The study of convex functions is an interesting area of research due to its huge applications in pure and applied mathematics special in optimization theory. The aim of this paper is to introduce and study a more generalized class of convex functions. We established Schur (S), Hermite-Hadamard (HH), and Fejér (F) type inequalities for introduced class of convex functions. The results presented in this paper extend and generalize many existing results of the literature.

1. Introduction

The study of convexity is very simple and ordinary concept and very important due to its massive applications in industry and business; convexity has a great influence on our daily life. In the solution of many real-world problems, the concept of convexity is very decisive. Problems faced in constrained control and estimation are convex. Geometrically, “the convex function is a real-valued function if the line segment joining any two of its points lies on or above the graph of the function in Euclidean space” [1, 2].

The inequality theory is a best way to understand and apply convexity and the following HH inequality for a C-function g is most famous and most studied inequality in recent years;

(1)

There are many interesting generalizations for HH inequality, for example, see [3, 4]. Alp et al. [5] established q–HH inequalities and gave quantum estimates for midpoint type inequalities via convex and quasi–C-functions. Rahman et al. [6] presented certain fractional proportional integral inequalities via C-functions. Liu et al. [7] established the several HH type inequality via C-function for ψ–Riemann–Liouville fractional integrals. In [8], Farid et al. established certain inequalities for Riemann–Liouville fractional integrals in more general form via C-functions and proposed some applications of their established results. Tunc [9] studied h–C-functions and presented Ostrowski-type inequalities via h–C-functions. He also proposed applications of his established Ostrowski-type inequalities to special means. Set et al. [10] established the HH’s inequality for some C-functions via fractional integrals. They also presented some other resulted results in the setting of fractional integrals.

Moreover, Avci, Kavurmaci, and Özdemir in [11] established the HH type for the class of s–C-functions in 2nd sense which is more generalized convexity. The exponentially m–C-functions were investigated by Rashid et al., in [12]. Different kinds of inequalities for exponentially m–C-functions were also established in the same paper. In [13], Baloch and Chu investigated harmonic C-functions and established Petrovic-type inequalities. Sun and Chu [14] studied g–C-functions and established inequalities for the generalized weighted mean values. Ali, Khan, and Khurshidi [15] investigated η–C-functions and established HH inequality for fractional integrals. Chu et al. [16] generalized the HH type inequalities for MT–C-functions and a variant of Jensen-type inequality for harmonic C-functions was given in [17].

The strongly C-function is one of the important functions of convexity, see [4, 18–21]. A function g : I⟶ℝ is strongly C-function if

(2)

holds ∀a₁, a₂ ∈ L, t ∈ [0, 1] and μ is modulus value.

Convexity is being generalized day by day using different methods, In this work, we discuss the convexity of higher order. For the generalization and different application of convexity, see for instance [22–24].

Convex function g : I = [a₁, a₂] ⊂ ℝ⟶ℝ in the classical sense is introduced as

(3)

where a₁, a₂ ∈ I and t ∈ [0, 1].

A function g on close set L is said to be a higher-order strongly convex, if there exists a constant μ ≥ 0, such that

(4)

where a₁, a₂ ∈ L, q ≥ 0, t ∈ [0, 1] and

(5)

If we put q = 2, then (4) becomes strongly C-function in classical sense with same ϕ(t) as defined in (5).

It is noticed that the function ϕ(.) in (5) is not modified. Characterizations of the higher-order strongly C-function discussed in [25] are not correct. Mohsen et al. [26] modify ϕ(.) for the higher-order strongly C-function. Noor and Noor [27] generalized the concept of higher-order strongly convex defined in [26] by higher-order strongly generalized C-functions. For more detailed survey, we refer the readers to [28] and references therein.

Saleem et al. [29] extended the concept of higher-order strongly generalized C-function by introducing generalized strongly p–C-functions of higher order. The generalized strongly modified (p, h)–C-functions of higher order are studied in this paper.

In this paper, we introduced the generalized strongly modified (p; h)–C-functions of higher order and establish some well-known inequalities. Motivation of this work is to improve and extend the existing results. As stated in the remarks of this paper, many existing results can be derived from our results.

