In this article, generalized versions of the k-fractional Hadamard and Fejér-Hadamard inequalities are constructed. To obtain the generalized versions of these inequalities, k-fractional integral operators including the well-known Mittag-Leffler function are utilized. The class of (p, h)-convex functions for Hadamard-type inequalities give the generalizations of results which have been proved in literature for p-convex, h-convex, and several functions deducible from these two classes.

1. Introduction

The subject of fractional calculus which includes the study of fractional-order integrals and derivatives has become a very prominent research topic in recent years. It has got the attention of researchers working in almost all fields of science and engineering. Integral operators are a very important part of fractional calculus; they play an important role in the mathematical treatment of different kinds of real-world problems. In the theory of mathematical inequalities, fractional integral operators appeared as a tool for the generalizations of classical inequalities. In the last two decades, a lot of such generalizations have been published by several authors.

Iscan and Wu [1] have generalized Hermite-Hadamard inequalities for harmonically convex functions by applying fractional integrals. Akkurt et al. [2] have obtained inequalities for (k − h)-Riemann-Liouville fractional integrals with the aid of synchronous and monotonic functions. Mubeen and Habibullah [3] have introduced k-analogue of Riemann-Liouville fractional integrals which are used for establishing k-fractional versions of well-known inequalities. A study of k-fractional integrals has been presented by Tunç et al. in [4], and fractional inequalities have been proved. Farid et al. in [5] demonstrated some novel Fejér-Hadamard and Hadamard inequalities for harmonically convex functions by using fractional integrals containing the Mittag-Leffler function.

Kunt and Iscan [6] presented Hermite-Hadamard-Fejér-type inequalities for p-convex functions using fractional integrals. Rashid et al. [7] demonstrated the Hadamard and the Fejér-Hadamard-type inequalities for the generalized fractional integral operator involving Mittag-Leffler functions for preinvexity and m-preinvexity. Klnç et al. [8] proved Hadamard and Fejér-Hadamard inequalities for (h, m)-strongly convex functions via generalized fractional integrals with Mittag-Leffler functions. Jia et al. [9] demonstrated new types of Hadamard and Fejér-Hadamard fractional integral inequalities for Riemann-Liouville fractional integrals. Yussouf et al. [10] presented generalized types of Hadamard and Fejér-Hadamard-type fractional integral inequalities by utilizing generalized fractional integrals including Mittag-Leffler functions. Faisal et al. [11] proved new Hermite-Hadamard-Jensen-Mercer-type inequalities for convex functions by using Riemann-Liouville fractional integrals. The other recent studies in this area were presented by Zhao et al. [12–14] and Chu et al. [15].

The purpose of this article is to investigate the Hadamard and the Fejér-Hadamard-type inequalities for (p, h)-convex functions via generalized k-fractional integrals including Mittag-Leffler functions. Inequalities for several kinds of convex functions are special cases of results of this paper. Also, several new inequalities can be deduced from presented results in specific settings. First, we give definitions of integral operators and some results which will set the foundation for this article.

Definition 1 (see [16].)Let w, α, l, γ, c, ∈ℂ, and with . Also, let σ ∈ L₁[ε, v] and ζ ∈ [ε, v]. In that case, the generalized fractional operators and are defined by

(1)

where

(2)

is the enlarged generalized Mittag-Leffler function and

is the expansion of beta function described as noted below:

(3)

where

Definition 2 (see [16].)Let σ, ς : [ε, υ]⟶ℝ with 0 < ε < υ be the functions such that σ is positive and σ ∈ L₁[ε, υ] and ς be differentiable and absolutely increasing. Also, let ϕ/ζ be an increasing function on [ε, ∞), and γ, c, w, α, l, ∈ℂ, with , and 0 < v ≤ μ + δ. In that case, for ζ ∈ [ε, υ], the fractional operators are defined by

(4)

Definition 3 (see [17].)Let σ, ς : [ε, υ]⟶ℝ with 0 < ε < υ be the functions such that σ is positive and σ ∈ L₁[ε, υ] and ς be differentiable and absolutely increasing. Also, let w, γ, c, α, l, ∈ℂ and with and 0 < v ≤ μ + δ. In that case, for ζ ∈ [ε, υ], the unified integral operators are defined by

(5)

Recently, Yue et al. [18] described generalized k-fractional operators including the further extension of the Mittag-Leffler function as noted below.

