1. Introduction and Preliminaries
The following inequality regarding square integrable functions is known as the Wirtinger inequality:
Theorem 1 ([1,2]).Let ϖ be a real-valued function with period and If then:with equality holding iff where For recently published papers of this type, see [3,4,5]. Beesack in [
6,
7] generalized (
1) as follows:
Theorem 2. Let ϖ be absolutely continuous on with then for all , we have:with equality holding iff and satisfies the following equation: The next functional is known as the Chebyshev functional (see [
8]):
Several bounds for
have been found by many authors, and many important applications have been given. For example, Alomari in [
9] obtained a bound for the Chebyshev functional. Maširević et al. in [
10] established new bounds on the Chebyshev functional for the
function class. Rahman et al. in [
11] derived certain new proportional and Hadamard proportional fractional integral inequalities. Khan et al. in [
12] investigated the Hirota equation using the modified double Laplace decomposition method. Rahman et al. in [
13] obtained the weighted fractional integral inequalities for Chebyshev functionals. Khan et al. in [
14] established applications of the fixed-point theory to investigate a system of factional-order differential equations. Ayub et al. in [
15] used new a Mittag–Leffler function and derived its applications. Iqbal et al. in [
16] found new generalized Pólya–Szegö- and Chebyshev-type inequalities with a general kernel and measure. Gul et al. in [
17] investigated a class of boundary-value problems under the ABC fractional derivative. Nisar et al. in [
18] derived the weighted fractional Pólya–Szegö- and Chebyshev-type integral inequalities concerning another function. Khan et al. in [
19] investigated the impulsive boundary-value problem with the Riemann–Liouville fractional-order derivative. Rahman et al. in [
20] established generalized fractional integral inequalities for the monotone weighted Chebyshev functionals. Srivastava et al. in [
21] obtained new Chebyshev-type inequalities via a general family of fractional integral operators with a modified Mittag–Leffler kernel. Set et al. in [
22] found Chebyshev-type inequalities by using generalized proportional Hadamard fractional integrals via the Polya–Szegö inequality with applications. Özdemir et al. in [
23] obtained some new Chebyshev-type inequalities for functions whose derivatives belong to
spaces. Akdemir et al. in [
24] found new general variants of Chebyshev-type inequalities via generalized fractional integral operators. Butt et al. in [
25] used Caputo fractional derivatives via exponential
s-convex functions.
The following important results were obtained by Alomari in [
9].
Lemma 1. Let and (the interior set of I). Assume that ϖ is an absolutely continuous function on I, where ϖ and are positive and If and then: The constant is the best possible for every
Lemma 2. Let and . Suppose that ϖ is an absolutely continuous function on I, where ϖ and are positive and If and then the inequality (4) holds. Let us denote where the space of all positive n-th differentiable functions whose n-th derivatives are positive locally absolutely continuous on with the condition that Then, the above Lemmas 1 and 2 are generalized as follows:
Lemma 3. Let If with , for all and then for all , we have: Lemma 4. Let If with , for all and then for all , the inequality (5) holds. Furthermore, we define with as the space of all positive differentiable functions whose first derivatives are positive locally absolutely continuous on and
Theorem 3. Let then for all , we have: Theorem 4. Let then for all , we obtain: The theory of convexity has played very important role in the development of the theory of inequalities. A wide class of inequalities can easily be obtained using the convexity property of the functions.
Let us recall the following definitions that are used in the sequel.
Definition 1 ([26]).A function is said to be convex, if: Definition 2 ([9]).A function is said to be P-convex, if: Definition 3 ([26]).A function is said to be quasi-convex, if: Definition 4 ([27]).A function is said to be s-convex for some fixed , if: Definition 5 ([28,29]).A function is said to be m-convex for some fixed , if: Definition 6 ([28,29]).A function is said to be -convex for some fixed , if: Motivated by the above results, the aim of this paper was to derive some new inequalities of the Beesack–Wirtinger type for different kinds of differentiable convex functions. Furthermore, we generalized our results for functions that are n-times differentiable convex. Finally, many interesting Ostrowski- and Chebyshev-type inequalities are given. Some conclusions and future research are provided as well. We hope that the ideas and techniques of this paper will inspire interested readers working in this fascinating field.
