Introduction
Simpson's rules (Thomas Simpson 1710-1761) are well-known methods in numerical analysis for the purpose of numerical integration and the numerical approximation of definite integrals. Two famous Simpson's rules are known in the literature, and one of them is the following estimation known as Simpson's inequality:
Theorem 1 Suppose that
is a four times continuously differentiable mapping on (ρ, σ), and let
Then, one has the inequality
Simpson's inequalities have been extensively studied by numerous researchers due to their wide range of applications in various fields of mathematics. Simpson's inequalities for different classes of functions have been studied, but one can find many inequality papers in the literature based on convex functions since convex functions are the basis of Simpson's inequality for integrals. For example, Alomari, et al. introduced some inequalities of Simpson's type based on functions whose absolute value of the first derivative is s-convex and concave in [1]. New inequalities of Simpson type for functions of bounded variation and their application to quadrature formulae in Numerical Analysis are given by Dragomir, et al. in [2]. In [3], Sarıkaya, et al. generalized the inequalities based on s-convex functions given by Alomari. In [4], Sarıkaay, et al. derived recent inequalities of Simpson type involving local fractional integrals for generalized convex functions are derived. What's more, An inequality of the Simpson type for an n-times continuously differentiable mapping is given by Liu in [5]. In addition to the references mentioned here, there are many articles on Simpson's inequality in the literature. Interested readers can find papers on Simpson-type inequalities in the literature for any class of function.
On the other hand, many studies have recently been carried out in the field of q -analysis, starting with Euler due to the high demand for mathematics that models quantum computing q -calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions, and other sciences quantum theory, mechanics, and the theory of relativity [6-12]. Apparently, Euler was the founder of this branch of mathematics, by using the parameter q in Newton's work of infinite series. Later, Jackson was the first to develop q -calculus that was known without limits calculus in a systematic way [9]. In 1908-1909, Jackson defined the general q -integral and q -difference operator [11]. In 1969, Agarwal described the q -fractional derivative for the first time [13]. In 1966-1967 Al-Salam introduced a q -analogue of the Riemann-Liouville fractional integral operator and q-fractional integral operator [14]. In 2004, Rajkovic gave a definition of the Riemann-type q -integral which was generalized of Jackson q-integral. In 2013, Tariboon introduced
-difference operator [15].
Many integral inequalities well known in classical analysis such as Hölder inequality, Hermite-Hadamard inequality, Ostrowski inequality, Cauchy-Bunyakovsky-Schwarz, Gruss, Gruss-Cebysev, and other integral inequalities have been proved and applied for q -calculus using classical convexity. For illustrate, Alp, et al. proved The fundamental q -Hermite–Hadamard inequality, some new q -Hermite–Hadamard inequalities, and generalized q -Hermite–Hadamard inequality for convex and quasi-convex functions in [16]. In addition, Noor, et al. investigated some new integral inequalities including q -integrals related to Hermite-Hadamard and Ostrowski-type integral inequalities by using different classes of mappings [17-19]. In [20], via newly defined quantum integrals, Simpson and Newton-type inequalities for convex functions are established by Budak, et al.
In light of all these studies, Simpson-type inequalities involving quantum integrals for functions whose absolute value of the first derivative is convex will be analyzed in this work. Actually, Tunç, et al. [21] obtained Simpson's type quantum integral inequalities. Unfortunately, there are many mistakes in the proofs. For example, in Lemma 4, Tunç found the equality
Here, for q ∈(0, 1),
For instance,
So, the proof of Lemma 4 is not correct. Lemma 5 also has the same errors. On the other hand, since Lemma 4 and Lemma 5 are used in the proof of Theorem 1, there are errors in this theorem. Moreover, Theorem 2 and 3 have the same mistakes. For instance, because of (9), the following equalities are also not true:
The integral boundaries that cause all these errors are chosen independently of q
In 2018 Tunç, et al. [21] obtained Simpson's type quantum integral inequalities. Unfortunately, there are many mistakes in the proofs. Many q -integrals are calculated incorrectly. Besides, the results of the lemma and theorems are also wrong. For example, in Lemma 4,
Here, for
For instance,
So, the proof of Lemma 4 is not correct. Lemma 5 also has the same errors. On the other hand, since Lemma 4 and Lemma 5 are used in the proof of Theorem 1, there are errors in this theorem. Moreover, Theorem 2 and 3 have the same mistakes. For instance, because of (9), the following equalities are also not true:
The integral boundaries that cause all these errors are chosen independently of q
Now, let us show the following Theorem 1 in [21] is not correct. For this, we give an example.
Theorem 2 Suppose that
is a q - differentiable function on (ρ, σ) and 0 < q < 1. If
is convex and integrable function on [ρ, σ], then we possess the inequality
Example 1 Let's choose
on [0 ,1] and x(t) satisfies the conditions of Theorem 2. On the other hand,
is convex and integrable on [0 ,1] Then we have
Also,
As we have seen, from (2) and (3) and for q = (0, 1) we write
For instance, choosing
we have
Therefore, Inequality (1) is not correct.
Similarly, other theorems can be shown to be false.