This paper is organized as follows: Section 2 contains definitions, Section 3 contains some basic results, and Section 4 contains HH, F, and S type inequalities for the generalized strongly modified (p, h)–C-functions of higher order.

2. Definitions and Basic Results

The first part of this paper contains basic definitions of convexity.

Definition 1 (Additive). A function η is additive when η(x₁, y₁) + η(x₂, y₂) = η(x₁ + x₂, y₁ + y₂)∀x₁, x₂, y₁, y₂ ∈ ℝ, see [30] for detail.

Definition 2 (Super multiplicative function) [31]. A mapping g : I ⊂ ℝ is super multiplicative function when

(6)

Definition 3. Non-negatively homogeneous. A function η is said to be non-negatively homogeneous if η(λa₁, λa₂) = λη(a₁, a₂)∀a₁, a₂ ∈ ℝ and λ ≥ 0.

The definition of h–C-function is given in [32], which was generalized by Noor et al. in [33] as modified h–C-function. Later on in [34], the class of generalized strongly modified h–C-function. The class of generalized strongly C-functions of higher order was introduced in [27] and generalized strongly p–C-functions of higher order were introduced in [29]. Motivated by these innovations, the extended version of higher order strongly convexity, which is generalized strongly modified (p, h)–C-functions of higher order, is introduced in this paper as follows:

Definition 4. Generalized strongly modified (p, h)–C-functions of higher order. A function g : I⟶ℝ is said to be generalized strongly modified (p, h)– convex of higher order if

(7)

holds for all a₁, a₂ ∈ I, μ ≥ 0, and t ∈ [0, 1].

The class of all generalized strongly modified (p, h)–C-functions of higher order is denoted by GSMPHCF.

Remark 5.

(1)
By taking μ = 0, p = 1, and η(a, b) = a − b in (7), we get the definition of modified h convex function
(2)
By taking μ = 0, p = 1, η(a, b) = a − b, and h(t) = t in (7), we get the definition of C-function
(3)
By taking h(t) = t and p = 1 in (7), we get the definition of generalized strongly convex function
(4)
By taking h(t) = t in (7), we get the definition of generalized strongly p convex function

Hence, in the light of the above remark, our definition is more generalized than all exiting notions.

Now, we establish some basic properties.

Proposition 6. Consider η is non-negatively homogeneous and additive and the functions g, h : I ⊂ ℝ⟶ℝ are in GSMPHCF, then a f +bg∀a, b ∈ ℝ⁺ is also in GSMPHCF.

Proof. It is cleared that (a + b)μ is always greater than μ because a, b ∈ ℝ⁺ and μ ≥ 0 using this fact we can show that af+bg ∀a, b ∈ ℝ⁺ is in GSMPHCF.

Proposition 7. Consider the functions h₁, h₂ on L are non-negative and h₂(t) ⩽ h₁(t). If g is generalized strongly modified (p, h₂)–C-function of higher order, then g is also generalized strongly modified (p, h₁)–C-function of higher order.

Proof. The proof is straight forward.

Proposition 8. Suppose a class of functions g_j : I ⊂ ℝ⟶ℝ be in GSMPHCF where λ_j ∈ ℝ⁺ for every j ∈ ℕ and η is non-negatively homogeneous and additive then their linear combination f : ℝ⟶ℝ is also in GSMPHCF.

Proof. Let the linear combination

(8)

Set , so

(9)

In this proof, we have used a straight forward result that μ is always greater than μ when λ_j ∈ ℝ⁺ for every j ∈ ℕ.

3. Hermite-Hadamard (HH), Fejér (F), and Schur (S) Type Inequalities

In this section, we present Hermite-Hadamard, Fejér, and Schur type inequalities for the new notion of convexity introduced in this paper.

3.1. S Type Inequality

Firstly, we present Schur type inequality.

Theorem 9. Let g : I⟶ℝ and h be the super multiplicative function, where η : N × N⟶M be a bi-function for appropriate A, B⊆ℝ. Then, for a₁, a₂, a₃ ∈ L, a₁ < a₂ < a₃ such that a₃ − a₁, a₃ − a₂, a₂ − a₁ ∈ L, the following inequality holds

(10)

if and only if g is in GSMPHCF.