2. Main Results
In this main section, by applying Lemmas 1–4, Theorems 3 and 4, and the fact that every convex function is absolutely continuous, we derive the following inequalities of the Beesack–Wirtinger type.
Theorem 5. Let and . Assume that ϖ is a differentiable function on I, where ϖ and are positive and If is a P-convex function on then for all , we have:where: Proof. From the
P-convexity of
on
we have:
Multiplying by
and using Lemma 1, we obtain the desired inequality (
8). □
Theorem 6. Let and . Suppose that ϖ is a differentiable function on I, where ϖ and are positive and If is a quasi-convex function on then for all , we have:where is defined as in Theorem 5. Proof. From the quasi-convexity of
on
we have:
Multiplying by
and using Lemma 1, we obtain the desired inequality (
9). □
Theorem 7. Let and . Assume that ϖ is a differentiable function on I, where ϖ and are positive and If is a convex function on then for all , we have:where is defined as in Theorem 5. Proof. From the convexity of
on
we have:
Multiplying by
and using Lemma 1, we obtain the desired inequality (
10). □
Theorem 8. Let and . Suppose that ϖ is a differentiable function on I, where ϖ and are positive and If is an s-convex function on then for all , we have:where is defined as in Theorem 5. Proof. From the
s-convexity of
on
we have:
Multiplying by
and using Lemma 1, we obtain the desired inequality (
11). □
Remark 1. Taking in Theorem 8, we obtain Theorem 7.
Theorem 9. Let and . Assume that ϖ is a differentiable function on I, where ϖ and are positive, and If is an m-convex function on then for all and we have:where is defined as in Theorem 5. Proof. From the
m-convexity of
on
we have:
Multiplying by
and using Lemma 1, we obtain the desired inequality (
12). □
Remark 2. Taking in Theorem 9, we obtain Theorem 7.
Theorem 10. Let and . Suppose that ϖ is a differentiable function on I, where ϖ and are positive and If is an -convex function on then for all and we have:where is defined as in Theorem 5. Proof. From the
-convexity of
on
we have:
Multiplying by
and using Lemma 1, we obtain the desired inequality (
13). □
Remark 3. Taking in Theorem 10, we obtain Theorem 9.
Remark 4. Our above results still hold if we apply Lemma 2, so we omit their proofs.
Theorem 11. Let and ϖ be an n-times differentiable function on such that are positive with for all If is a P-convex function on then for all , we have:where: Proof. From the
P-convexity of
on
we have:
Multiplying by
and using Lemma 3, we obtain the desired inequality (
14). □
Theorem 12. Let and ϖ be an n-times differentiable function on such that are positive with for all If is a quasi-convex function on then for all , we have:where is defined as in Theorem 11. Proof. From the quasi-convexity of
on
we have:
Multiplying by
and using Lemma 3, we obtain the desired inequality (
15). □
Theorem 13. Let and ϖ be an n-times differentiable function on such that are positive with for all If is a convex function on then for all , we have:where is defined as in Theorem 11. Proof. From the convexity of
on
we have:
Multiplying by
and using Lemma 3, we obtain the desired inequality (
16). □
Theorem 14. Let and ϖ be an n-times differentiable function on such that are positive with for all If is an s-convex function on then for all , we have:where is defined as in Theorem 11. Proof. From the
s-convexity of
on
we have:
Multiplying by
and using Lemma 3, we obtain the desired inequality (
17). □
Remark 5. Taking in Theorem 14, we obtain Theorem 13.
