1 Introduction

The concept of fractional calculus was pioneered by Niels Henrik Abel, and the foundation of fractional calculus as an independent subject was laid by Liouville [1]. Fractional calculus is pivotal in applied mathematics and mathematical analysis [2]. It makes it possible to find the solution of fractional derivatives and fractional integrals of any order.

In mathematical analysis, convex analysis plays a crucial role in exploring the properties and applications of convex functions [3, 4]. Convex functions are pivotal in many fields such as engineering [5], economics [6], geometry [7] and mathematical optimization [8]. The applications of convex functions to the special functions are presented in [9].

In the field of mathematics, inequalities are very crucial. In convex analysis, Hermite-Hadamard (H-H) inequalities, which were established by Charles Hermite and Jacques Hadamard in 1885 [5], are of utmost importance.

For a convex function , the H-H inequality [10] is given as, H-H inequalities can establish maximum and minimum values of functions over an interval. This behaviour makes them applicable in multiple fields. For example, in the field of probability theory, H-H inequalities establish bounds for the occurrence of a particular event [11]. Also, in image processing, H-H inequalities are utilized to ensure image quality by fixing pixels within particular limits [12]. Also, the H-H type inequalities for subadditive functions is provided in [13].

Motivated by the applications of H-H inequalities in multiple disciplines and the work done by Angulo, H., Giménez, J., Moros, A. M., & Nikodem, K on strongly h-convex functions in [14], Feng, B., et al on modified (p, h)-convex functions in [15], Zhang et al. on p-convex functions in [16, 17], and Toader G., on the family of modified h-convex function in [18], we have introduced a novel class of convex functions which generalizes these motivated classes and have provided its applications in the form of Schur inequality and H-H inequalities.

To achieve the goals, the paper is structured in the following order: Some important definitions are reviewed in Section 2. The notion of strongly modified (p, h)-convex function is introduced in Section 3. Also, some basic properties are proved for this novel class of convexity. The Schur inequality is presented in Section 4 for this newly defined class of convex functions. Section 5 demonstrates the proofs of the H-H inequalities for strongly modified (p, h)-convex function. Lastly, Section 6 provides a comprehensive summary of the entire research work.

2 Preliminaries

The following are some definitions which are useful in the results.

Strongly convex function [19, 20]:

Suppose l₁ is a positive real number. A function is said to be strongly convex function with modulus l₁, if holds, ∀u₁, u₂ ∈ B₁, and r ∈ [0, 1].

Super multiplicative function [21]:

A function ξ: is called super multiplicative, if

Gamma function [22]:

Integral form of Gamma function is

Riemann-Liouville (R-L) integrals [23]:

Suppose ξ ∈ L₁[c, d], we can define R − L integrals and of order β > 0 as,

Caputo fractional derivatives (CFD) [24–26]:

Suppose ACⁿ[a, b] be the space of functions that have nth derivatives absolutely continuous, ξ ∈ ACⁿ[a, b], where n = [β] + 1, then we can define CFD and of order β > 0 as,

3 Main results

The novel class of convex functions named as strongly modified (p, h)-convex functions is introduced in this section.

Let be non-zero, non-negative function, and l₁ be a positive real number. A function is said to be strongly modified (p, h)-convex function with modulus l₁, if (1) holds, ∀u₁, u₂ ∈ B₁, p ≥ 1, and r ∈ (0, 1).

Remark 1 (a) By choosing l₁ = 0, p = 1, and h(r) = r in inequality (1), one obtains the convex function.

(b) If we choose l₁ = 0, and h(r) = r in inequality (1), one obtains the p-convex function.

(c) By choosing l₁ = 0, and p = 1 in inequality (1), one obtains the modified h-convex function (see [27]).

(d) If we put l₁ = 0 in inequality (1), one obtains the modified (p, h)-convex function.

The validity of this novel concept of convexity is presented in the following example:

Example 2.1 Suppose u₁, u₂ ∈ [1, ∞), r ∈ (0, 1), p ≥ 1, l₁ > 0 and h(r) = r⁵, then the function ξ(u) = u⁴ is strongly modified (p, h)-convex function.

Particularly, if we choose u₁, u₂ ∈ [1, ∞) with u₁ < u₂, r = 1/2, p = 2 and l₁ = 1/2, in inequality (1), we get (2)

Fig 1 presents the validity of inequality (2).