Proof. Let a₁, a₂, a₃ ∈ L ⊂ ℝ, such that (a₃ − a₂)/(a₃ − a₁) ∈ (0, 1)⊆L, (a₂ − a₁)/(a₃ − a₁) ∈ (0, 1)⊆L, and ((a₃ − a₂)/(a₃ − a₁)) + ((a₂ − a₁)/(a₃ − a₁)) = 1, then

(11)

as h is super multiplicative.

Suppose , by definition of g, we have

(12)

Inserting , x = a₁, and y = a₃ in inequality (12), we obtain,

(13)

Multiplying (13) by and using the fact that h is super multiplicative, we get

(14)

For conversely insert a₁ = x, , a₃ = y, and in inequality (10) and using the fact that h is super multiplicative function such that h(t(y^p − x^p)) ≥ h(y^p − x^p)h(t), we get

(15)

implies

(16)

The prove is completed.

Remark 10.

(1)
For h(t) = t, (10) reduced to S type inequality for function defined in [29].
(2)
For μ = 0, p = 1, and η(a, b) = a − b, (10) reduced to S type inequality for modified h–C-function, see [33].

3.2. HH Type Inequality

Here we present Hermite–Hadamard type inequality.

Theorem 11. Let g : I⟶ℝ in GSMPHCF defined on [a₁, a₂] with a₁ < a₂, then

(17)

Proof. Choose and , then

(18)

Implies

(19)

Integrating with respect to t on [0,1], we get

(20)

Implies

(21)

Putting x = (1 − t)a₁ + ta₂, we get

(22)

Implies

(23)

Now,

(24)

Taking we get

(25)

Similarly,

(26)

From inequalities (25) and (26), we have

(27)

This implies

(28)

Implies

(29)

Inequalities (23) and (29) assure to proof.

Remark 12.

(1)
For p = 1, μ = 0, and η(a, b) = a − b, (17) reduced to HH type inequality for modified h–C-function, see [33].
(2)
For h(t) = t, (17) reduced to HH type inequality for function defined in [29].
(3)
For μ = 0, p = 1, η(a, b) = a − b, and h(t) = t,(17) reduced to HH type inequality for C-function in classical sense

3.3. F Type Inequality

This subsection contains Fejér type inequality.

Lemma 13. Consider g be in GSMPHCF and η(x, y) = −η(y, x), then

(30)

where

and t ∈ [0, 1].

Proof. Let g : I⟶ℝ be in GSMPHCF then for , we get

(31)

Theorem 14. Let g : [a₁, a₂]⟶ℝ be in GSMPHCF and w : [a₁, a₂]⟶ℝ be integrable and symmetric w.r.t a₁ + a₂/2, where w ≥ 0, then we have

(32)

where M_η = [η(g(a₁), g(a₂)) + η(g(a₂), g(a₁))]/2.

Proof. Let g be a generalized strongly modified (p, h)–C-function of higher order where and , then

(33)

where

, so

(34)

Now, if , then

(35)

So,

(36)

Similarly, if then

(37)

Adding inequalities (36) and (37), where w is symmetric, then

(38)

Now, for , we have

(39)

Inequalities (34) and (39) assure to proof.

Remark 15.

(1)
For h(t) = t, this F type inequality reduced to F type inequality for the function defined in [29].
(2)
For μ = 0, p = 1, and η(a, b) = a − b, this F Type inequality reduced to F type inequality for the function defined in [33].
(3)
For μ = 0, p = 1, η(a, b) = a − b, and h(t) = t, this F type inequality reduced to F type inequality for C-function in classical sense

4. Conclusions

In this paper, we introduced the notion of generalized strongly modified (p; h)–C-functions of higher order and established HH, F, and S type inequalities. Many existing results can be derived from our results. Our results are applicable in pure and applied mathematics especially in optimization theory. For applications point, we refer to the readers [35–38].

5. Future Directions

It will be interesting to introduce a new and more generalized version of the convexity. The inequalities established in this paper can be further extended for different versions of fractional integral operators.

Conflicts of Interest

We do not have any competing interests.

Authors’ Contributions

Y.M. wrote the final version of the paper and verified the results and arranged the funding for this paper. M.S.S. proposed the problem and supervised this work. I.B. proved the main results of the paper and Y.X. wrote the first version of the paper.