Theorem 15. Let and ϖ be an n-times differentiable function on such that are positive with for all If is an m-convex function on then for all and we have:where is defined as in Theorem 11. Proof. From the
m-convexity of
on
we have:
Multiplying by
and using Lemma 3, we obtain the desired inequality (
18). □
Remark 6. Taking in Theorem 15, we obtain Theorem 13.
Theorem 16. Let and ϖ be an n-times differentiable function on such that are positive with for all If is an -convex function on then for all and we have:where is defined as in Theorem 11. Proof. From the
-convexity of
on
we have:
Multiplying by
and using Lemma 3, we obtain the desired inequality (
19). □
Remark 7. Taking in Theorem 16, we obtain Theorem 15.
Remark 8. Our above results still holds if we apply Lemma 4, so we omit their proofs.
3. Inequalities of Ostrowski Type
The Ostrowski inequality [
30] is remarkable and has the following representation:
Theorem 17. Let be a differentiable function on , with and If for all then: For other recent results of this type, please see [
9,
30,
31] and the references therein.
Theorem 18. Let and ϖ be a differentiable function on , where ϖ and are positive with If is a P-convex function on then for all , we have:where: Proof. From the
P-convexity of
on
we have:
Multiplying by
and using Theorem 3, we obtain the desired inequality (
21). □
Theorem 19. Let and ϖ be a differentiable function on , where ϖ and are positive with If is a quasi-convex function on then for all , we have:where is defined as in Theorem 18. Proof. From the quasi-convexity of
on
we have:
Multiplying by
and using Theorem 3, we obtain the desired inequality (
22). □
Theorem 20. Let and ϖ be a differentiable function on , where ϖ and are positive with If is a convex function on then for all , we have:where is defined as in Theorem 18. Proof. From the convexity of
on
we have:
Multiplying by
and using Theorem 3, we obtain the desired inequality (
23). □
Theorem 21. Let and ϖ be a differentiable function on , where ϖ and are positive with If is an s-convex function on then for all , we have:where is defined as in Theorem 18. Proof. From the
s-convexity of
on
we have:
Multiplying by
and using Theorem 3, we obtain the desired inequality (
24). □
Remark 9. Taking in Theorem 21, we obtain Theorem 20.
Theorem 22. Let and ϖ be a differentiable function on , where ϖ and are positive with If is an m-convex function on then for all and we have:where is defined as in Theorem 18. Proof. From the
m-convexity of
on
we have:
Multiplying by
and using Theorem 3, we obtain the desired inequality (
25). □
Remark 10. Taking in Theorem 22, we obtain Theorem 20.
Theorem 23. Let and ϖ be a differentiable function on , where ϖ and are positive with If is an -convex function on then for all and we have:where is defined as in Theorem 18. Proof. From the
-convexity of
on
we have:
Multiplying by
and using Theorem 3, we obtain the desired inequality (
26). □
Remark 11. Taking in Theorem 23, we obtain Theorem 22.
Theorem 24. Let and ϖ be an n-times differentiable function on , where are positive with for all If is a P-convex function on then for all , we have:where: Proof. From the
P-convexity of
on
we have:
Multiplying by
and using Theorem 4, we obtain the desired inequality (
27). □
Theorem 25. Let and ϖ be an n-times differentiable function on , where are positive with for all If is a quasi-convex function on then for all , we have:where is defined as in Theorem 24. Proof. From the quasi-convexity of
on
we have:
Multiplying by
and using Theorem 4, we obtain the desired inequality (
28). □
Theorem 26. Let and ϖ be an n-times differentiable function on , where are positive with for all If is a convex function on then for all , we have:where is defined as in Theorem 24. Proof. From the convexity of
on
we have:
Multiplying by
and using Theorem 4, we obtain the desired inequality (
29). □
Theorem 27. Let and ϖ be an n-times differentiable function on , where are positive with for all If is an s-convex function on then for all , we have:where is defined as in Theorem 24. Proof. From the
s-convexity of
on
we have:
Multiplying by
and using Theorem 4, we obtain the desired inequality (
30). □
Remark 12. Taking in Theorem 27, we obtain Theorem 26.