Definition 4. Let σ, ς : [ε, υ]⟶ℝ with 0 < ε < υ be the functions such that σ be positive and σ ∈ L₁[ε, υ] and ς be differentiable and absolutely increasing. Let α > k and γ, c, w, α, l, ∈ℝ, α, l > 0, and c > γ > 0, with and 0 < v ≤ μ + δ. In that case, for ζ ∈ [ε, υ], the left-right generalized k-fractional operators and are defined by

(6)

(7)

Next, we define convex and related functions.

Definition 5 (see [19].)A function σ : [ε, υ]⟶ℝ is said to be convex, if

(8)

for all ζ, η ∈ [ε, υ] and t ∈ [0, 1].

Definition 6 (see [20].)Let I ⊂ (0, ∞) be a real interval and p ∈ ℝ\{0}. A function σ : I⟶ℝ is said to be a p-convex function, if

(9)

for all ζ, η ∈ I and t ∈ [0, 1].

Definition 7 (see [6].)Let p ∈ ℝ\{0}. In that case, a function σ : [ε, υ] ⊂ (0, ∞)⟶ℝ is called p-symmetric in accordance with , if

(10)

for t ∈ [0, 1].

Definition 8 (see [21].)Let h : J⟶ℝ be a nonnegative and nonzero function and p ∈ ℝ\{0}. We say that σ : I⟶ℝ is a (p, h)-convex function, if σ is nonnegative and

(11)

for all ζ, η ∈ I and t ∈ (0, 1). If the above inequality is reversed, then σ is said to be a (p, h)-concave function.

Remark 9.

(i)
By taking h(t) = t in Definition 8, we obtain the definition of p-convex function
(ii)
By taking h(t) = t and p = 1 in Definition 8, we obtain the definition of convex function
(iii)
By taking h(t) = t^s and p = 1 in Definition 8, we obtain the definition of s-convex function
(iv)
By taking h(t) = t⁻¹ and p = 1 in Definition 8, we obtain the definition of Godunova-Levin-type convex function
(v)
By taking p = 1 in Definition 8, we obtain the definition of h-convex function
(vi)
By taking h(t) = 1 and p = 1 in Definition 8, we obtain the definition of p-function

The following inequality is the well-known Hadamard inequality.

Theorem 10 (see [22].)Let σ : [ε, υ]⟶ℝ be a convex function for ε < υ. Then, the following inequality holds:

(12)

The Fejér-Hadamard inequality is a weighted type of the Hadamard inequality presented by Fejér in [23].

Theorem 11. Let σ : [ε, υ]⟶ℝ be a convex function and ς : [ε, υ]⟶ℝ be nonnegative, integrable, and symmetric with respect to ((ε + υ))/2. In that case, the below inequality takes:

(13)

For other classes of functions defined after motivating from convex function, the above inequalities have been studied extensively (see [1, 5, 6, 24–26]).

In the upcoming section, first, we construct the Hadamard inequality for (p, h)-convex function via generalized k-fractional integrals. Then, an identity is used to construct the Fejér-Hadamard inequality for (p, h)-convex function via generalized k-fractional integrals. After, another version of the Hadamard inequality and the Fejér-Hadamard inequality is presented. Moreover, the presented results generalize many already published results.

2. k-Fractional Integral Inequalities of Hadamard and Fejér-Hadamard Type

The generalized k-fractional Hadamard inequality is stated and proved in the following theorem.

