function
ϝ
:
[
π
1
,
π
2
]
→
R
, the
q
π
1
-derivative of
ϝ
at
x
∈
[
π
1
,
π
2
]
is characterized by the expression:
(4)
D
q
π
1
ϝ
(
x
)
=
ϝ
(
x
)
−
ϝ
(
q
x
+
(
1
−
q
)
π
1
)
(
1
−
q
)
(
x
−
π
1
)
,
x
≠
π
1
.
If
x
=
π
1
, we define
D
q
π
1
ϝ
(
π
1
)
=
lim
x
→
π
1
D
q
π
1
ϝ
(
x
)
if it exists, and it is finite.
Definition 2
[11] For a function
ϝ
:
[
π
1
,
π
2
]
→
R
, the
q
π
2
-derivative of
ϝ
at
x
∈
[
π
1
,
π
2
]
is characterized by the expression:
(5)
D
q
π
2
ϝ
(
x
)
=
ϝ
(
q
x
+
(
1
−
q
)
π
2
)
−
ϝ
(
x
)
(
1
−
q
)
(
π
2
−
x
)
,
x
≠
π
2
.
If
x
=
π
2
, we define
D
q
π
2
ϝ
(
π
2
)
=
lim
x
→
π
2
D
q
π
2
ϝ
(
x
)
if it exists and it is finite.
Definition 3
[10] Let
ϝ
:
[
π
1
,
π
2
]
→
R
be a function. Then, the
q
π
1
-definite integral on
[
π
1
,
π
2
]
is defined as follows:
(6)
∫
π
1
π
2
ϝ
(
x
)
d
q
π
1
x
=
(
1
−
q
)
(
π
2
−
π
1
)
∑
n
=
0
∞
q
n
ϝ
(
q
n
π
2
+
(
1
−
q
n
)
π
1
)
=
(
π
2
−
π
1
)
∫
0
1
ϝ
(
(
1
−
t
)
π
1
+
t
π
2
)
d
q
t
.
Definition 4
[11] Let
ϝ
:
[
π
1
,
π
2
]
→
R
be a function. Then, the
q
π
2
-definite integral on
[
π
1
,
π
2
]
is defined as follows:
(7)
∫
π
1
π
2
ϝ
(
x
)
d
q
π
2
x
=
(
1
−
q
)
(
π
2
−
π
1
)
∑
n
=
0
∞
q
n
ϝ
(
q
n
π
1
+
(
1
−
q
n
)
π
2
)
=
(
π
2
−
π
1
)
∫
0
1
ϝ
(
t
π
1
+
(
1
−
t
)
π
2
)
d
q
t
.
In [11], Bermudo et al. established the following quantum HH type inequality.
Theorem 1
For the convex mapping
ϝ
:
[
π
1
,
π
2
]
→
R
, the following inequality holds
(8)
ϝ
π
1
+
π
2
2
≤
1
2
(
π
2
−
π
1
)
∫
π
1
π
2
ϝ
(
x
)
d
q
π
1
x
+
∫
π
1
π
2
ϝ
(
x
)
d
q
π
2
x
≤
ϝ
(
π
1
)
+
ϝ
(
π
2
)
2
.
In [33], Budak et al. proved the following Ostrowski inequality by using the concepts of quantum derivatives and integrals.
Theorem 2
Let
ϝ
:
[
π
1
,
π
2
]
⊂
R
→
R
be a function and
D
q
π
2
ϝ
and
d
q
π
1
ϝ
be two continuous and integrable functions on
[
π
1
,
π
2
]
. If
∣
D
q
π
2
ϝ
(
t
)
∣
,
∣
D
q
π
1
ϝ
(
t
)
∣
≤
M
for all
t
∈
[
π
1
,
π
2
]
, then we have the following quantum quantum Ostrowski inequality:
(9)
ϝ
(
x
)
−
1
π
2
−
π
1
∫
π
1
x
ϝ
(
t
)
d
q
π
1
t
+
∫
x
π
2
ϝ
(
t
)
d
q
π
2
t
≤
q
M
(
π
2
−
π
1
)
(
x
−
π
1
)
2
+
(
π
2
−
x
)
2
[
2
]
q
.