Acknowledgments

This research is funded by the School of General Education, Dalian Neusoft University of Information, Dalian 116023, China. This research is supported by Department of Mathematics, Univeristy of Okara, Okara Pakistan.

Open Research

Data Availability

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

References

1 Butt S. I., Jaksic R., Kvesic L., and Pecaric J., n-Exponential convexity of weighted Hermite-Hadamard’s inequality, Journal of Mathematical Inequalities. (2007) 8, no. 2, 299–311, https://doi.org/10.7153/jmi-08-21, 2-s2.0-84902439554.
10.7153/jmi-08-21
Google Scholar
2 Khan S., Khan M. A., Butt S. I., and Chu Y. M., A new bound for the Jensen gap pertaining twice differentiable functions with applications, Advances in Difference Equations. (2020) 2020, no. 1, 333, https://doi.org/10.1186/s13662-020-02794-8.
10.1186/s13662-020-02794-8
Web of Science® Google Scholar
3 Awan M. U., Talib S., Chu Y. M., Noor M. A., and Noor K. I., Some new refinements of Hermite–Hadamard-type inequalities involving -Riemann–Liouville fractional integrals and applications, Mathematical Problems in Engineering. (2020) 2020, 10, https://doi.org/10.1155/2020/3051920.
10.1155/2020/3051920
Web of Science® Google Scholar
4 Lara T., Merentes N., and Nikodem K., Strong -convexity and separation theorems, International Journal of Analysis. (2016) 2016, 5, https://doi.org/10.1155/2016/7160348.
10.1155/2016/7160348
Google Scholar
5 Alp N., Sarikaya M. Z., Kunt M., and Iscan I., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, Journal of King Saud University-Science. (2018) 30, no. 2, 193–203, https://doi.org/10.1016/j.jksus.2016.09.007, 2-s2.0-84992361871.
10.1016/j.jksus.2016.09.007
Web of Science® Google Scholar
6 Rahman G., Nisar K. S., Abdeljawad T., and Ullah S., Certain fractional proportional integral inequalities via convex functions, Mathematics. (2020) 8, no. 2, https://doi.org/10.3390/math8020222.
10.3390/math8020222
Web of Science® Google Scholar
7 Liu K., Wang J., and O’Regan D., On the Hermite-Hadamard type inequality for ψ–Riemann-Liouville fractional integrals via convex functions, Journal of Inequalities and Applications. (2019) 2019, no. 1, 27, https://doi.org/10.1186/s13660-019-1982-1, 2-s2.0-85060750949.
10.1186/s13660-019-1982-1
Web of Science® Google Scholar
8 Farid G., Nazeer W., Saleem M. S., Mehmood S., and Kang S. M., Bounds of Riemann-Liouville fractional integrals in general form via convex functions and their applications, Mathematics. (2018) 6, no. 11, https://doi.org/10.3390/math6110248, 2-s2.0-85056429900.
10.3390/math6110248
Web of Science® Google Scholar
9 Tunç M., Ostrowski-type inequalities via h-convex functions with applications to special means, Journal of Inequalities and Applications. (2013) 2013, no. 1, 326, https://doi.org/10.1186/1029-242X-2013-326, 2-s2.0-84894484238.
10.1186/1029-242X-2013-326
Web of Science® Google Scholar
10 Set E., Sarikaya M. Z., Özdemir M. E., and Yildirim H., The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results, 2011, http://arxiv.org/abs/1112.6176.
Google Scholar
11 Avci M., Kavurmaci H., and Özdemir M. E., New inequalities of Hermite-Hadamard type via s -convex functions in the second sense with applications, Applied Mathematics and Computation. (2011) 217, no. 12, 5171–5176, https://doi.org/10.1016/j.amc.2010.11.047, 2-s2.0-79551643706.
10.1016/j.amc.2010.11.047
Web of Science® Google Scholar
12 Rashid S., Noor M. A., and Noor K. I., Fractional exponentially m-convex functions and inequalities, International Journal of Analysis and Applications. (2019) 17, no. 3, 464–478, https://doi.org/10.28924/2291-8639-17-2019-464.
