Abstract
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 -convex and -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.
1. Introduction
Integral inequality is a modern model of approximation theory that describes the growth rate of competing mathematical analysis. This model is also used in various fields such as ordinary differential equations [1–5] and fractional calculus [6–17].
Among the several known inequalities, the most simple is Simpson’s type, which has been successfully applied in several models of ordinary differential equations [18–29] and fractional differential equations [30–32]. Simpson’s integral inequality is as follows: for any four times continuously differentiable function on , Simpson’s integral inequality is defined as follows:where .
If the function is neither four times differentiable nor is the fourth derivative bounded on , then we cannot apply the classical Simpson quadrature formula.
The following literature results obtained by Alomari et al. [18] and Sarikaya et al. [23] become a special case in our findings in Sections 2 and 3.
Lemma 1. (see [18]). Let be twice differentiable function on with , then we have
Lemma 2. (see [23]). Let be twice differentiable function on such that , where with , then we havewhereThrough this paper, represents the set of real numbers and be an interval in and be a bifunction apart from some special cases.
This paper deals with the notations of -convex and -quasiconvex functions which were introduced by Gordji et al. [33] as follows.
Definition 1. A function is called convex with respect to (or briefly -convex), iffor all and . Furthermore, is called -quasiconvex, iffor all and .
Remark 1. (i)It is easy to see the definition that every -convex function is -quasiconvex(ii)If we take in Definition 1, then the definitions of -convex and -quasiconvex are reduced to the definition of convex function and quasiconvex function, respectivelyNext, we will give examples for the above definitions.
Example 1. Let , then is convex and -convex with ; indeed,
Example 2. Let , then is not convex but is -convex with ; indeed,
Example 3. Let , with . We observe that is convex on and therefore -quasiconvex with .
Example 4. Let , with . We observe that is convex on and therefore -quasiconvex with .
Example 5. Let , with . We obviously see that is -quasiconvex with .
The essential object of this study is to establish new Simpson’s integral inequalities for the -convex and -quasiconvex functions in the second derivative sense at certain powers.
2. Simpson’s Inequality for -Convex
In this section, we give a new refinement of Simpson integral inequality for twice differentiable functions.
Theorem 1. Let be a twice differentiable function on such that , where with . If is -convex on , then we have
Proof. By making the use of Lemma 2 and the -convexity of , we find thatwhereA simple rearrangement gives us the proof.
Corollary 1. Theorem 1 with gives the following new inequality:
Remark 2. Inequality (9) with becomesMoreover, inequality (12) with becomesThese are both obtained by Sarikaya et al. [23] in Theorem 2.2 and Corollary 2.3, respectively.
Theorem 2. Let be a twice differentiable function on such that , where with . If is -convex on and , then we havewhere .
Proof. Let , then by using Lemma 2, we haveBy making the use of the Hölder’s inequality for the above integrals, we haveBy -convexity of for the last two integrals, we haveBy substituting (18) and (19) into (17), we havewhere we used the identityThus, we are done.
Corollary 2. Theorem 2 with gives the following new inequality:
Remark 3. Inequality (15) with becomesMoreover, inequality (22) with becomesThese are both obtained by Sarikaya et al. [23] in Theorem 2.5 and Corollary 2.6, respectively.
Remark 4. Theorem 2 and Corollary 2 with become Theorem 1 and Corollary 1, respectively.
3. Simpson’s Inequality for -Quasiconvex
Theorem 3. Let be a twice differentiable function on provided , where with . If is -quasiconvex on , then we have
Proof. By making use of -quasiconvexity of and Lemma 2, we getwhereA simple rearrangement completes the proof.
Corollary 3. Theorem 3 with becomes
Theorem 4. Let be a twice differentiable function on provided , where with . If is -quasiconvex on and , then we havewhere .
Proof. Let , then by using Lemma 2, we haveBy making the use of the Hölder’s inequality for the above integrals, we haveBy -quasiconvexity of for the last two integrals, we haveBy substituting (32) and (33) into (31), we havewhere we used the following identityThus we are done.
Corollary 4. Theorem 4 with becomes
Remark 5. Theorem 4 and Corollary 4 with become Theorem 3 and Corollary 3, respectively.