Recently, Asawasamrit et al. [36] gave the following generalizations of inequalities (8) and (9) using the s-convexity.
Theorem 3
Assume that the mapping
ϝ
:
[
0
,
∞
)
→
R
is s-convex in the second sense and
π
1
,
π
2
∈
[
0
,
∞
)
with
π
1
<
π
2
, then the following inequality holds for
s
∈
(
0
,
1
]
:
(10)
2
s
−
1
ϝ
π
1
+
π
2
2
≤
1
2
(
π
2
−
π
1
)
∫
π
1
π
2
ϝ
(
x
)
d
q
π
1
x
+
∫
π
1
π
2
ϝ
(
x
)
d
q
π
2
x
≤
ϝ
(
π
1
)
+
ϝ
(
π
2
)
[
s
+
1
]
q
.
Theorem 4
Let
ϝ
:
[
π
1
,
π
2
]
⊂
R
→
R
be function and
D
q
π
2
ϝ
and
D
q
π
1
ϝ
be two continuous and integrable functions on
[
π
1
,
π
2
]
. If
∣
D
q
π
2
ϝ
(
t
)
∣
,
∣
D
q
π
1
ϝ
(
t
)
∣
≤
M
for all
t
∈
[
π
1
,
π
2
]
, then we have the following quantum Ostrowski inequality for s-convex functions in the second sense:
(11)
ϝ
(
x
)
−
1
π
2
−
π
1
∫
π
1
x
ϝ
(
t
)
d
q
π
1
t
+
∫
x
π
2
ϝ
(
t
)
d
q
π
2
t
≤
M
q
π
2
−
π
1
1
[
s
+
2
]
q
+
Θ
11
[
(
x
−
π
1
)
2
+
(
π
2
−
x
)
2
]
,
where
Θ
11
=
∫
0
1
t
(
1
−
t
)
d
q
s
t
.
3 Post-quantum calculus and some inequalities
In this section, we review some fundamental notions and notations of
(
p
,
q
)
-calculus.
The
[
n
]
p
,
q
is said to be (
p
,
q
)-integers and expressed as follows:
[
n
]
p
,
q
=
p
n
−
q
n
p
−
q
with
0
<
q
<
p
≤
1
.
The
[
n
]
p
,
q
!
and
n
k
!
are called (
p
,
q
)-factorial and (
p
,
q
)-binomial, respectively, and expressed as follows:
[
n
]
p
,
q
!
=
∏
k
=
1
n
[
k
]
p
,
q
,
n
≥
1
,
[
0
]
p
,
q
!
=
1
,
n
k
!
=
[
n
]
p
,
q
!
[
n
−
k
]
p
,
q
!
[
k
]
p
,
q
!
.
Definition 5
[12] The
(
p
,
q
)
-derivative of mapping
ϝ
:
[
π
1
,
π
2
]
→
R
is given as follows:
D
p
,
q
ϝ
(
x
)
=
ϝ
(
p
x
)
−
ϝ
(
q
x
)
(
p
−
q
)
x
,
x
≠
0
with
0
<
q
<
p
≤
1
.
Definition 6
[14] The
(
p
,
q
)
π
1
-derivative of mapping
ϝ
:
[
π
1
,
π
2
]
→
R
is given as follows:
(12)
D
p
,
q
π
1
ϝ
(
x
)
=
ϝ
(
p
x
+
(
1
−
p
)
π
1
)
−
ϝ
(
q
x
+
(
1
−
q
)
π
1
)
(
p
−
q
)
(
x
−
π
1
)
,
x
≠
π
1
with
0
<
q
<
p
≤
1
. For
x
=
π
1
, we state
D
p
,
q
π
1
ϝ
(
π
1
)
=
lim
x
→
π
1
D
p
,
q
π
1
ϝ
(
x
)
if it exists and it is finite.