Theorem 12. Let h : J⟶ℝ be nonnegative, nonzero, and integrable function. Also, let σ, ς : [ε, υ]⟶ℝ with 0 < ε < υ be the functions such that σ is positive and σ ∈ L₁[ε, υ], and ς is differentiable and absolutely increasing. If σ is (p, h)-convex, and p ∈ R\{0}, in that case, the below inequalities for k-fractional operators (6) and (7) occur:

(i)
If p > 0, in that case,

(14)

where

and θ(t) = ς^1/p(t) for t ∈ [ε^p, υ^p]

(ii)
If p < 0, in that case,

(15)

where

and θ(t) = ς^1/p(t) for t ∈ [υ^p, ε^p]

Proof. We demonstrate claim (i) as noted below:

(i)
Since σ is (p, h)-convex function on [ε, υ], for ζ, η ∈ I, we get

(16)

Taking and in the above inequality, we get

(17)

Multiplying both sides of (17) by and integrating over [0, 1], we get

(18)

By choosing ς(ζ) = tς^p(ε) + (1 − t)ς^p(υ) and ς(η) = tς^p(υ) + (1 − t)ς^p(ε) in (18), we get

(19)

By utilizing k-fractional operators (6) and (7), the first side of (14) is acquired.

Now, to demonstrate the second side of (14), once again, (p, h)-convexity of f over [ε, υ], and for t ∈ [0, 1], we get

(20)

Multiplying both sides of (20) by and integrating over [0, 1], we get

(21)

Taking ς(ζ) = tς^p(ε) + (1 − t)ς^p(υ) and ς(η) = tς^p(υ) + (1 − t)ς^p(ε) in (21), in that case, by utilizing k-fractional operators (6) and (7), the second side of (14) is acquired.

(ii)
Proof is the same as the proof of (i)

Corollary 13. By utilizing (14) and (15), some more k-fractional inequalities are offered as noted below:

(i)
By choosing ς = I and , we acquire

(22)

(ii)
By choosing ς = I, , we acquire

(23)

(iii)
By choosing p = −1 and ς = I, we acquire

(24)

(iv)
By choosing p = −1, , and ς = I, we acquire

(25)

(v)
By choosing p = −1, we acquire

(26)

Remark 14. The well-known k-fractional inequalities are further noted as below:

(i)
By choosing h(t) = t and k = 1 in Corollary 13 (i), Theorem 9 of [6] is acquired
(ii)
By choosing h(t) = t and k = 1 in Corollary 13 (ii), Theorem 2.1 of [24] is acquired
(iii)
By choosing h(t) = t and k = 1 in Corollary 13 (iii), Theorem 2.1 of [5] is acquired
(iv)
By choosing h(t) = t and k = 1 in Corollary 13 (iv), Theorem 4 of [1] is acquired
(v)
By choosing h(t) = t and k = 1 in Corollary 13 (v), Theorem 2.1 of [27] is acquired

The below lemma is beneficial to offer the Fejér-Hadamard inequality for generalized k-fractional integrals.

Lemma 15 (see [18].)Let σ, ς : [ε, υ]⟶ℝ with 0 < ε < υ be the functions such that σ is positive and σ ∈ L₁[ε, υ], and ς is differentiable and absolutely increasing. If p ∈ R\{0} and , in the case for generalized k-fractional operators (6) and (7), we have

(i)
If p > 0, in that case,

(27)

with θ(t) = ς^1/p(t) for all t ∈ [ε^p, υ^p]

(ii)
If p < 0, in that case,

(28)

with θ(t) = ς^1/p(t) for all t ∈ [υ^p, ε^p]

The Fejér-Hadamard inequality for generalized k-fractional integrals is stated and proved in the following theorem.

Theorem 16. Let h : J⟶ℝ be a nonnegative and nonzero function. Also, let σ, ς, r : [ε, υ]⟶ℝ, 0 < ε < υ, be the functions such that σ is positive and σ ∈ L₁[ε, υ], ς is differentiable and absolutely increasing, and r is a nonnegative and integrable function. If σ is (p, h)-convex with p ∈ R\{0}, and , then the following inequalities for generalized k-fractional integral operators (6) and (7) hold:

(i)
If p > 0, in that case,

(29)

where

, and θ(t) = ς^1/p(t) for t ∈ [ε^p, υ^p]

(ii)
If p < 0, in that case,

(30)

where

, and θ(t) = ς^1/p(t) for t ∈ [υ^p, ε^p]

Proof. We demonstrate claim (i) as noted below:

(i)
Multiplying both sides of (17) by and then integrating over [0, 1], we have

(31)

By choosing ς(x) = tς^p(ε) + (1 − t)ς^p(υ), that is, ς^p(ε) + ς^p(υ) − ς(ζ) = tς^p(υ) + (1 − t)ς^p(ε), in (31) and using , we have

(32)

This implies

(33)

By using Lemma 15 (i) in the above inequality, we get the first inequality of (29).

