Convexity plays a vital role in pure and applied mathematics specially in optimization theory, but the classical convexity is not enough to fulfil the needs of modern mathematics; hence, it is important to study generalized notion of convexity. Fraction integral operators also become an important tool for solving problems of model physical and engineering processes that are found to be best described by fractional differential equations. The aim of this paper is to study MT-h-convex functions via fractional integral operators. We establish several Hermite–Hadamard-type inequalities for MT-h-convex function via classical and generalized fractional integrals. We also obtain special means related to our results and present some error estimates for the trapezoidal formulas.

1. Introduction

One of the most important notions in mathematics is convex functions which are very important for both pure and applied mathematicians. Convex functions are helpful in solving problems of optimization theory and many other problems of applied nature.

Definition 1. Let and J^o be interior of J, a mapping said to be convex on J, if the following inequality holds for all c, d ∈ J and λ ∈ [0, 1],

(1)

The mapping ψ is said to be concave if −ψ is convex.

The theory of inequalities got the attention of many researchers, and the new inequalities are always appreciable not only in real analysis but also the researchers working in applied sciences use inequalities as a very effective tool for analyzing different practical problems and to study various properties of solution of different equations [1]. Jensen-type inequalities, Hardy-type inequalities [2], Gagliardo-Nirenberg-type inequalities [3], Grüss-type inequalities [4], Ostrowski-type inequality [5, 6], etc. are extensively studied in the literature. The most famous inequality in literature is known as Hermite–Hadamard inequality, which has fundamental role in convex analysis. The Hermite–Hadamard-type inequalities for different classes of convex function can be found in [7, 8] and references therein.

Let

be a convex mapping defined on the interval J of real numbers and c, d ∈ J. Then, the Hermite–Hadamard inequality is as follows:

(2)

with c < d and c, d ∈ J. If ψ is concave, then both the inequalities reverse their direction.

Hermite–Hadamard inequality is the most important inequality so far in inequality theory, and several extensions of this inequality are given by researchers in recent years [9]. Our motivation is to establish a generalized version of Hermite–Hadamard-type inequality for MT-h-convex functions. It is worthy to mention here that several results of literature can be obtained from our established results as a particular case by taking suitable values of involved parameters.

Since classical notion of convexity is not enough for solving today’s problems, so this notion has been generalized by several researchers to meet the needs of modern mathematics. Now, we present some generalized notions of convexity.

Definition 2 (see [10].)Let be a nonnegative mapping, h ≠ 0. The mapping is said to be h-convex, if ψ is nonnegative and for all c, d ∈ J, λ ∈ (0, 1), the following inequality holds:

(3)

The mapping is said to be h-concave if inequality (3) is reversed.

Definition 3 (see [11], [12].)A mapping is said to be MT-convex on J if it is nonnegative and satisfies the following inequality:

(4)

Motivated by the above two notions, we introduce the following notion of MT-h-convex function.

Definition 4. A mapping is said to be MT-h-convex on J, if for all c, d ∈ J and h(λ) ∈ (0, 1), it is nonnegative and satisfies the following inequality:

(5)

Fraction integral operators also become an important tool for solving problems of model physical and engineering processes that are found to be best described by fractional differential equations. Fractional calculus creates a diversity in inequality theory of convex analysis [13, 14].

The following generalized fractional integral operators were introduced by Ertugral and Sariaya [15] as follows:

Let ρ : [0, ∞)⟶[0, ∞) be such that

(6)

then the left-hand-sided and right-hand-sided generalized fractional integral operators are defined as:

(7)

(8)

respectively.

The following remark justifies the generality of the above fractional integral operators.

Remark 5. (1) If ρ(λ) = λ, then (7) and (8) convert to usual Riemann fractional integral, respectively

(9)

(2) If ρ(λ) = λ^η/Γ(η), then (7) and (8) reduce to the Riemann-Liouville integral [14, 16]

(10)

Here, and .

Note that, for η = 1, the Riemann-Liouville integral converts to the classical integrals. For the other interesting special cases of (7) and (8), we refer to the readers [17, 18].

The aim of this paper is to establish Hermite–Hadamard-type inequalities for the proposed notion of MT-h-convex function in the setting of classical and generalized fractional integral operators. As applications of our results, we present special means related with our results. We also establish some error estimates for the trapezoidal formula.