Definition 7
[15] The
(
p
,
q
)
π
2
-derivative of mapping
ϝ
:
[
π
1
,
π
2
]
→
R
is given as follows:
(13)
D
p
,
q
π
2
ϝ
(
x
)
=
ϝ
(
q
x
+
(
1
−
q
)
π
2
)
−
ϝ
(
p
x
+
(
1
−
p
)
π
2
)
(
p
−
q
)
(
π
2
−
x
)
,
x
≠
π
2
.
with
0
<
q
<
p
≤
1
. For
x
=
π
2
, we state
D
p
,
q
π
2
ϝ
(
π
2
)
=
lim
x
→
π
2
D
p
,
q
π
2
ϝ
(
x
)
if it exists and it is finite.
Remark 1
It is clear that if we use
p
=
1
in (12) and (13), then the equalities (12) and (13) reduce to (4) and (5), respectively.
Definition 8
[14] The definite
(
p
,
q
)
π
1
-integral of mapping
ϝ
:
[
π
1
,
π
2
]
→
R
on
[
π
1
,
π
2
]
is stated as follows:
(14)
∫
π
1
x
ϝ
(
τ
)
d
p
,
q
π
1
τ
=
(
p
−
q
)
(
x
−
π
1
)
∑
n
=
0
∞
q
n
p
n
+
1
ϝ
q
n
p
n
+
1
x
+
1
−
q
n
p
n
+
1
π
1
with
0
<
q
<
p
≤
1
.
Definition 9
[15] The definite
(
p
,
q
)
π
2
-integral of mapping
ϝ
:
[
π
1
,
π
2
]
→
R
on
[
π
1
,
π
2
]
is stated as follows:
(15)
∫
x
π
2
ϝ
(
τ
)
d
p
,
q
π
2
τ
=
(
p
−
q
)
(
π
2
−
x
)
∑
n
=
0
∞
q
n
p
n
+
1
ϝ
q
n
p
n
+
1
x
+
1
−
q
n
p
n
+
1
π
2
with
0
<
q
<
p
≤
1
.
Remark 2
It is evident that if we pick
p
=
1
in (14) and (15), then the equalities (14) and (15) change into (6) and (7), respectively.
Remark 3
If we take
π
1
=
0
and
x
=
π
2
=
1
in (14), then we have
∫
0
1
ϝ
(
τ
)
d
p
,
q
0
τ
=
(
p
−
q
)
∑
n
=
0
∞
q
n
p
n
+
1
ϝ
q
n
p
n
+
1
.
Similarly, by taking
x
=
π
1
=
0
and
π
2
=
1
in (15), then we obtain that
∫
0
1
ϝ
(
τ
)
d
p
,
q
1
τ
=
(
p
−
q
)
∑
n
=
0
∞
q
n
p
n
+
1
ϝ
1
−
q
n
p
n
+
1
.
Lemma 1
[15] We have the following equalities:
∫
π
1
π
2
(
π
2
−
x
)
α
d
p
,
q
π
2
x
=
(
π
2
−
π
1
)
α
+
1
[
α
+
1
]
p
,
q
∫
π
1
π
2
(
x
−
π
1
)
α
d
p
,
q
π
1
x
=
(
π
2
−
π
1
)
α
+
1
[
α
+
1
]
p
,
q
,
where
α
∈
R
−
{
−
1
}
.
Recently, Vivas-Cortez et al. [15] proved the following HH type inequalities for convex functions using the
(
p
,
q
)
π
2
-integral:
Theorem 5
[15] For a convex mapping
ϝ
:
[
π
1
,
π
2
]
→
R
, which is differentiable on
[
π
1
,
π
2
]
, the following inequalities hold for
(
p
,
q
)
π
2
-integral:
(16)
ϝ
p
π
1
+
q
π
2
[
2
]
p
,
q
≤
1
p
(
π
2
−
π
1
)
∫
p
π
1
+
(
1
−
p
)
π
2
π
2
ϝ
(
x
)
d
p
,
q
π
2
x
≤
p
ϝ
(
π
1
)
+
q
ϝ
(
π
2
)
[
2
]
p
,
q
,
where
0
<
q
<
p
≤
1
.