Now, to demonstrate the second side of (29), multiplying both sides of (20) with , and next integrating over [0, 1], we obtain

(34)

Setting ς(ζ) = tς^p(ε) + (1 − t)ς^p(υ) and using in (34), we have

(35)

By utilizing Lemma 15 (i) in the above inequality, we get the second side of (29).

(ii)
The proof is similar to the proof of (i)

Corollary 17. By utilizing (29) and (30), some more k-fractional inequalities are offered as noted below:

(i)
By choosing ς = I and , we acquire

(36)

(ii)
By choosing ς = I and , we acquire

(37)

(iii)
By choosing , r(ζ) = 1, ς = I, and p = −1, we acquire

(38)

(iv)
By choosing ς = I, p = −1, and r(ζ) = 1, we acquire

(39)

(v)
By choosing , r(ζ) = 1, p = −1, and ς = I, we acquire

(40)

(vi)
By choosing p = −1, we acquire

(41)

Remark 18. The mentioned k-fractional inequalities are further connected with foreknown conclusions as noted below:

(i)
By setting h(t) = t and k = 1 in Corollary 17 (ii), Theorem 9 of [6] is obtained
(ii)
By setting h(t) = t and k = 1 in Corollary 17 (iii), Theorem 2.1 of [24] is obtained
(iii)
By setting h(t) = t and k = 1 in Corollary 17 (iv), Theorem 2.1 of [5] is obtained
(iv)
By setting h(t) = t and k = 1 in Corollary 17 (v), Theorem 4 of [1] is obtained
(v)
By setting h(t) = t and k = 1 in Corollary 17 (vi), Theorem 2.5 of [27] is obtained

In the next theorem, we offer another type of the Hadamard inequality.

Theorem 19. Let h : J⟶ℝ is a nonnegative and nonzero function. Also, let σ, ς : [ε, υ]⟶ℝ, 0 < ε < υ, be the functions such that σ is positive and σ ∈ L₁[ε, υ] and ς is differentiable and absolutely increasing. If σ is (p, h)-convex, p ∈ R\{0}, then for generalized k-fractional integral operators (6) and (7), we have the following:

(i)
If p > 0, in that case,

(42)

where

, and θ(t) = ς^1/p(t) for t ∈ [ε^p, υ^p]

(ii)
If p < 0, in that case,

(43)

where

, and θ(t) = ς^1/p(t) for t ∈ [υ^p, ε^p]

Proof. We demonstrate claim (i) as noted below:

(i)
Choosing and in (16), we have

(44)

Multiplying both sides of (44) by and then integrating over [0, 1], we have

(45)

By choosing and in (45), then by using k-fractional integral operators (6) and (7), the first inequality of (42) is obtained.

Now, to demonstrate the second side of (42), once again (p, h)-convexity of σ over [ε, υ] and for t ∈ [0, 1], we obtain

(46)

Multiplying both sides of (46) by and then integrating over [0, 1], we have

(47)

Choosing ς(ζ) = (t/2)ς^p(ε) + (2 − t/2)ς^p(υ) and ς(η) = (t/2)ς^p(υ) + (2 − t/2)ς^p(ε) in (47) and by utilizing k-fractional operators (6) and (7), the second side of (42) is acquired.

(ii)
Proof is the same as the proof of (i)

Corollary 20. By utilizing (42) and (43), some more k-fractional inequalities are offered as noted below:

(i)
By choosing ς = I and , we acquire

(48)

(ii)
By choosing p = −1 and ς = I, we acquire

(49)

(iii)
By choosing p = −1, we acquire

(50)

Remark 21. The mentioned k-fractional inequalities are farther connected with foreknown conclusions as noted below:

(i)
By choosing h(t) = t and k = 1 in Corollary 20 (i), Theorem 2.3 of [24] is acquired
(ii)
By choosing h(t) = t and k = 1 in Corollary 20 (ii), Theorem 2.3 of [5] is acquired
(iii)
By choosing h(t) = t and k = 1 in Corollary 20 (iii), Theorem 2.7 of [27] is acquired

The second type of the Fejér-Hadamard inequality for generalized k-fractional integrals is given as noted below.