Corollary 5. Theorem 4 with becomes
4. Applications
Some applications for our findings are presented.
4.1. Applications to Special Means
The special means are itemized as follows:(i)The arithmetic mean:(ii)The harmonic mean:(iii)The logarithmic mean: for .(iv)The -logarithmic mean: for .
We know that is a monotonic nondecreasing function over with . In particular, we can say that .
Now, using our findings in Section 2, we conclude the following new inequalities.
Proposition 1. Let with . Then, we have
Proof. The assertion follows from Theorem 1 with and a simple computation, where is -convex function with (see Example 1).
Proposition 2. Let , . Then, we have
Proof. The assertion follows from Theorem 1 and a simple computation applied to , where is -convex function with (see Example 2).
The following proposition is a particular case of Corollary 11 in [34] when (see Remark 12 in [34]).
Proposition 3. Let , . Then, we have
Proof. The assertion follows from Theorem 3 and a simple computation applied to , where is -quasiconvex function with (see Example 5).
Proposition 4. Let , . Then, for all , we have
Proof. The assertion follows from Theorem 4 and a simple computation applied to , where is -quasiconvex function with (see Example 4).
4.2. Applications to Simpson’s Formula
Let be a partition of the interval ; that is ; and consider Simpson’s formula:
We know that if is differentiable such that exists on and . Then, we havewhere the approximation error satisfies
It is clear that if the function is not four times differentiable or is not bounded on , then (47) cannot be applied.
Theorem 5. Let be a twice differentiable function on such that , where with . If is -convex on , then for every division of we have
Proof. By applying Theorem 1 on the subintervals of the division to getBy summing over from 0 to and taking into account that is -convex to getwhich completes our proof.
Corollary 6. Theorem 5 with becomes
Theorem 6. Let be a twice differentiable function on such that , where with . If is -quasiconvex on , then for every division of we have
Proof. By applying Theorem 3 and by the same method used for proof of the previous theorem, we can produce the desired result.
Proposition 5. Let be a twice differentiable function on such that , where with . If is -convex on , then we havewhere
Proof. The proof follows from Theorem 2 directly.
Proposition 6. Let be a twice differentiable function on such that , where with . If is -quasiconvex on , then we have
Proof. The proof follows from Theorem 4 directly.
4.3. Applications to the Midpoint Formula
Let be a partition as before. Here we consider the midpoint formula:
Suppose that the function is differentiable with existing on and , and then, we havewhere the approximation error satisfies
Proposition 7. Let be a twice differentiable function on , with . If is -convex on , then for any division of , we have
Proof. By applying Corollary 1 on the subintervals of the division , to getBy summing over from 0 to to getwhich completes our proof.
Proposition 8. Let be a twice differentiable function on , with . If is -convex on and , then for any division of , we havewhere
Proof. By applying Corollary 2 on the subintervals of the division to getwhereBy summing over from 0 to to getwhich completes our proof.
Proposition 9. Let be a twice differentiable function on , with . If is -quasiconvex on , then for any division of , we have
Proof. By applying Corollary 3 on the subintervals of the division to getBy summing over from 0 to to getwhich completes our proof.
Proposition 10. Let be a twice differentiable function on , with . If is -quasiconvex on and , then in (30), for every division of , we have
Proof. By applying Corollary 4 on the subintervals of the division , we getBy summing over from 0 to to getwhich rearranges to the proof.
5. Illustrative Plots
Finally, we present two three-dimensional plots to demonstrate the validity of the inequalities (42) and (44) in the case of -convex and -quasiconvex functions, respectively.
From inequality (42), we can define
Thus, Figure 1 represents the plot of inequality (42) and .
(a)
(b)
From inequality (44), we can define
Thus, Figure 2 represents the plot of inequality (44) and .
(a)
(b)
6. Conclusion
In this study, we have considered Simpson’s type integral inequalities for the -convex and -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
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors want to give thanks to the Dirección de investigación from Pontificia Universidad Católica del Ecuador for technical support to our research project entitled: “Algunas desigualdades integrales para funciones convexas generalizadas y aplicaciones”. The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.