Theorem 6
[15] For a convex function
ϝ
:
[
π
1
,
π
2
]
→
R
, the following inequality holds:
(17)
ϝ
π
1
+
π
2
2
≤
1
2
p
(
π
2
−
π
1
)
∫
π
1
p
π
2
+
(
1
−
p
)
π
1
ϝ
(
x
)
d
p
,
q
π
1
x
+
∫
p
π
1
+
(
1
−
p
)
π
2
π
2
ϝ
(
x
)
d
p
,
q
π
2
x
≤
ϝ
(
π
1
)
+
ϝ
(
π
2
)
2
,
where
0
<
q
<
p
≤
1
.
4 Hermite-Hadamard inequalities
In this section, we prove HH inequalities for s-convex functions in the second kind using the post-quantum integrals.
Theorem 7
Assume that the mapping
ϝ
:
[
0
,
∞
)
→
R
is s-convex in the second sense and
π
1
,
π
2
∈
[
0
,
∞
)
with
π
1
<
π
2
, then the following inequality holds for
s
∈
(
0
,
1
]
:
(18)
2
s
−
1
ϝ
π
1
+
π
2
2
≤
1
2
p
(
π
2
−
π
1
)
∫
π
1
π
2
ϝ
(
x
)
d
p
,
q
π
1
x
+
∫
π
1
π
2
ϝ
(
x
)
d
p
,
q
π
2
x
≤
ϝ
(
π
1
)
+
ϝ
(
π
2
)
[
s
+
1
]
p
,
q
.
Proof
We have s-convexity, as we know from s-convexity
(19)
2
s
ϝ
x
+
y
2
≤
ϝ
(
x
)
+
ϝ
(
y
)
.
We obtain the following by putting
x
=
t
π
2
+
(
1
−
t
)
π
1
and
y
=
t
π
1
+
(
1
−
t
)
π
2
in (19)
2
s
ϝ
π
1
+
π
2
2
≤
ϝ
(
t
π
2
+
(
1
−
t
)
π
1
)
+
ϝ
(
t
π
1
+
(
1
−
t
)
π
2
)
.
From Definitions 8 and 9, we have
2
s
−
1
ϝ
π
1
+
π
2
2
≤
1
2
p
(
π
2
−
π
1
)
∫
π
1
p
π
2
+
(
1
−
p
)
π
1
ϝ
(
x
)
d
p
,
q
π
1
x
+
∫
p
π
1
+
(
1
−
p
)
π
2
π
2
ϝ
(
x
)
d
p
,
q
π
2
x
,
and the first inequality in (18) is proved.
To prove the second inequality, we use the s-convexity, and we have
(20)
ϝ
(
t
π
2
+
(
1
−
t
)
π
1
)
≤
t
s
ϝ
(
π
2
)
+
(
1
−
t
)
s
ϝ
(
π
1
)
and
(21)
ϝ
(
t
π
1
+
(
1
−
t
)
π
2
)
≤
t
s
ϝ
(
π
1
)
+
(
1
−
t
)
s
ϝ
(
π
2
)
.