2. Hermite–Hadamard-Type Inequalities via Classical Fractional Integral

First, we present the following identity that has been obtained in [19] which will play a crucial role in the proof of our main results.

Lemma 6. Let be a differentiable function on J^o and c, d ∈ J with c < d. If ψ^′ ∈ L₁[c, d], then the following inequality holds:

(11)

for each v ∈ [c, d].

Theorem 7. Let be a differentiable function on J^o such that ψ^′ ∈ L₁[c, d] where c, d ∈ J. If |ψ^′| is MT-h-convex function on [c, d] and |ψ^′| ≤ Q where v ∈ [c, d], then we have

(12)

where

(13)

with h(λ)/h(1 − λ) < ∞ is finite.

Proof. From Lemma 6, we have

(14)

Applying mode on both sides,

(15)

Employing MT-h-convexity of |ψ^′|, we have

(16)

Since |ψ^′(v)| ≤ Q, so

(17)

The proof is completed.

Corollary 8. In Theorem 7, if we substitute h(λ) = λ, we get [20] Theorem 7.

Remark 9. If we take v = (c + d)/2 in Theorem 7, then we obtain

(18)

Theorem 10. Let be a differentiable function on J^o such that ψ^′ ∈ L₁[c, d] where c, d ∈ J. If is MT-h-convex mapping on [c, d] and q > 1, (1/p) + (1/q) = 1 also |ψ^′| ≤ Q and v ∈ [c, d], then we have

(19)

Proof. Assume that p > 1 and using Lemma 6, we have

(20)

Using Hölder inequality,

(21)

Since is MT-h-convex mapping and |ψ(v)| ≤ Q, so we have

(22)

Some small calculations yield that

(23)

This implies that

(24)

The proof is completed.

Corollary 11. In Theorem 10, if we substitute h(λ) = λ, we get [20] Theorem 2.4.

Remark 12. If we take v = (c + d)/2 in Theorem 10, then we obtain

(25)

Theorem 13. Let be a differentiable function on J^o such that ψ^′ ∈ L₁[c, d] with c < d where c, d ∈ J. If is MT-h-convex function on [c, d] with q > 1 and |ψ^′(v)| ≤ Q where v ∈ [c, d], then we have

(26)

where

(27)

with h(λ)/h(1 − λ) < ∞ is finite.

Proof. Using Lemma 6, Hölder inequality, and MT-h-convexity of , we have

(28)

The proof is completed.

Corollary 14. In Theorem 13, if we substitute h(λ) = λ, we get [20] Theorem 2.6.

Remark 15. If we take v = (c + d)/2 in Theorem 13, then we obtain

(29)

3. Hermite–Hadamard-Type Inequalities via Generalized Fractional Operators

To establish Hermite–Hadamard-type inequalities via generalized fractional operators for MT-h-convex function, we need the following identity [21].

Lemma 16. Let ψ: [c,d] be differentiable function on (c, d) with c < d such that ψ ∈ L¹[c, d]. Then, for each t ∈ (0, 1), we have

(30)

where v = tc + (1 − t)d, and

(31)

Theorem 17. Let be a differentiable function on (c, d) and ψ^′ ∈ L¹[c, d] with 0 ≤ c < d and η > 0. Then, the following inequality holds for each t ∈ (0, 1). If |ψ^′| is MT-h-convex on [c, d],

(32)

where the constants M₁, M₂, N₁, and N₂ are

(33)

Proof. From Lemma 16, we have

(34)

Using mode property on both sides, we obtain

(35)

Since|ψ^′| is MT-h-convex, so we have

(36)

After simplification, we obtain desired result

(37)

Corollary 18. In Theorem 17, if we substitute h(λ) = λ, we get [21] Theorem 2.1.

Theorem 19. Let be a differentiable function over (c, d) and ψ^′εL¹[c, d] with 0 ≤ c < d and η > 0. If is MT-h-convex with q > 1. Then for each t ∈ (0, 1), the following inequality holds:

(38)

where (1/p) + (1/q) = 1 and p > 1.

Proof. By using Lemma 16, Hölder’s inequality, and MT-h-convexity of , we get

(39)

Since,

(40)

this implies that

(41)

The proof is completed.

Corollary 20. If we substitute h(λ) = λ in Theorem 19, we obtain [21] Theorem 7.

