Fuzzy Sets and Their Applications in MathematicsView this Special Issue
Schur, Hermite-Hadamard, and Fejér Type Inequalities for the Class of Higher-Order Generalized Convex Functions
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.
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 is most famous and most studied inequality in recent years;
There are many interesting generalizations for HH inequality, for example, see [3, 4]. Alp et al.  established –HH inequalities and gave quantum estimates for midpoint type inequalities via convex and quasi–C-functions. Rahman et al.  presented certain fractional proportional integral inequalities via C-functions. Liu et al.  established the several HH type inequality via C-function for –Riemann–Liouville fractional integrals. In , 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  studied –C-functions and presented Ostrowski-type inequalities via –C-functions. He also proposed applications of his established Ostrowski-type inequalities to special means. Set et al.  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  established the HH type for the class of –C-functions in 2nd sense which is more generalized convexity. The exponentially –C-functions were investigated by Rashid et al., in . Different kinds of inequalities for exponentially –C-functions were also established in the same paper. In , Baloch and Chu investigated harmonic C-functions and established Petrovic-type inequalities. Sun and Chu  studied –C-functions and established inequalities for the generalized weighted mean values. Ali, Khan, and Khurshidi  investigated –C-functions and established HH inequality for fractional integrals. Chu et al.  generalized the HH type inequalities for MT–C-functions and a variant of Jensen-type inequality for harmonic C-functions was given in .
The strongly C-function is one of the important functions of convexity, see [4, 18–21]. A function is strongly C-function if holds and 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 in the classical sense is introduced as where and .
A function on close set is said to be a higher-order strongly convex, if there exists a constant , such that where , , and
If we put , then (4) becomes strongly C-function in classical sense with same as defined in (5).
It is noticed that the function in (5) is not modified. Characterizations of the higher-order strongly C-function discussed in  are not correct. Mohsen et al.  modify for the higher-order strongly C-function. Noor and Noor  generalized the concept of higher-order strongly convex defined in  by higher-order strongly generalized C-functions. For more detailed survey, we refer the readers to  and references therein.
Saleem et al.  extended the concept of higher-order strongly generalized C-function by introducing generalized strongly –C-functions of higher order. The generalized strongly modified –C-functions of higher order are studied in this paper.
In this paper, we introduced the generalized strongly modified –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 –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 is additive when see  for detail.
Definition 2 (Super multiplicative function) . A mapping is super multiplicative function when
Definition 3. Non-negatively homogeneous. A function is said to be non-negatively homogeneous if and
The definition of –C-function is given in , which was generalized by Noor et al. in  as modified –C-function. Later on in , the class of generalized strongly modified –C-function. The class of generalized strongly C-functions of higher order was introduced in  and generalized strongly –C-functions of higher order were introduced in . Motivated by these innovations, the extended version of higher order strongly convexity, which is generalized strongly modified –C-functions of higher order, is introduced in this paper as follows:
Definition 4. Generalized strongly modified –C-functions of higher order. A function is said to be generalized strongly modified – convex of higher order if holds for all , and .
The class of all generalized strongly modified –C-functions of higher order is denoted by GSMPHCF.
Remark 5. (1)By taking , and in (7), we get the definition of modified convex function(2)By taking and in (7), we get the definition of C-function(3)By taking and in (7), we get the definition of generalized strongly convex function(4)By taking in (7), we get the definition of generalized strongly p convex functionHence, 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 is non-negatively homogeneous and additive and the functions are in GSMPHCF, then f is also in GSMPHCF.
Proof. It is cleared that is always greater than because and using this fact we can show that fg is in GSMPHCF.
Proposition 7. Consider the functions , on are non-negative and . If is generalized strongly modified –C-function of higher order, then is also generalized strongly modified –C-function of higher order.
Proof. The proof is straight forward.
Proposition 8. Suppose a class of functions be in GSMPHCF where for every and is non-negatively homogeneous and additive then their linear combination is also in GSMPHCF.
Proof. Let the linear combination Set , so In this proof, we have used a straight forward result that is always greater than when for every .
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 and be the super multiplicative function, where be a bi-function for appropriate . Then, for , such that , the following inequality holds if and only if is in GSMPHCF.
Proof. Let , such that and , then
as is super multiplicative.
Suppose , by definition of , we have Inserting , and in inequality (12), we obtain, Multiplying (13) by and using the fact that is super multiplicative, we get For conversely insert , , and in inequality (10) and using the fact that is super multiplicative function such that we get implies The prove is completed.
Remark 10. (1)For , (10) reduced to S type inequality for function defined in .(2)For , and , (10) reduced to S type inequality for modified –C-function, see .
3.2. HH Type Inequality
Here we present Hermite–Hadamard type inequality.
Theorem 11. Let in GSMPHCF defined on with , then
Proof. Choose and , then Implies Integrating with respect to on [0,1], we get Implies Putting , we get Implies Now, Taking we get Similarly, From inequalities (25) and (26), we have This implies Implies Inequalities (23) and (29) assure to proof.