By adding (20) and (21), from Definitions 8 and 9, we have
1
2
p
(
π
2
−
π
1
)
∫
π
1
p
π
2
+
(
1
−
p
)
π
1
ϝ
(
x
)
d
p
,
q
π
1
x
+
∫
p
π
1
+
(
1
−
p
)
π
2
π
2
ϝ
(
x
)
d
p
,
q
π
2
x
≤
ϝ
(
π
1
)
+
ϝ
(
π
2
)
[
s
+
1
]
p
,
q
,
and the proof is completed.□
Example 1
For s-convex function
f
(
x
)
=
x
s
, from inequality (18) with
a
=
s
=
1
,
b
=
2
,
p
=
1
2
, and
q
=
1
4
,
we have
2
s
−
1
ϝ
π
1
+
π
2
2
=
3
2
,
1
2
p
(
π
2
−
π
1
)
∫
π
1
π
2
ϝ
(
x
)
d
p
,
q
π
1
x
+
∫
π
1
π
2
ϝ
(
x
)
d
p
,
q
π
2
x
=
1
2
−
1
4
∑
n
=
0
∞
1
4
n
1
2
n
+
1
1
4
n
1
2
n
+
1
2
+
1
−
1
4
n
1
2
n
+
1
+
1
2
−
1
4
∑
n
=
0
∞
1
4
n
1
2
n
+
1
1
4
n
1
2
n
+
1
+
1
−
1
4
n
1
2
n
+
1
2
=
3
and
ϝ
(
π
1
)
+
ϝ
(
π
2
)
[
1
+
s
]
p
,
q
=
1
+
2
1
4
+
1
2
=
4
.
Thus,
3
2
<
3
<
4
,
which shows that the inequality proved in Theorem 7 is true.
Remark 4
If we set
s
=
1
in Theorem 7, then we recapture the inequality (17).
Remark 5
In Theorem 7, if we take the limit as
p
=
1
, then inequality (18) becomes the inequality (10).
Remark 6
In Theorem 7, if we take
p
=
1
and later take the limit as
q
→
1
−
, then inequality (18) becomes the inequality (2).
5 Ostrowski’s inequalities
In this section, we prove post-quantum Ostrowski type inequalities for s-convex functions in the second sense.
We begin with the following identity.
Lemma 2
Let
ϝ
:
[
π
1
,
π
2
]
⊂
R
→
R
be a function. If
D
p
,
q
π
2
ϝ
and
π
1
D
p
,
q
ϝ
are two continuous and integrable functions on
[
π
1
,
π
2
]
, then for all
x
∈
[
π
1
,
π
2
]
, we have
(22)
ϝ
(
x
)
−
1
p
(
π
2
−
π
1
)
∫
π
1
p
x
+
(
1
−
p
)
π
1
ϝ
(
t
)
d
p
,
q
π
1
t
+
∫
p
x
+
(
1
−
p
)
π
2
π
2
ϝ
(
t
)
d
p
,
q
π
2
t
=
q
(
x
−
π
1
)
2
π
2
−
π
1
∫
0
1
t
D
p
,
q
π
1
ϝ
(
t
x
+
(
1
−
t
)
π
1
)
d
p
,
q
0
t
−
q
(
π
2
−
x
)
2
π
2
−
π
1
∫
0
1
t
D
p
,
q
π
2
ϝ
(
t
x
+
(
1
−
t
)
π
2
)
d
p
,
q
1
t
.
Proof
From Definitions 6 and 7, we have
D
p
,
q
π
1
ϝ
(
t
x
+
(
1
−
t
)
π
1
)
=
ϝ
(
p
t
x
+
(
1
−
p
t
)
π
1
)
−
ϝ
(
q
t
x
+
(
1
−
q
t
)
π
1
)
t
(
x
−
π
1
)
(
p
−
q
)
and
D
p
,
q
π
2
ϝ
(
t
x
+
(
1
−
t
)
π
2
)
=
ϝ
(
q
t
x
+
(
1
−
q
t
)
π
2
)
−
ϝ
(
p
t
x
+
(
1
−
p
t
)
π
2
)
t
(
π
2
−
x
)
(
p
−
q
)
.