10.28924/2291-8639-17-2019-464
Web of Science® Google Scholar
13 Abbas Baloch I. and Chu Y. M., Petrović-type inequalities for Harmonic -convex functions, Journal of Function Spaces. (2020) 2020, 7, https://doi.org/10.1155/2020/3075390.
10.1155/2020/3075390
Web of Science® Google Scholar
14 Sun M. B. and Chu Y. M., Inequalities for the generalized weighted mean values of g-convex functions with applications, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matemáticas. (2020) 114, no. 4, 1–12, https://doi.org/10.1007/s13398-020-00908-1.
10.1007/s13398-020-00908-1
Web of Science® Google Scholar
15 Ali T., Khan M. A., and Khurshidi Y., Hermite-Hadamard inequality for fractional integrals via eta-convex functions, Acta Mathematica Universitatis Comenianae. (2017) 86, no. 1, 153–164.
Google Scholar
16 Chu Y. M., Khan M. A., Khan T. U., and Ali T., Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, Journal of Nonlinear Sciences and Applications. (2016) 9, no. 6, 4305–4316, https://doi.org/10.22436/jnsa.009.06.72.
10.22436/jnsa.009.06.72
Google Scholar
17 Baloch I. A., Mughal A. A., Chu Y. M., Haq A. U., and De La Sen M., A variant of Jensen-type inequality and related results for harmonic convex functions, Aims Mathematics. (2020) 5, no. 6, 6404–6418, https://doi.org/10.3934/math.2020412.
10.3934/math.2020412
Web of Science® Google Scholar
18 Murty K. G. and Yu F. T., Linear Complementarity, Linear and Nonlinear Programming (Vol. 3), 1988, Heldermann, Berlin.
Google Scholar
19 Nikodem K. and Pales Z., Characterizations of inner product spaces by strongly convex functions, Banach Journal of Mathematical Analysis. (2011) 5, no. 1, 83–87, https://doi.org/10.15352/bjma/1313362982, 2-s2.0-78649794965.
10.15352/bjma/1313362982
Web of Science® Google Scholar
20 Poracká-Divis Z., Existence theorem and convergence of minimizing sequences in extremum problems, SIAM Journal on Mathematical Analysis. (1971) 2, no. 4, 505–510, https://doi.org/10.1137/0502051.
10.1137/0502051
Google Scholar
21 Qu G. and Li N., On the exponential stability of primal-dual gradient dynamics, IEEE Control Systems Letters. (2019) 3, no. 1, 43–48, https://doi.org/10.1109/LCSYS.2018.2851375, 2-s2.0-85057641496.
10.1109/LCSYS.2018.2851375
Google Scholar
22 Barani A., Ghazanfari A. G., and Dragomir S. S., Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, Journal of Inequalities and Applications. (2012) 2012, no. 1, 247, https://doi.org/10.1186/1029-242X-2012-247, 2-s2.0-84873334641.
10.1186/1029-242X-2012-247
Google Scholar
23 Bessenyei M. and Páles Z., Characterizations of convexity via Hadamard’s inequality, Mathematical Inequalities & Applications. (1998) 9, no. 1, 53–62, https://doi.org/10.7153/mia-09-06, 2-s2.0-33644537093.
10.7153/mia-09-06
Google Scholar
24 Zalinescu C., Convex Analysis in General Vector Spaces, 2002, World scientific, https://doi.org/10.1142/5021.
10.1142/5021
Web of Science® Google Scholar
25 Gilányi A., Merentes N., Nikodem K., and Páles Z., Characterizations and decomposition of strongly Wright-convex functions of higher order, Opuscula Mathematica. (2015) 35, no. 1, https://doi.org/10.7494/OpMath.2015.35.1.37, 2-s2.0-84911894059.
10.7494/OpMath.2015.35.1.37
Google Scholar
26 Mohsen B. B., Noor M. A., Noor K. I., and Postolache M., Strongly convex functions of higher order involving bifunction, Mathematics. (2019) 7, no. 11, https://doi.org/10.3390/math7111028.
10.3390/math7111028
Web of Science® Google Scholar
27 Noor M. A. and Noor K. I., Higher order strongly generalized convex functions, Applied Mathematics & Information Sciences. (2020) 14, no. 1, 133–139, https://doi.org/10.18576/amis/140117.