On the other hand, in [16], Alp, et al. showed that
and proved the following true q -Hermite-Hadamard inequalities convex functions on quantum integral:
Theorem 3 If
be a convex differentiable function on [ρ, σ] and and 0 < q < 1. Then, q -Hermite-Hadamard inequalities
As can be seen, q -Hermite-Hadamard inequality includes the value of
. For this, Simpson's type quantum integral inequalities must also contain the value
In this paper, motivated by the above results, by choosing the appropriate integral bounds we establish correct Simpson-type quantum integral inequalities obtained by using quantum integrals for functions whose absolute value of the q -derivatives are convex. Later, similar results for mappings whose powers of the absolute value of q -derivatives are convex are obtained via Hölder's inequality. Also, relations between special cases of these results and inequalities presented in the earlier works are examined.
Preliminaries and Definitions of q-Calculus
Throughout this paper, let ρ < σ and 0 < q < 1 be a constant. The following definitions and theorems for q - derivative and q - integral of a function x on [ρ, σ] are given in [15,21].
Definition 1 [15,21]. For a continuous function
then q - derivative of x at
is characterized by the expression
Since
is a continuous function, thus we have
. The function x is said to be q - differentiable on [ρ, σ] if
exists for all
. If ρ = 0 in (4), then
, where
is familiar q -derivative of x at
defined by the expression ([12,23])
Definition 2 [21, 22]. Let
be a continuous function. Then the q -definite integral on
is delineated as
for
.
If ρ = 0 in (6), then
, where
is familiar q -definite integral on
defined by the expression (see [14])
If
, then the q -definite integral on
is expressed as
notation
Lemma 1 [21] For
, the following formula holds:
Main results
For convenience, we begin with some notations which will be used in what follows.
and
In order to easily prove our main results we first give a new identity including quantum integrals in the following.
Lemma 2 Let
be a q - differentiable function on (ρ, σ) and 0<q <1. If
is continuous and integrable on [ρ, σ], then we possess the identity
where
Proof. From the basic properties of quantum integral and the definition of
, it follows that
Also, using the definition of q -derivative, we find that
Now, if we calculate the first quantum integral on the right side of the above equality by considering the definition 2, then we obtain
Similarly, we have
Multiplying the resulting equality by q(σ – ρ) after substituting the identities (16) and (17) in (15), the desired result can be readily attained.
Corollary 4 Under the assumptions of Lemma 2 with
, one has
which was presented by Alomari, et al. in [3]. Here,
is defined by
Now, we examine how the results come out when we use a function whose quantum derivatives in modulus are convex.
Theorem 5 Suppose that
is a q - differentiable function on (ρ, σ) and0 <q < 1. If
is convex and integrable function on [ρ, σ], then we possess the inequality
where
are defined as in (10)-(13), respectively.
Proof. If we take the absolute value of both sides of (14), then we have
For the first expression on the right side of the inequality (19), seeing that
is convex on [
ρ, σ], it follows that
Now, calculating the quantum integrals on the right side of the above inequality by considering the case when ρ = 0 of Lemma 1, we find that
and
Then, one has the result
If similar operations are applied for the other expression in (19), due to the convexity of
then one possesses the inequality
Substituting the inequalities (20) and (21) in (19), the desired result can be readily attained. Hence, the proof is completed.
Corollary 6 If we take the limit of both sides of (18) as
, then the inequality (18) yields the result
which is Simpson type inequality for functions whose absolute values of derivatives are convex. This result was provided by Alomari, et al. in [1].
We observe how the inequalities come out when we use mappings whose q -derivatives in modulus at certain powers are convex.
Theorem 7 Assume that
is a q -differentiable function on (ρ, σ) and0 <q < 1. If
is convex and integrable function on [ρ, σ] where s >1 with
, then one has the result
Proof. We reconsider the inequality (19). Applying Hölder's inequality to the first integral on the right side of (19), due to the convexity of
it is found that
We also observe that
Similarly, using Hölder's inequality for the second integral on the right side of (19), owing to the convexity of
, we find that
We also have
Finally, if we substitute the above results in (19), then we obtain the desired inequality.
Theorem 8 Supposing that
is a q - differentiable function on (ρ, σ) and0 <q < 1. If
is convex and integrable function on [ρ, σ] where s ≥1, then one has the result
where
are defined as in (10)-(13), respectively.
Proof. We consider the inequality (19). Applying power mean inequality to the first integral on the right side of (19), we find that
The operations which have been used in the proof of theorem 5 are applied by considering that
is convex on [ρ, σ] it is easy to see that
where
and
are defined as in (10) and (11), respectively. Also, from the definition of quantum integral, we observe that
Thus, we obtain the inequality
Similarly, for the second integral on the right side of (19), we have
Should we substitute the inequalities (23) and (24) in (19), and we then capture the desired result which finishes the proof.
Corollary 9 If we take the limit of both sides of (22) as
, then the inequality (22) reduces to the result
which was presented by Alomari, et al. in [3].
Conclusion
In this work, Simpson-type quantum integral inequalities found to be incorrect by Tunç were corrected. New and correct quantum integral inequalities were thus developed by using mappings whose absolutes value of q -derivatives are convex. Also, relations between special cases of these results and inequalities given in the earlier works are observed. Also, this paper describes how to find quantum integral inequalities for convex functions.