By using Definition 9, we have
(23)
I
1
=
∫
0
1
t
D
p
,
q
π
2
ϝ
(
t
x
+
(
1
−
t
)
π
2
)
d
p
,
q
1
t
=
1
(
π
2
−
x
)
(
p
−
q
)
∫
0
1
[
ϝ
(
q
t
x
+
(
1
−
q
t
)
π
2
)
−
ϝ
(
p
t
x
+
(
1
−
p
t
)
π
2
)
]
d
p
,
q
1
t
=
1
π
2
−
x
∑
n
=
0
∞
q
n
p
n
+
1
ϝ
q
n
+
1
p
n
+
1
x
+
1
−
q
n
+
1
p
n
+
1
π
2
−
∑
n
=
0
∞
q
n
p
n
+
1
ϝ
q
n
p
n
x
+
1
−
q
n
p
n
π
2
=
1
π
2
−
x
1
q
∑
n
=
0
∞
q
n
+
1
p
n
+
1
ϝ
q
n
+
1
p
n
+
1
x
+
1
−
q
n
+
1
p
n
+
1
π
2
−
1
p
∑
n
=
0
∞
q
n
p
n
ϝ
q
n
p
n
x
+
1
−
q
n
p
n
π
2
=
1
π
2
−
x
1
q
−
1
p
∑
n
=
0
∞
q
n
p
n
ϝ
q
n
p
n
x
+
1
−
q
n
p
n
π
2
−
1
q
ϝ
(
x
)
=
1
π
2
−
x
p
−
q
p
q
∑
n
=
0
∞
q
n
p
n
ϝ
q
n
p
n
x
+
1
−
q
n
p
n
π
2
−
1
q
ϝ
(
x
)
=
1
π
2
−
x
1
p
q
(
π
2
−
x
)
∫
p
x
+
(
1
−
p
)
π
2
π
2
ϝ
(
x
)
d
p
,
q
π
2
x
−
1
q
ϝ
(
x
)
.
Similarly, from Definition 8, we have
(24)
I
2
=
∫
0
1
t
D
p
,
q
π
1
ϝ
(
t
x
+
(
1
−
t
)
π
1
)
d
p
,
q
0
t
=
1
x
−
π
1
1
q
ϝ
(
x
)
−
1
p
q
(
x
−
π
1
)
∫
π
1
p
x
+
(
1
−
p
)
π
1
ϝ
(
x
)
d
p
,
q
π
1
x
.
Thus, we obtain the resultant equality (22) by subtracting (23) from (24).□
Remark 7
In Lemma 2, if we set
p
=
1
, then we obtain the equality:
ϝ
(
x
)
−
1
(
π
2
−
π
1
)
∫
π
1
x
ϝ
(
t
)
d
q
π
1
t
+
∫
x
π
2
ϝ
(
t
)
d
q
π
2
t
=
q
(
x
−
π
1
)
2
π
2
−
π
1
∫
0
1
t
D
q
π
1
ϝ
(
t
x
+
(
1
−
t
)
π
1
)
d
q
0
t
−
q
(
π
2
−
x
)
2
π
2
−
π
1
∫
0
1
t
D
q
π
2
ϝ
(
t
x
+
(
1
−
t
)
π
2
)
d
q
1
t
,
which is proved by Budak et al. in [33].
Remark 8
In Lemma 2, if we set
p
=
1
and later taking the limit as
q
→
1
−
, then we obtain [37, Lemma 1].
Theorem 8
Assume that the mapping
ϝ
:
I
⊂
[
0
,
∞
)
→
R
is differentiable and
π
1
,
π
2
∈
I
with
π
1
<
π
2
. If
∣
D
p
,
q
π
1
ϝ
∣
and
∣
D
p
,
q
π
2
ϝ
∣
are s-convex mappings in the second sense, then the following inequality holds:
(25)
ϝ
(
x
)
−
1
p
(
π
2
−
π
1
)
∫
π
1
p
x
+
(
1
−
p
)
π
1
ϝ
(
t
)
d
p
,
q
π
1
t
+
∫
p
x
+
(
1
−
p
)
π
2
π
2
ϝ
(
t
)
d
p
,
q
π
2
t
≤
q
(
x
−
π
1
)
2
π
2
−
π
1
1
[
s
+
2
]
p
,
q
∣
D
p
,
q
π
1
ϝ
(
x
)
∣
+
Θ
1
∣
D
p
,
q
π
1
ϝ
(
π
1
)
∣
+
q
(
π
2
−
x
)
2
π
2
−
π
1
1
[
s
+
2
]
p
,
q
∣
D
p
,
q
π
2
ϝ
(
x
)
∣
+
Θ
2
∣
D
p
,
q
π
2
ϝ
(
π
2
)
∣
,
where
Θ
1
=
∫
0
1
t
(
1
−
t
)
s
d
p
,
q
0
t
and
Θ
2
=
∫
0
1
t
(
1
−
t
)
s
d
p
,
q
1
t
.