Remark 12. (1)For and , (17) reduced to HH type inequality for modified –C-function, see .(2)For , (17) reduced to HH type inequality for function defined in .(3)For , and ,(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 be in GSMPHCF and , then where and .
Proof. Let be in GSMPHCF then for , we get
Theorem 14. Let be in GSMPHCF and be integrable and symmetric w.r.t , where , then we have where
Proof. Let be a generalized strongly modified –C-function of higher order where and , then where , so Now, if , then So, Similarly, if then Adding inequalities (36) and (37), where is symmetric, then Now, for , we have Inequalities (34) and (39) assure to proof.
Remark 15. (1)For , this F type inequality reduced to F type inequality for the function defined in .(2)For and , this F Type inequality reduced to F type inequality for the function defined in .(3)For , , and , this F type inequality reduced to F type inequality for C-function in classical sense
In this paper, we introduced the notion of generalized strongly modified –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.
All data required for this paper is included within this paper.
Conflicts of Interest
We do not have any competing interests.
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.
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.
S. I. Butt, R. Jaksic, L. Kvesic, and J. Pecaric, “n-Exponential convexity of weighted Hermite-Hadamard’s inequality,” Journal of Mathematical Inequalities, vol. 8, no. 2, pp. 299–311, 2007.View at: Publisher Site | Google Scholar
S. Khan, M. A. Khan, S. I. Butt, and Y. M. Chu, “A new bound for the Jensen gap pertaining twice differentiable functions with applications,” Advances in Difference Equations, vol. 2020, no. 1, Article ID 333, 2020.View at: Publisher Site | Google Scholar
M. U. Awan, S. Talib, Y. M. Chu, M. A. Noor, and K. I. Noor, “Some new refinements of Hermite–Hadamard-type inequalities involving -Riemann–Liouville fractional integrals and applications,” Mathematical Problems in Engineering, vol. 2020, 10 pages, 2020.View at: Publisher Site | Google Scholar
T. Lara, N. Merentes, and K. Nikodem, “Strong -convexity and separation theorems,” International Journal of Analysis, vol. 2016, 5 pages, 2016.View at: Publisher Site | Google Scholar
N. Alp, M. Z. Sarikaya, M. Kunt, and I. Iscan, “q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions,” Journal of King Saud University-Science, vol. 30, no. 2, pp. 193–203, 2018.View at: Publisher Site | Google Scholar
G. Rahman, K. S. Nisar, T. Abdeljawad, and S. Ullah, “Certain fractional proportional integral inequalities via convex functions,” Mathematics, vol. 8, no. 2, p. 222, 2020.View at: Publisher Site | Google Scholar
K. Liu, J. Wang, and D. O’Regan, “On the Hermite-Hadamard type inequality for –Riemann-Liouville fractional integrals via convex functions,” Journal of Inequalities and Applications, vol. 2019, no. 1, Article ID 27, 2019.View at: Publisher Site | Google Scholar
G. Farid, W. Nazeer, M. S. Saleem, S. Mehmood, and S. M. Kang, “Bounds of Riemann-Liouville fractional integrals in general form via convex functions and their applications,” Mathematics, vol. 6, no. 11, p. 248, 2018.View at: Publisher Site | Google Scholar
M. Tunç, “Ostrowski-type inequalities via h-convex functions with applications to special means,” Journal of Inequalities and Applications, vol. 2013, no. 1, Article ID 326, 2013.View at: Publisher Site | Google Scholar
E. Set, M. Z. Sarikaya, M. E. Özdemir, and H. Yildirim, “The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results,” 2011, http://arxiv.org/abs/1112.6176.View at: Google Scholar
M. Avci, H. Kavurmaci, and M. E. Özdemir, “New inequalities of Hermite-Hadamard type via s -convex functions in the second sense with applications,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5171–5176, 2011.View at: Publisher Site | Google Scholar
S. Rashid, M. A. Noor, and K. I. Noor, “Fractional exponentially m-convex functions and inequalities,” International Journal of Analysis and Applications, vol. 17, no. 3, pp. 464–478, 2019.View at: Publisher Site | Google Scholar
I. Abbas Baloch and Y. M. Chu, “Petrović-type inequalities for Harmonic -convex functions,” Journal of Function Spaces, vol. 2020, 7 pages, 2020.View at: Publisher Site | Google Scholar
M. B. Sun and Y. M. Chu, “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, vol. 114, no. 4, pp. 1–12, 2020.View at: Publisher Site | Google Scholar
T. Ali, M. A. Khan, and Y. Khurshidi, “Hermite-Hadamard inequality for fractional integrals via eta-convex functions,” Acta Mathematica Universitatis Comenianae, vol. 86, no. 1, pp. 153–164, 2017.View at: Google Scholar
Y. M. Chu, M. A. Khan, T. U. Khan, and T. Ali, “Generalizations of Hermite-Hadamard type inequalities for MT-convex functions,” Journal of Nonlinear Sciences and Applications, vol. 9, no. 6, pp. 4305–4316, 2016.View at: Publisher Site | Google Scholar
I. A. Baloch, A. A. Mughal, Y. M. Chu, A. U. Haq, and M. De La Sen, “A variant of Jensen-type inequality and related results for harmonic convex functions,” Aims Mathematics, vol. 5, no. 6, pp. 6404–6418, 2020.View at: Publisher Site | Google Scholar