10.18576/amis/140117
Google Scholar
28 Butt S. I., Klaricic Bakula M., Pecaric D., and Pecaric J., Jensen-Grüss inequality and its applications for the Zipf-Mandelbrot law, Mathematical Methods in the Applied Sciences. (2021) 44, no. 2, 1664–1673, https://doi.org/10.1002/mma.6869.
10.1002/mma.6869
Web of Science® Google Scholar
29 Saleem M. S., Chu Y. M., Jahangir N., Akhtar H., and Jung C. Y., On generalized strongly p-convex functions of higher order, Journal of Mathematics. (2020) 2020, 8, https://doi.org/10.1155/2020/8381431.
10.1155/2020/8381431
Web of Science® Google Scholar
30 Delavar M. R., On ϕ-convex functions, Journal of Mathematical Inequalities. (2016) 10, no. 3, 173–183.
Web of Science® Google Scholar
31 Finol C. E. and Wojtowicz M., Multiplicative properties of real functions with applications to classical functions, Aequationes Mathematicae. (2000) 59, no. 1, 134–149, https://doi.org/10.1007/PL00000120, 2-s2.0-0141591693.
10.1007/PL00000120
Google Scholar
32 Varosanec S., On h -convexity, Journal of Mathematical Analysis and Applications. (2007) 326, no. 1, 303–311, https://doi.org/10.1016/j.jmaa.2006.02.086, 2-s2.0-33750610962.
10.1016/j.jmaa.2006.02.086
Web of Science® Google Scholar
33 Noor M., Noor K., and Awan U., Hermite-Hadamard type inequalities for modified h-convex functions, Journal of Applied Mathematics and Mechanics. (2014) 6.
Google Scholar
34 Zhao T., Saleem M. S., Nazeer W., Bashir I., and Hussain I., On generalized strongly modified h-convex functions, Journal of Inequalities and Applications. (2020) 2020, no. 1, 11, https://doi.org/10.1186/s13660-020-2281-6.
10.1186/s13660-020-2281-6
Web of Science® Google Scholar
35 Agarwal P., Baleanu D., Chen Y., Momani S., and Machado J. A. T., Fractional Calculus: ICFDA 2018, Amman, Jordan, July 16-18 (Vol. 303), 2019, Springer Nature, https://doi.org/10.1007/978-981-15-0430-3.
10.1007/978-981-15-0430-3
Google Scholar
36 P. Agarwal, S. S. Dragomir, M. Jleli, and B. Samet, Advances in Mathematical Inequalities and Applications, 2018, Springer Singapore, https://doi.org/10.1007/978-981-13-3013-1.
10.1007/978-981-13-3013-1
Google Scholar
37 Salahshour S., Ahmadian A., Senu N., Baleanu D., and Agarwal P., On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem, Entropy. (2015) 17, no. 2, 885–902, https://doi.org/10.3390/e17020885, 2-s2.0-84923359923.
10.3390/e17020885
Web of Science® Google Scholar
38 Tariboon J., Ntouyas S. K., and Agarwal P., New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations, Advances in Difference Equations. (2015) 2015, no. 1, 18, https://doi.org/10.1186/s13662-014-0348-8, 2-s2.0-84922041655.
10.1186/s13662-014-0348-8
Google Scholar

All articles

Schur, Hermite-Hadamard, and Fejér Type Inequalities for the Class of Higher-Order Generalized Convex Functions

Abstract

1. Introduction

2. Definitions and Basic Results

3. Hermite-Hadamard (HH), Fejér (F), and Schur (S) Type Inequalities

3.1. S Type Inequality

3.2. HH Type Inequality

3.3. F Type Inequality

4. Conclusions

5. Future Directions

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Schur, Hermite-Hadamard, and Fejér Type Inequalities for the Class of Higher-Order Generalized Convex Functions

Abstract

1. Introduction

2. Definitions and Basic Results

3. Hermite-Hadamard (HH), Fejér (F), and Schur (S) Type Inequalities

3.1. S Type Inequality

3.2. HH Type Inequality

3.3. F Type Inequality

4. Conclusions

5. Future Directions

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

References

Related

Information