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Fig 1. The graphical presentations of inequality (2).

https://doi.org/10.1371/journal.pone.0311386.g001

Green and Blue colours represent the right hand side and the left hand side of inequality (2) respectively.

Example 2.2 Consider u₁, u₂ ∈ [1, ∞), r ∈ (0, 1), then the function ξ(u) = u³ is convex function but not strongly modified (p, h)-convex function for h(r) = r⁵, p ≥ 1, and l₁ > 0.

Particularly, if we choose u₁, u₂ ∈ [1, ∞) with u₁ < u₂, r = 1/2, p = 2 and l₁ = 1/2, in inequality (1), we get (3)

Fig 2 presents the graph of inequality (3).

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Fig 2. The graphical presentations of inequality (3).

https://doi.org/10.1371/journal.pone.0311386.g002

Green and Blue colours represent the right hand side and the left hand side of inequality (3) respectively.

Now, some basic properties of strongly modified (p, h)-convex functions are proved.

Lemma 2.1 Let ξ and Φ be strongly modified (p, h)-convex functions, then their sum is also strongly modified (p, h)-convex function.

Proof 1 For u₁, u₂ ∈ B₁, p ≥ 1, l₁ > 0 and r ∈ (0, 1), we have

Since ξ and Φ are strongly modified (p, h)-convex functions,

Lemma 2.2 Let ξ be strongly modified (p, h)-convex function, then for scalar n > 0, nξ is also strongly modified (p, h)-convex function.

Proof 2 For u₁, u₂ ∈ B₁, p ≥ 1, l₁ > 0 and r ∈ (0, 1), we have

Lemma 2.3 Let h₁, h₂ be two non-zero, non-negative functions on such that h₂(r) ≤ h₁(r). If is strongly modified h₂-convex function, then ξ is also strongly modified h₁-convex function.

Proof 3 For u₁, u₂ ∈ B₁, p ≥ 1, l₁ > 0 and r ∈ (0, 1), we have

Remark 2 Let h₁, h₂ be two non-zero, non-negative functions on such that h₁(r) ≤ h₂(r) and if is strongly modified h₁-convex function, then ξ is also strongly modified h₂-convex function.

Lemma 2.4 Let be strongly modified (p, h)-convex functions for and , then their linear combination (4) for all l ∈ B₁ is also strongly modified (p, h)-convex function.

Proof 4 For u₁, u₂ ∈ B₁, p ≥ 1, l₁ > 0, r ∈ (0, 1) and in (4), we get

Lemma 2.5 Let be non-empty collection of strongly modified (p, h)-convex functions such that for all l ∈ B₁, exists in . The function defined by for all l ∈ B₁ is also strongly modified (p, h)-convex function.

Proof 5 For u₁, u₂ ∈ B₁, p ≥ 1, l₁ > 0 and r ∈ (0, 1), we have (5)

By choosing in inequality (5), we get

Lemma 2.6 Let ξ be strongly modified (p, h)-convex function, then for all w ∈ B₁, p ≥ 1 and r ∈ [0, 1]. Where, .

Proof 6 For w ∈ B₁ and r ∈ (0, 1), we have

4 Schur inequality

The next theorem presents the Schur inequality for the strongly modified (p, h)-convex function.

Theorem 1 Let be strongly modified (p, h)-convex function and be non-zero, non-negative, super multiplicative function, then for u₁, u₂, u₃ ∈ B₁ such that u₁ < u₂ < u₃, u₃ − u₁, u₃ − u₂, u₂ − u₁ ∈ B₁ and r ∈ (0, 1), we have (6)

Proof 7 Let u₁, u₂, u₃ ∈ B₁ be such that and , then we have

Suppose h(u₃ − u₂) > 0, then by definition of ξ, we get (7)

By choosing , s = u₁ and w = u₃ in (7), we get

Conversely, (8)

By choosing, , s = u₁ and w = u₃ in (8), we get

Thus, ξ is strongly modified (p, h)-convex function.