Theorem 22. Let h : J⟶ℝ be a nonnegative and nonzero function. Also, let σ, ς, r : [ε, υ]⟶ℝ with 0 < ε < υ be the functions such that σ is positive and σ ∈ L₁[ε, υ], ς is differentiable and absolutely increasing, and r is a nonnegative and integrable function. If σ is (p, h)-convex, p ∈ R\{0} and , then the following inequalities for generalized k-fractional integral operators (6) and (7) hold:

(i)
If p > 0, in that case,

(51)

where

, and θ(t) = ς^1/p(t) for t ∈ [ε^p, υ^p]

(ii)
If p < 0, in that case,

(52)

where

, and θ(t) = ς^1/p(t) for t ∈ [υ^p, ε^p]

Proof. We demonstrate the first claim as noted below:

(i)
Multiplying (44) by and then integrating over [0, 1], we have

(53)

By choosing ς(ζ) = (t/2)ς^p(ε) + (2 − t/2)ς^p(υ) and ς(η) = (t/2)ς^p(υ) + (2 − t/2)ς^p(ε), that is, ς^p(ε) + ς^p(υ) − ς(ζ) = (t/2)ς^p(υ) + (2 − t/2)ς^p(ε), in (53), in that case, by utilizing and k-fractional integral operators (6) and (7), the first side of (51) is obtained.

Now, to demonstrate the second side of (51), multiplying both sides of (46) with and integrating over [0, 1], we obtain

(54)

By choosing ς(ζ) = (t/2)ς^p(ε) + (2 − t/2)ς^p(υ) and ς(η) = (t/2)ς^p(υ) + (2 − t/2)ς^p(ε), that is, ς^p(ε) + ς^p(υ) − ς(ζ) = (t/2)ς^p(υ) + (2 − t/2)ς^p(ε), in (54), then by using and k-fractional integral operators (6) and (7), the second inequality of (51) is acquired.

(ii)
Proof is the same as the proof of (i)

Corollary 23. By utilizing (51) and (52), some more k-fractional inequalities are offered as noted below:

(i)
By choosing ς = I and , we acquire

(55)

(ii)
By choosing p = −1 and ς = I, we acquire

(56)

(iii)
By choosing p = −1, we acquire

(57)

Remark 24. The well-known k-fractional inequalities are further noted as below:

(i)
By choosing h(t) = t and k = 1 in Corollary 23 (i), Theorem 2.6 of [24] is acquired
(ii)
By choosing h(t) = t and k = 1 in Corollary 23 (ii), Theorem 2.6 of [5] is acquired
(iii)
By choosing h(t) = t and k = 1 in Corollary 23 (iii), Theorem 2.10 of [27] is acquired

3. Conclusion

In this article, generalized versions of k-fractional Hadamard and Fejér-Hadamard inequalities are presented. To obtain these inequalities, k-fractional integral operators including the Mittag-Leffler function have been used for (p, h)-convex functions. Many published results in the literature are directly connected with the findings of this paper. Some corollaries have been formulated as new k-fractional Hadamard and Fejér-Hadamard inequalities.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The study was funded by the Science & Technology Bureau of Chengdu 2020-YF09-00005-SN supported by the Sichuan Science and Technology program 2021YFH0107 Erasmus+ SHYFTE Project 598649-EPP-1-2018-1-FR-EPPKA2-CBHE-JP.

Open Research

Data Availability

There is no data required for this paper.