Proof
From Lemma 2 and properties of the modulus, we have
(26)
ϝ
(
x
)
−
1
p
(
π
2
−
π
1
)
∫
π
1
p
x
+
(
1
−
p
)
π
1
ϝ
(
t
)
d
p
,
q
π
1
t
+
∫
p
x
+
(
1
−
p
)
π
2
π
2
ϝ
(
t
)
d
p
,
q
π
2
t
≤
q
(
x
−
π
1
)
2
π
2
−
π
1
∫
0
1
t
∣
D
p
,
q
π
1
ϝ
(
t
x
+
(
1
−
t
)
π
1
)
∣
d
p
,
q
0
t
+
q
(
π
2
−
x
)
2
π
2
−
π
1
∫
0
1
t
∣
D
p
,
q
π
2
ϝ
(
t
x
+
(
1
−
t
)
π
2
)
∣
d
p
,
q
1
t
.
Since the mapping
∣
D
p
,
q
π
1
ϝ
∣
and
∣
D
p
,
q
π
2
ϝ
∣
are s-convexities in the second sense, therefore
(27)
∫
0
1
t
∣
D
p
,
q
π
1
ϝ
(
t
x
+
(
1
−
t
)
π
1
)
∣
d
p
,
q
0
t
≤
∫
0
1
t
s
+
1
∣
D
p
,
q
π
1
ϝ
(
x
)
∣
d
p
,
q
0
t
+
∫
0
1
t
(
1
−
t
)
s
∣
D
p
,
q
π
1
ϝ
(
π
1
)
∣
d
p
,
q
0
t
=
1
[
s
+
2
]
p
,
q
∣
D
p
,
q
π
1
ϝ
(
x
)
∣
+
Θ
1
∣
D
p
,
q
π
1
ϝ
(
π
1
)
∣
and
(28)
∫
0
1
t
∣
D
p
,
q
π
2
ϝ
(
t
x
+
(
1
−
t
)
π
2
)
∣
d
p
,
q
1
t
≤
∫
0
1
t
s
+
1
∣
D
p
,
q
π
2
ϝ
(
x
)
∣
d
p
,
q
1
t
+
∫
0
1
t
(
1
−
t
)
s
∣
D
p
,
q
π
2
ϝ
(
π
2
)
∣
d
p
,
q
1
t
=
1
[
s
+
2
]
p
,
q
∣
D
p
,
q
π
2
ϝ
(
x
)
∣
+
Θ
2
∣
D
p
,
q
π
2
ϝ
(
π
2
)
∣
.
We obtain the resultant inequality (25) by putting (27) and (28) in (26).□
Corollary 1
If we set
s
=
1
in Theorem 8, then we obtain the following new Ostrowski type inequality for convex functions:
ϝ
(
x
)
−
1
p
(
π
2
−
π
1
)
∫
π
1
p
x
+
(
1
−
p
)
π
1
ϝ
(
t
)
d
p
,
q
π
1
t
+
∫
p
x
+
(
1
−
p
)
π
2
π
2
ϝ
(
t
)
d
p
,
q
π
2
t
≤
q
(
x
−
π
1
)
2
π
2
−
π
1
1
[
3
]
p
,
q
∣
D
p
,
q
π
1
ϝ
(
x
)
∣
+
[
3
]
p
,
q
−
[
2
]
p
,
q
[
3
]
p
,
q
[
2
]
p
,
q
∣
D
p
,
q
π
1
ϝ
(
π
1
)
∣
+
q
(
π
2
−
x
)
2
π
2
−
π
1
1
[
3
]
p
,
q
∣
D
p
,
q
π
2
ϝ
(
x
)
∣
+
[
3
]
p
,
q
−
[
2
]
p
,
q
[
3
]
p
,
q
[
2
]
p
,
q
∣
D
p
,
q
π
2
ϝ
(
π
2
)
∣
.
