Hermite–Hadamard Type Inequalities via Generalized Harmonic Exponential Convexity and Applications

Butt, Saad Ihsan; Tariq, Muhammad; Aslam, Adnan; Ahmad, Hijaz; Nofal, Taher A.

doi:https://doi.org/10.1155/2021/5533491

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Developments in Geometric Function Theory

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Volume 2021 | Article ID 5533491 | https://doi.org/10.1155/2021/5533491

Hermite–Hadamard Type Inequalities via Generalized Harmonic Exponential Convexity and Applications

Saad Ihsan Butt,¹Muhammad Tariq,¹Adnan Aslam,²Hijaz Ahmad,³and Taher A. Nofal⁴

Academic Editor: Gangadharan Murugusundaramoorthy

Received05 Jan 2021

Revised23 Jan 2021

Accepted31 Jan 2021

Published12 Feb 2021

Abstract

In this work, we introduce the idea and concept of –polynomial –harmonic exponential type convex functions. In addition, we elaborate the newly introduced idea by examples and some interesting algebraic properties. As a result, several new integral inequalities are established. Finally, we investigate some applications for means. The amazing techniques and wonderful ideas of this work may excite and motivate for further activities and research in the different areas of science.

1. Introduction

Theory of convexity present an active, fascinating, and attractive field of research and also played prominence and amazing act in different fields of science, namely, mathematical analysis, optimization, economics, finance, engineering, management science, and game theory. Many researchers endeavor, attempt, and maintain his work on the concept of convex functions and extend and generalize its variant forms in different ways using innovative ideas and fruitful techniques. Convexity theory provides us with a unified framework to develop highly efficient, interesting, and powerful numerical techniques to tackle and to solve a wide class of problems which arise in pure and applied sciences. In recent years, the concept of convexity has been improved, generalized, and extended in many directions. The concept of convex functions also played prominence and meaningful act in the advancement of the theory of inequalities. A number of studies have shown that the theory of convex functions has a close relationship with the theory of inequalities.

The integral inequalities are useful in optimization theory, functional analysis, physics, and statistical theory. In diverse and opponent research, inequalities have a lot of applications in statistical problems, probability, and numerical quadrature formulas [1–3]. So eventually due to many generalizations, variants, extensions, widespread views, and applications, convex analysis and inequalities have become an attractive, interesting, and absorbing field for the researchers and for attention; the reader can refer to [4–6]. Recently Kadakal and Iscan [7] introduced a generalized form of convexity, namely, –polynomial convex functions.

It is well known that the harmonic mean is the special case of power mean. It is often used for the situations when the average rates is desired and have a lot of applications in different field of sciences which are statistics, computer science, trigonometry, geometry, probability, finance, and electric circuit theory. Harmonic mean is the most appropriate measure for rates and ratios because it equalizes the weights of each data point. Harmonic mean is used to define the harmonic convex set. In 2003, first time harmonic convex set was introduced by Shi [8]. Harmonic and –harmonic convex function was for the first time introduced and discussed by Anderson et al. [9] and Noor et al. [10], respectively. Awan et al. [11] keeping his work on generalizations, introduced a new class called –polynomial harmonically convex function. Motivated and inspired by the ongoing activities and research in the convex analysis field, we found out that there exists a special class of function known as exponential convex function, and nowadays, a lot of people working are in this field [12, 13]. Dragomir [14] introduced the class of exponential type convexity. After Dragomir, Awan et al. [15] studied and investigated a new class of exponentially convex functions. Kadakal and İşcan introduced a new definition of exponential type convexity in [16]. Recently, Geo et al. [17] introduced –polynomial harmonic exponential type convex functions. The fruitful benefits and applications of exponential type convexity is used to manipulate for statistical learning, information sciences, data mining, stochastic optimization and sequential prediction [7, 18, 19] and the references therein. Before we start, we need the following necessary known definitions and literature.

2. Preliminaries

In this section, we recall some known concepts.

Definition 1 (see [5]). Let be a real valued function. A function is said to be convex, if holds for all and

Definition 2 (see [20]). A function is said to be harmonic convex, if holds for all and .

For the harmonic convex function, İşcan [20] provided the Hermite–Hadamard type inequality.

Definition 3 (see [21]). A function is said to be –harmonic convex, if holds for all and .
Note that in (3), we get the following inequality: holds for all .

The function is called Jensen –harmonic convex function.

If we put and , then –harmonic convex sets and –harmonic convex functions collapse to classical convex sets, harmonic convex sets, and harmonic convex functions, respectively.

Definition 4 (see [17]). A function is called –polynomial harmonic exponential type convex function, if holds for every and .

Motivated by the above results, literature, and ongoing activities and research, we organise the paper in the following way. Firstly, we will give the idea and its algebraic properties of –polynomial –harmonic exponential type convex functions. Secondly, we will derive the new sort of (H–H) and refinements of (H–H) type inequalities by using the newly introduced idea. Finally, we will give some applications for means and conclusion.