References

1 Iscan I. and Wu S., Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Applied Mathematics and Computation. (2014) 238, 237–244, https://doi.org/10.1016/j.amc.2014.04.020, 2-s2.0-84899632101.
10.1016/j.amc.2014.04.020
Web of Science® Google Scholar
2 Akkurt A., Yildirim M. E., and Yildirim H., On some integral inequalities for (k, h)-Riemann-Liouville fractional integral, New Trends in Mathematical Sciences. (2016) 4, no. 2, 138–146, https://doi.org/10.20852/ntmsci.2016217824.
10.20852/ntmsci.2016217824
Google Scholar
3 Mubeen S. and Habibullah G. M., k-fractional integrals and applications, International Journal of Contemporary Mathematical Sciences. (2012) 7, no. 2, 89–94.
Google Scholar
4 Tunç T., Budak H., Usta F., and Sarikaya M. Z., On new generalized fractional integral operators and related fractional inequalities, Konuralp Journal of Mathematics (KJM). (2020) 8, no. 2, 268–278.
Google Scholar
5 Farid G., Rehman A. U., and Mehmood S., Hadamard and Fejer-Hadamard type integral inequalities for harmonically convex functions via an extended generalized Mittag-Leffler function, Journal of Mathematical and Computational Science. (2018) 8, 630–643, https://doi.org/10.28919/jmcs/3880.
10.28919/jmcs/3880
Google Scholar
6 Kunt M. and Iscan I., Hermite-Hadamard-Fejér type inequalities for p-convex functions via fractional integrals, Iranian Journal of Science and Technology, Transactions A: Science. (2018) 42, no. 4, 2079–2089, https://doi.org/10.1007/s40995-017-0352-4, 2-s2.0-85055534272.
10.1007/s40995-017-0352-4
Web of Science® Google Scholar
7 Rashid S., Akdemir A. O., Noor M. A., and Noor K. I., Generalization of inequalities analogous to preinvex functions via extended generalized Mittag-Leffler functions, 2019 International Conference on Applied and Engineering Mathematics (ICAEM), 2019, 256–263, https://doi.org/10.1109/ICAEM.2019.8853807, 2-s2.0-85073685636.
10.1109/ICAEM.2019.8853807
Google Scholar
8 Klnç S., Akkurt A., and Yldrm H., Generalization Mittag-Leffler function associated with of the Hadamard and Fejér-Hadamard inequalities for (h − m)-strongly convex functions via fractional integrals, Journal of Universal Mathematics. (2019) 2, no. 1, 8–15, https://doi.org/10.33773/jum.506507.
10.33773/jum.506507
Google Scholar
9 Jia W., Yussouf M., Farid G., and Khan K. A., Hadamard and Fejer–Hadamard inequalities for (α, h − m) − p-convex functions via Riemann–Liouville fractional integrals, Mathematical Problems in Engineering. (2021) 2021, 12, 9945114, https://doi.org/10.1155/2021/9945114.
10.1155/2021/9945114
Web of Science® Google Scholar
10 Yussouf M., Farid G., Khan K. A., and Jung C. Y., Hadamard and Fejer–Hadamard inequalities for further generalized fractional integrals involving Mittag-Leffler functions, Journal of Mathematics. (2021) 2021, 13, 5589405, https://doi.org/10.1155/2021/5589405.
10.1155/2021/5589405
Web of Science® Google Scholar
11 Faisal S., Khan M. A., Khan T. U., Saeed T., Alshehri A. M., and Nwaeze E. R., New “Conticrete” Hermite-Hadamard-Jensen-Mercer fractional inequalities, Symmetry. (2022) 14, no. 2, https://doi.org/10.3390/sym14020294.
10.3390/sym14020294
Web of Science® Google Scholar
12 Zhao T. H., He Z. Y., and Chu Y. M., On some refinements for inequalities involving zero-balanced hypergeometric function, AIMS Mathematics. (2020) 5, no. 6, 6479–6495, https://doi.org/10.3934/math.2020418.
10.3934/math.2020418
Web of Science® Google Scholar
13 Zhao T. H., Wang M. K., and Chu Y. M., Monotonicity and convexity involving generalized elliptic integral of the first kind, RACSAM. (2021) 115, no. 2, 1–13, https://doi.org/10.1007/s13398-020-00992-3.
10.1007/s13398-020-00992-3
Google Scholar