Remark 10
In Corollary 1, if we set
p
=
1
, then we obtain the following inequality:
ϝ
(
x
)
−
1
π
2
−
π
1
∫
π
1
x
ϝ
(
t
)
d
q
π
1
t
+
∫
x
π
2
ϝ
(
t
)
d
q
π
2
t
≤
q
(
π
2
−
π
1
)
[
2
]
q
[
3
]
q
[
(
x
−
π
1
)
2
(
[
2
]
q
∣
D
q
π
1
ϝ
(
x
)
∣
+
q
2
∣
D
q
π
1
ϝ
(
π
1
)
∣
)
+
(
π
2
−
x
)
2
(
[
2
]
q
∣
D
q
π
2
ϝ
(
x
)
∣
+
q
2
∣
D
q
π
2
ϝ
(
π
2
)
∣
)
]
,
which is given by Budak et al. in [33].
Corollary 2
If we assume
∣
D
p
,
q
π
1
ϝ
(
x
)
∣
,
∣
D
p
,
q
π
1
ϝ
(
x
)
∣
≤
M
in Theorem
8, then we have following post-quantum Ostrowski type inequality for s-convex functions in the second sense:
(29)
ϝ
(
x
)
−
1
p
(
π
2
−
π
1
)
∫
π
1
p
x
+
(
1
−
p
)
π
1
ϝ
(
t
)
d
p
,
q
π
1
t
+
∫
p
x
+
(
1
−
p
)
π
2
π
2
ϝ
(
t
)
d
p
,
q
π
2
t
≤
M
q
(
x
−
π
1
)
2
π
2
−
π
1
1
[
s
+
2
]
p
,
q
+
Θ
1
+
M
q
(
π
2
−
x
)
2
π
2
−
π
1
1
[
s
+
2
]
p
,
q
+
Θ
2
.
Remark 12
If we set
s
=
p
=
1
in Corollary 2, then we recapture inequality (9).
Remark 13
In Corollary 2, if we set
p
=
1
and later take the limit as
q
→
1
−
, then Corollary 2 reduces to [38, Theorem 2].
Theorem 9
Assume that the mapping
ϝ
:
I
⊂
[
0
,
∞
)
→
R
is differentiable and
π
1
,
π
2
∈
I
with
π
1
<
π
2
. If
∣
D
p
,
q
π
1
ϝ
∣
p
1
and
∣
D
p
,
q
π
2
ϝ
∣
p
1
,
p
1
≥
1
are s-convex mappings in the second sense, then the following inequality holds:
(30)
ϝ
(
x
)
−
1
p
(
π
2
−
π
1
)
∫
π
1
p
x
+
(
1
−
p
)
π
1
ϝ
(
t
)
d
p
,
q
π
1
t
+
∫
p
x
+
(
1
−
p
)
π
2
π
2
ϝ
(
t
)
d
p
,
q
π
2
t
≤
q
π
2
−
π
1
1
[
2
]
p
,
q
1
−
1
p
1
(
x
−
π
1
)
2
1
[
s
+
2
]
p
,
q
∣
D
p
,
q
π
1
ϝ
(
x
)
∣
p
1
+
Θ
1
∣
D
p
,
q
π
1
ϝ
(
π
1
)
∣
p
1
1
p
1
+
(
π
2
−
x
)
2
1
[
s
+
2
]
p
,
q
∣
D
p
,
q
π
2
ϝ
(
x
)
∣
p
1
+
Θ
2