Theorem 21. Let be a differentiable function on (c, d) and ψ^′ ∈ L¹[c, d] with 0 ≤ c < d and η > 0. If function is MT-h-convex on [c, d] for q > 1, then for each tε(0, 1), we have

(42)

Proof. By using Lemma 16, power mean integral inequality, and MT-h-convexity of , we have

(43)

The proof is completed.

Corollary 22. If we substitute identity function h(λ) = λ in Theorem 21, we obtain [21] Theorem 2.3.

4. Application to Special Means

In this section, we present applications of our results in special means. Firstly, we give definitions of special means.

(1)
The arithmetic mean
(44)
(2)
The logarithmic mean
(45)
(3)
The generalized logarithmic mean
(46)

Now, using the results of §2, we give some applications to special means of real numbers.

Proposition 23. Assume that and n ∈ ℤ, |n| ≥ 2. Then, ∀ q ≥1 following inequality holds:

(47)

Proof. For ψ(v) = vⁿ the statement follows by Remark 12 and 15 where .

Proposition 24. Assume that . Then, ∀ q ≥1, we have

(48)

Proof. The statement follows by Remark 12 and 15 for ψ(v) = 1/v.

5. Estimates of Error for Trapezoidal Formula

Assume that f is a division c = v₀ < v₁ < ⋯<v_n−1 < v_n = d of interval [c, d] and consider the quadrature formula

(49)

where

(50)

for the trapezoidal version E_r(ψ, f) donates the associated approximation error.

Proposition 25. Let be a differentiable function on J^o such that ψ^′ ∈ L₁[c, d], where c, d ∈ J with c < d and is MT-h-convex on [c, d]. Then, in 24, for every division f of [c, d] and |ψ^′(v)| ≤ Q, v ∈ [c, d], the trapezoidal error estimate satisfies

(51)

where p > 1, (1/p) + (1/q) = 1.

Proof. On applying Remark 12 on the subinterval [v_k, v_k+1](k = 0, 1, 2, ⋯, n − 1) of the division, we have

(52)

With this, the proof is completed.

Proposition 26. Let be a differentiable function on J^o such that ψ^′ ∈ L₁[c, d], where c, d ∈ J with c < d and is MT-h-convex on [c, d] where q ≥ 1, for every division f of [c, d] and |ψ^′(v)| ≤ Q, v ∈ [c, d], the trapezoidal error estimate satisfies

(53)

Proof. By using Remark 15, the proof comes the same as Proposition 25.

6. Conclusion

Convexity and fractional integral operators are the most important notions to deal with the problems of today’s world. In the present paper, we introduced a more general notion of convexity, called as MT-h-convexity. The classical and generalized fractional integral operators are used to establish the most famous and most studied Hermite–Hadamard-type inequalities for the proposed class of convex functions. Applications of presented results to special means are also given. Corollaries and remarks presented in this paper justify the generality of our results. It is interesting to establish Hermite–Hadamard-type inequalities for the other variants of fractional integral operators, like Caputo fractional integral operators and Atangana fractional integral operators.

Conflicts of Interest

The authors do not have any conflict of interests.

Authors’ Contributions

Hengxiao Qi designed the paper and proved the main results of this paper; Waqas Nazeer analyzed the results, wrote the final version of the paper, and relate this paper with the existing results to justify that the results of this paper are more generalized; Fatima Abbas proposed the problem and supervised this work; and Wenbo Liao wrote the first version of this paper.

Acknowledgments

This work was sponsored by the innovative engineering scientific research supportive project of the Communist Party School of the Shandong Provincial CCP Committee (Shandong Administration College) (2021cx035) and the innovative project of the Shandong Administrative College.

Open Research

Data Availability

All data required for this research is included within this paper.

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Some Inequalities of Hermite–Hadamard Type for MT-h-Convex Functions via Classical and Generalized Fractional Integrals

Abstract

1. Introduction

2. Hermite–Hadamard-Type Inequalities via Classical Fractional Integral

3. Hermite–Hadamard-Type Inequalities via Generalized Fractional Operators

4. Application to Special Means

5. Estimates of Error for Trapezoidal Formula

6. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

References

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Some Inequalities of Hermite–Hadamard Type for MT-h-Convex Functions via Classical and Generalized Fractional Integrals

Abstract

1. Introduction

2. Hermite–Hadamard-Type Inequalities via Classical Fractional Integral

3. Hermite–Hadamard-Type Inequalities via Generalized Fractional Operators

4. Application to Special Means

5. Estimates of Error for Trapezoidal Formula

6. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

References

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