14 Zhao T. H., Wang M. K., and Chu Y. M., Concavity and bounds involving generalized elliptic integral of the first kind, Journal of Mathematical Inequalities. (2007) 15, no. 2, 701–724.
Google Scholar
15 Chu H. H., Zhao T. H., and Chu Y. M., Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means, Mathematica Slovaca. (2020) 70, no. 5, 1097–1112.
10.1515/ms-2017-0417
Web of Science® Google Scholar
16 Farid G., A unified integral operator and further its consequences, Open Journal of Mathematical Analysis. (2020) 4, no. 1, 1–7, https://doi.org/10.30538/psrp-oma2020.0047.
10.30538/psrp-oma2020.0047
Google Scholar
17 Yussouf M., Farid G., Khan K. A., and Jung C. Y., Hadamard and Fejér-Hadamard inequalities for further generalized fractional integrals involving Mittag-Leffler functions, Journal of Mathematics. (2021) 2021, 13, 5589405, https://doi.org/10.1155/2021/5589405.
10.1155/2021/5589405
Web of Science® Google Scholar
18 Yue Y., Farid G., Demirel A. K., Nazeer W., and Zhao Y., Hadamard and Fejér-Hadamard inequalities for generalized k-fractional integrals involving further extension of Mittag-Leffler functions, AIMS Mathematics. (2021) 7, no. 1, 681–703.
10.3934/math.2022043
Google Scholar
19 Niculescu C. P. and Persson L. E., Convex Functions and Their Applications A Contemporary Approach, 2006, Springer, New York, https://doi.org/10.1007/0-387-31077-0.
10.1007/0-387-31077-0
Google Scholar
20 Iscan I., Ostrowski type inequalities for p-convex functions, New Trends in Mathematical Sciences. (2016) 4, no. 3, 140–150, https://doi.org/10.20852/ntmsci.2016318838.
10.20852/ntmsci.2016318838
Google Scholar
21 Fang Z. B. and Shi R., On the (p,h)-convex function and some integral inequalities, Journal of Inequalities and Applications. (2014) 2014, no. 1, 16, https://doi.org/10.1186/1029-242X-2014-45, 2-s2.0-84899829113.
10.1186/1029-242X-2014-45
Google Scholar
22 Pachpatte B. G., Mathematical Inequalities, 2005, 67, Elsevier.
Google Scholar
23 Fejér L., Uberdie Fourierreihen II, Math Naturwiss Anz Ungar. Akad. Wiss. (1906) 24, 369–390.
Google Scholar
24 Abbas G. and Farid G., Hadamard and Fejér-Hadamard type inequalities for harmonically convex functions via generalized fractional integrals, Journal of Analysis. (2017) 25, no. 1, 107–119, https://doi.org/10.1007/s41478-017-0032-y.
10.1007/s41478-017-0032-y
Google Scholar
25 Ullah H., Khan M. A., and Saeed T., Determination of bounds for the Jensen gap and its applications, Mathematics. (2021) 9, no. 23, https://doi.org/10.3390/math9233132.
10.3390/math9233132
Web of Science® Google Scholar
26 Khan M. A., Anwar S., Khalid S., and Sayed Z. M. M. M., Inequalities of the type Hermite-Hadamard-Jensen-Mercer for strong convexity, Mathematical Problems in Engineering. (2021) 2021, 16, 5386488, https://doi.org/10.1155/2021/5386488.
10.1155/2021/5386488
Web of Science® Google Scholar
27 Qiang X., Farid G., Yussouf M., Khan K. A., and Rehman A. U., A note on the boundedness of sublinear operators on grand variable Herz spaces, Journal of Inequalities and Applications. (2020) 2020, no. 1, 13, https://doi.org/10.1186/s13660-019-2265-6.
10.1186/s13660-019-2265-6
Web of Science® Google Scholar

Citing Literature

All articles

(p, h)-Convex Functions Associated with Hadamard and Fejér-Hadamard Inequalities via k-Fractional Integral Operators

Abstract

1. Introduction

2. k-Fractional Integral Inequalities of Hadamard and Fejér-Hadamard Type

3. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

(p, h)-Convex Functions Associated with Hadamard and Fejér-Hadamard Inequalities via k-Fractional Integral Operators

Abstract

1. Introduction

2. k-Fractional Integral Inequalities of Hadamard and Fejér-Hadamard Type

3. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Related

Information