The validity of schur inequality is presented in context of previously proved example below:

Example 3.1 Assuming, ξ(u) = u⁴, u₁ = 1, u₂ = 2, u₃ = 3, p = 2, and h(r) = r⁵ in inequality (6), we get (9)

By solving inequality (9), one gets

5 Hermite-Hadamard inequalities

The Hermite-Hadamard Inequalities for this novel class of convex functions is given in next theorem.

Theorem 2 Suppose is a non-zero, non-negative function and is a strongly modified (p, h)-convex function with u₁ < u₂, then

Proof 8 For s, w ∈ B₁, p ≥ 1, l₁ > 0 and n₁ ∈ [0, 1], we have (10)

Put, n₁ = 1/2 in (10), to get (11)

Put, and in (11), to get (12)

By integrating (12) with respect to r from 0 to 1, we get (13)

Put, in first integral of (13), and in second integral of (13), to get (14)

Put on right hand side of (14) to get (15)

From (14) and (15), we get

The following remark presents that the Theorem 2 generalizes the results that already exist in literature.

Remark 3 (a) Assume h(r) = r in Theorem 2, to get Theorem 2.1 of [28].

(b) Assume h(r) = r and p = 1 in Theorem 2, to obtain Theorem 6 of [29].

(c) Assume l₁ = 0 in Theorem 2, to get Theorem 3 of [18].

(d) Assume l₁ = 0 and h(r) = r in Theorem 2, to obtain H-H inequalities for convex function (see [5]).

The following example validate the Theorem 2.

Example 4.1 Assuming ξ(u) = u⁴, u₁ = 1, u₂ = 2, p = 2, and h(r) = r⁵ in inequality (6), we get

The next theorem presents the H-H inequalities for this novel concept of convexity by utilizing Riemann-Liouville integrals.

Theorem 3 Let be a strongly modified (p, h)-convex function with u₁ < u₂ for any u₁, u₂ ∈ [0, 1], then and

Proof 9 Since ξ is strongly modified (p, h)-convex function, therefore (16)

By putting r = 1/2 in (16), we get (17)

Assuming, and in (17), to get (18)

Multiply (18) by r^β − 1 and then integrate from 0 to 1 with respect to r, to get (19)

Use in first integral of (19) and in second integral of (19), to get (20)

Since,

Therefore, (20) become (21)

Also, ξ is strongly modified (p, h)-convex function, therefore (22) and (23)

By adding (22) and (23), we get (24)

Multiply (24) by r^β − 1 and then integrate from 0 to 1 with respect to r, to get (25)

Use in first integral of (25), and in second integral of (25), to get (26)

From (21) and (26), we get

Remark 4 Choose h(r) = r and p = 1 in Theorem 3, to get the result for strongly convex function (see [30]).

The H-H inequalities in context of Caputo Fractional derivatives for the strongly modified (p, h)-convex function is proved in following theorem.

Theorem 4 Let be a strongly modified (p, h)-convex function with u₁ < u₂ for any u₁, u₂ ∈ [0, 1], then and

Proof 10 Since ξⁿ is strongly modified (p, h)-convex function therefore, (27)

By taking r = 1/2 in (27), we get (28)

Assume and in (28), to get (29)

Multiply (29) by r^n−β−1 and then integrate from 0 to 1 with respect to r, to get, (30)

Assume in first integral of (30), and in second integral of (30), to get (31)

Since,

Therefore, (31) become, (32)

Also, ξⁿ is strongly modified (p, h)-convex function, therefore (33) and (34)

By adding (33) and (34), we get (35)

Multiply (35) by r^n−β−1 and then integrate from 0 to 1 with respect to r, to get (36)

Use in first integral of (36), and in second integral of (36), to get (37)

From (32) and (37), we get and

6 Conclusion

The paper introduced the concept of strongly modified (p, h)-convex functions which generalizes the notion of strongly convex functions [19, 29] and provided a thorough examination of their properties. Furthermore, the study has explored Schur inequality and H-H inequalities for this new class of convexity. Some special cases of H-H inequalities are proved in [5, 18, 28–30]. Several illustrations and graphs have been demonstrated to check the validity of the proved inequalities. In future, it is possible to extend the H-H integral inequalities using fractional operators and fractional difference operators given in [31] for the obtained class of convexity.

Acknowledgments

The authors are grateful to anonymous referees for their valuable suggestions, which significantly improved this manuscript.