Simpson’s Integral Inequalities for Twice Differentiable Convex Functions

Integral inequality is an interesting mathematical model due to its wide and signiﬁcant applications in mathematical analysis and fractional calculus. In the present research article, we obtain new inequalities of Simpson’s integral type based on the φ -convex and φ -quasiconvex functions in the second derivative sense. In the last sections, some applications on special functions are provided and shown via two ﬁgures to demonstrate the explanation of the readers.

Lemma 1 (see [18]). Let F: J ⟶ R be twice differentiable function on J with F ″ ∈ L 1 [ξ 1 , ξ 2 ], then we have Lemma 2 (see [23]). Let F: J ⟶ R be twice differentiable function on J such that F ″ ∈ L 1 [ξ 1 , ξ 2 ], where ξ 1 , ξ 2 ∈ J with ξ 1 < ξ 2 , then we have where Through this paper, R represents the set of real numbers and J be an interval in R and φ: R × R ⟶ R be a bifunction apart from some special cases.
This paper deals with the notations of φ-convex and φ-quasiconvex functions which were introduced by Gordji et al. [33] as follows.

Remark 1.
(i) It is easy to see the definition that every φ-convex function is φ-quasiconvex (ii) If we take φ(ξ 1 , ξ 2 ) � ξ 1 − ξ 2 in Definition 1, then the definitions of φ-convex and φ-quasiconvex are reduced to the definition of convex function and quasiconvex function, respectively Next, we will give examples for the above definitions.
e essential object of this study is to establish new Simpson's integral inequalities for the φ-convex and φ-quasiconvex functions in the second derivative sense at certain powers.

Simpson's Inequality for φ-Convex
In this section, we give a new refinement of Simpson integral inequality for twice differentiable functions.
, then we have

Mathematical Problems in Engineering
Proof. By making the use of Lemma 2 and the φ-convexity of |F ″ |, we find that where A simple rearrangement gives us the proof.
□ Mathematical Problems in Engineering Corollary 1.
and q ≥ 1, then we have where Proof. Let q ≥ 1, then by using Lemma 2, we have By making the use of the Hölder's inequality for the above integrals, we have By φ-convexity of |F ″ | q for the last two integrals, we have By substituting (18) and (19) into (17), we have where we used the identity us, we are done.
gives the following new inequality: Mathematical Problems in Engineering Moreover, inequality (22) ese are both obtained by Sarikaya et al. [23] in eorem 2.5 and Corollary 2.6, respectively.

Simpson's Inequality for φ-Quasiconvex
Proof. By making use of φ-quasiconvexity of |F ″ | and Lemma 2, we get 6 Mathematical Problems in Engineering A simple rearrangement completes the proof.
Proof. Let q ≥ 1, then by using Lemma 2, we have Mathematical Problems in Engineering 7 By making the use of the Hölder's inequality for the above integrals, we have By φ-quasiconvexity of |F ″ | q for the last two integrals, we have By substituting (32) and (33) into (31), we have Remark 5. eorem 4 and Corollary 4 with q � 1 become eorem 3 and Corollary 3, respectively.

Applications
Some applications for our findings are presented.

Mathematical Problems in Engineering
Proposition 3. Let ξ 1 , ξ 2 ∈ R, 0 < ξ 1 < ξ 2 . en, we have Proof. e assertion follows from eorem 3 and a simple computation applied to Proof. e assertion follows from eorem 4 and a simple computation applied to F(

Applications to Simpson's Formula.
Let P be a partition of the interval [ξ 1 , ξ 2 ]; that is P: ξ 1 � s 0 < s 1 < · · · < s n− 1 < s n � ξ 2 ; h i � (s i+1 − s i )/2 and consider Simpson's formula: We know that if F: where the approximation error E s (F, L) satisfies It is clear that if the function F is not four times differentiable or F (4) is not bounded on (ξ 1 , ξ 2 ), then (47) cannot be applied.
Proof. By applying eorem 1 on the subintervals [s i , s i+1 ], (i � 0, 1, 2, . . . , n − 1) of the division P to get By summing over i from 0 to n − 1 and taking into account that |F ″ | is φ-convex to get which completes our proof. □ Corollary 6.
Proof. By applying eorem 3 and by the same method used for proof of the previous theorem, we can produce the desired result. where Proof. e proof follows from eorem 2 directly.
Proof. e proof follows from eorem 4 directly.

Applications to the Midpoint Formula.
Let P be a partition as before. Here we consider the midpoint formula: Suppose that the function F: and K � sup x∈(ξ 1 ,ξ 2 ) |F ″ (x)| < ∞, and then, we have where the approximation error E M (F, P) satisfies Proof. By applying Corollary 1 on the subintervals [s i , s i+1 ], (i � 0, 1, . . . , n − 1) of the division P, to get By summing over i from 0 to n − 1 to get which completes our proof. □ Proposition 8. Let F: J ⟶ R be a twice differentiable function on J, ξ 1 , ξ 2 ∈ J with ξ 1 < ξ 2 . If |F ″ | q is φ-convex on [ξ 1 , ξ 2 ] and q ≥ 1, then for any division P of [ξ 1 , ξ 2 ], we have where

Mathematical Problems in Engineering
Proof. By applying Corollary 2 on the subintervals [s i , s i+1 ], (i � 0, 1, . . . , n − 1) of the division P to get where By summing over i from 0 to n − 1 to get which completes our proof.
Proof. By applying Corollary 3 on the subintervals [s i , s i+1 ], (i � 0, 1, . . . , n − 1) of the division P to get By summing over i from 0 to n − 1 to get which completes our proof.
and q ≥ 1, then in (30), for every division P of [ξ 1 , ξ 2 ], we have By summing over i from 0 to n − 1 to get E M (F, P) ≤ 1 81 which rearranges to the proof.
From inequality (44), we can define us, Figure 2 represents the plot of inequality (44) and W(x, y) − w(x, y).

Conclusion
In this study, we have considered Simpson's type integral inequalities for the φ-convex and φ-quasiconvex functions in the second derivative sense. Some special cases of our findings are investigated to show the powerfulness of our results. Also, the proposed inequalities can be applied to other mathematical and statistical models, as we have shown in Section 4.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.