K. G. Murty and F. T. Yu, Linear Complementarity, Linear and Nonlinear Programming (Vol. 3), Heldermann, Berlin, 1988.
K. Nikodem and Z. Pales, “Characterizations of inner product spaces by strongly convex functions,” Banach Journal of Mathematical Analysis, vol. 5, no. 1, pp. 83–87, 2011.View at: Publisher Site | Google Scholar
Z. Poracká-Divis, “Existence theorem and convergence of minimizing sequences in extremum problems,” SIAM Journal on Mathematical Analysis, vol. 2, no. 4, pp. 505–510, 1971.View at: Publisher Site | Google Scholar
G. Qu and N. Li, “On the exponential stability of primal-dual gradient dynamics,” IEEE Control Systems Letters, vol. 3, no. 1, pp. 43–48, 2019.View at: Publisher Site | Google Scholar
A. Barani, A. G. Ghazanfari, and S. S. Dragomir, “Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex,” Journal of Inequalities and Applications, vol. 2012, no. 1, Article ID 247, 2012.View at: Publisher Site | Google Scholar
M. Bessenyei and Z. Páles, “Characterizations of convexity via Hadamard’s inequality,” Mathematical Inequalities & Applications, vol. 9, no. 1, pp. 53–62, 1998.View at: Publisher Site | Google Scholar
C. Zalinescu, Convex Analysis in General Vector Spaces, World scientific, 2002.View at: Publisher Site
A. Gilányi, N. Merentes, K. Nikodem, and Z. Páles, “Characterizations and decomposition of strongly Wright-convex functions of higher order,” Opuscula Mathematica, vol. 35, no. 1, p. 37, 2015.View at: Publisher Site | Google Scholar
B. B. Mohsen, M. A. Noor, K. I. Noor, and M. Postolache, “Strongly convex functions of higher order involving bifunction,” Mathematics, vol. 7, no. 11, p. 1028, 2019.View at: Publisher Site | Google Scholar
M. A. Noor and K. I. Noor, “Higher order strongly generalized convex functions,” Applied Mathematics & Information Sciences, vol. 14, no. 1, pp. 133–139, 2020.View at: Publisher Site | Google Scholar
S. I. Butt, M. Klaricic Bakula, D. Pecaric, and J. Pecaric, “Jensen-Grüss inequality and its applications for the Zipf-Mandelbrot law,” Mathematical Methods in the Applied Sciences, vol. 44, no. 2, pp. 1664–1673, 2021.View at: Publisher Site | Google Scholar
M. S. Saleem, Y. M. Chu, N. Jahangir, H. Akhtar, and C. Y. Jung, “On generalized strongly p-convex functions of higher order,” Journal of Mathematics, vol. 2020, 8 pages, 2020.View at: Publisher Site | Google Scholar
M. R. Delavar, “On -convex functions,” Journal of Mathematical Inequalities, vol. 10, no. 3, pp. 173–183, 2016.View at: Google Scholar
C. E. Finol and M. Wojtowicz, “Multiplicative properties of real functions with applications to classical functions,” Aequationes Mathematicae, vol. 59, no. 1, pp. 134–149, 2000.View at: Publisher Site | Google Scholar
S. Varosanec, “On h -convexity,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 303–311, 2007.View at: Publisher Site | Google Scholar
M. Noor, K. Noor, and U. Awan, “Hermite-Hadamard type inequalities for modified h-convex functions,” Journal of Applied Mathematics and Mechanics, vol. 6, p. 2014, 2014.View at: Google Scholar
T. Zhao, M. S. Saleem, W. Nazeer, I. Bashir, and I. Hussain, “On generalized strongly modified h-convex functions,” Journal of Inequalities and Applications, vol. 2020, no. 1, Article ID 11, 2020.View at: Publisher Site | Google Scholar
P. Agarwal, D. Baleanu, Y. Chen, S. Momani, and J. A. T. Machado, Fractional Calculus: ICFDA 2018, Amman, Jordan, July 16-18 (Vol. 303), Springer Nature, 2019.View at: Publisher Site
P. Agarwal, S. S. Dragomir, M. Jleli, and B. Samet, Eds., Advances in Mathematical Inequalities and Applications, Springer Singapore, 2018.View at: Publisher Site
S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, and P. Agarwal, “On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem,” Entropy, vol. 17, no. 2, pp. 885–902, 2015.View at: Publisher Site | Google Scholar
J. Tariboon, S. K. Ntouyas, and P. Agarwal, “New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations,” Advances in Difference Equations, vol. 2015, no. 1, Article ID 18, 2015.View at: Publisher Site | Google Scholar