Theorem 28. Let and ϖ be an n-times differentiable function on , where are positive with for all If is an m-convex function on then for all and we have:where is defined as in Theorem 24. Proof. From the
m-convexity of
on
we have:
Multiplying by
and using Theorem 4, we obtain the desired inequality (
31). □
Remark 13. Taking in Theorem 28, we obtain Theorem 26.
Theorem 29. Let and ϖ be an n-times differentiable function on , where are positive with for all If is an -convex function on then for all and we have:where is defined as in Theorem 24. Proof. From the
-convexity of
on
we have:
Multiplying by
and using Theorem 4, we obtain the desired inequality (
32). □
Remark 14. Taking in Theorem 29, we obtain Theorem 28.
Theorem 30. Let , and Assume that ϖ is an n-times differentiable function on , where are positive with for all If is a P-convex function on then for all , we have:where is defined as in Theorem 24. Proof. Taking the modulus, applying the triangle inequality, and then, using the Hölder inequality, we obtain:
From the
P-convexity of
on
and applying Theorem 24, we obtain the desired inequality (
33). □
Theorem 31. Let , and Suppose that ϖ is an n-times differentiable function on , where are positive with for all If is a quasi-convex function on then for all , we have:where is defined as in Theorem 24. Proof. From Inequality (
34), the quasi-convexity of
on
, and applying Theorem 25, we obtain the desired inequality (
35). □
Theorem 32. Let , and Assume that ϖ is an n-times differentiable function on , where are positive with for all If is a convex function on then for all , we have:where is defined as in Theorem 24. Proof. From Inequality (
34), the convexity of
on
, and applying Theorem 26, we obtain the desired inequality (
36). □
Theorem 33. Let , and Suppose that ϖ is an n-times differentiable function on , where are positive with for all If is an s-convex function on then for all , we have:where is defined as in Theorem 24. Proof. From Inequality (
34), the
s-convexity of
on
, and applying Theorem 27, we obtain the desired inequality (
37). □
Remark 15. Taking in Theorem 33, we obtain Theorem 32.
Theorem 34. Let , and Assume that ϖ is an n-times differentiable function on , where are positive with for all If is an m-convex function on then for all and we have:where is defined as in Theorem 24. Proof. From Inequality (
34), the
m-convexity of
on
, and applying Theorem 28, we obtain the desired inequality (
38). □
Remark 16. Taking in Theorem 34, we obtain Theorem 32.
Theorem 35. Let , and Suppose that ϖ is an n-times differentiable function on , where are positive with for all If is an -convex function on then for all and we have:where is defined as in Theorem 24. Proof. From Inequality (
34), the
-convexity of
on
, and applying Theorem 29, we obtain the desired inequality (
39). □
Remark 17. Taking in Theorem 35, we obtain Theorem 34.
5. Conclusions
In this paper, via different kinds of differentiable convex functions, some new inequalities of the Beesack–Wirtinger type were proven. Furthermore, we generalized our results for functions that are
n-times differentiable convex. Finally, many interesting Ostrowski- and Chebyshev-type inequalities were derived as well. It is worth mentioning that from our results, several interesting inequalities using special means, modified Bessel functions of the first and second kind,
q-digamma function where
and some error estimations for quadrature formulas can be found; see [
15,
32,
33,
34,
35,
36,
37] for details. Since the different kinds of convex functions that we used to obtain our results have large applications in many mathematical areas, then they can be applied to derive several new important results in convex analysis, quantum mechanics, and related optimization theory and may stimulate further research in different areas of pure and applied sciences. Studies relating convexity may have useful applications in interdisciplinary studies, such as maximizing the likelihood from multiple linear regressions involving the Gauss–Laplace distribution. For more details, see [
38,
39,
40,
41,
42,
43,
44,
45].