3. Generalized Exponential Type Convex Functions and Its Properties

We are going to introduce a new definition called –polynomial –harmonic exponential type convex function and will study some of their algebraic properties. Throughout the paper, one thing gets in mind represents finite , –poly –harmonic exp convex function represents –polynomial –harmonic exponential type convex function and (H–H) represents Hermite–Hadamard.

Definition 5. A function is called –poly –harmonic exp convex, if holds for every and .

Remark 6. (i)Taking in Definition 5, we obtain the following new definition about –harmonically exp convex function:(ii)Taking in Definition 5, we obtain the following new definition about 2–poly –harmonically exp convex function:(iii)Taking in Definition 5, then, we get a definition, namely, –poly harmonically exp convex function which is defined by Geo et al. [17](iv)Taking in Definition 5, we obtain the following new definition about –poly exp convex function:(v)Taking and in Definition 5, we obtain the following new definition about harmonically exp type convex function:(vi)Taking and in Definition 5, then, we get a definition, namely, exponential type convex function which is defined by Kadakal et al. [16]That is the beauty of this newly introduce definition if we put the values of and , then, we obtain new inequalities and also found some results which connect with previous results.

Lemma 7. The following inequalities and are hold. If for all .

Proof. The rest of the proof is clearly seen.

Proposition 8. Every –harmonic convex function is –poly –harmonic exp convex function.

Proof. Using the definition of –harmonic convex function and from the Lemma 7, since and for all , we have

Proposition 9. Every –poly –harmonic exp convex function is –harmonic –convex function with .

Proof.

Remark 10. (i)If in Proposition 9, then as a result, we get harmonically convex function, which is introduced by Noor et al. [22](ii)If in Proposition 9, then as a result, we get –convex function, which is defined by Varošanec [6]

Now, we make and investigate some examples by way of newly introduced definition.

Example 11. If , is –harmonic convex function, then according to Proposition 8, it is an –poly –harmonic exp convex function.

Example 12. If , is –harmonic convex function, then according to Proposition 8, it is an –poly –harmonic exp convex function.

Now, we will discuss and investigate some of its algebraic properties.

Theorem 13. Let If and are two –poly –harmonic exp convex functions, then (i) is an –poly –harmonic exp convex function(ii)For , is an –poly –harmonic exp convex function

Proof. (i)Let and be an –poly –harmonic exp convex, then(ii)Let be an –pol –harmonic exp convex, thenwhich completes the proof.

Remark 14. (i)If in Theorem 13, then as a result, we get the and are –harmonic exp convex functions(ii)If in Theorem 13, then as a result, we get Theorem 3.2 in [17](iii)If in Theorem 13, then as a result, we get the and are harmonic exp convex functions(iv)If in Theorem 13, then as a result, we get the and are –poly exp convex functions(v)If and in Theorem 13, then as a result, we get Theorem 2.1 in [16]

Theorem 15. Let be –harmonic convex function and is nondecreasing and –poly exp convex function. Then, the function is an –poly –harmonic exp convex function.

Proof. and we have

Remark 16. (i)In case of , we investigate the following new inequality:(ii)In case of , the above Theorem 15 collapses to Theorem 3.3 in [17](iii)In case of , as a result, we obtain the following new inequality:(iv)In case of , then, the above Theorem 15 collapses to the following new inequality:(v)In case of and , as a result, the above Theorem 15 collapses to the Theorem (2.2) in [16]

Theorem 17. Let be a class of –poly –harmonic exp convex functions and . Then, is an –poly –harmonic exp convex function and is an interval.

Proof. Let and then which completes the proof.

Remark 18. (i)In case of , as a result, we get Theorem 3.4 in [17](ii)In case of and in Theorem 17, as a result, we get Theorem 2.3 in [16]

Theorem 19. If is an –poly –harmonic exp convex then is bounded on

Proof. Let and , then, there exist such that Thus, since and , we have The above proof clearly shows that is bounded above from For bounded below, the readers using the identical concept as in Theorem 2.4 in [16].

Remark 20. (i)In case of , we obtain Theorem 3.5 in [17](ii)In case of and , we obtain Theorem 2.4 in [16]

4. (H–H) Type Inequality via Generalized Exponential Type Convexity

The main object of this section is to investigate and prove a new version of (H–H) type inequality using –poly –harmonic exp convexity.

Theorem 21. Let be an –poly –harmonic exp convex function. If , then

Proof. Since is an –poly –harmonic exp convex function, we have which lead to Using the change of variables, we get Integrating the above inequality with respect to on we obtain which completes the left side inequality.
For the right side inequality, first of all, we change the variable of integration by and using Definition 5 for the function , we have which completes the proof.

Corollary 22. In case of in Theorem 21, then, we get the following new (H–H) type inequality for –harmonic exp convex functions:

Corollary 23. In case of in Theorem 21, then as a result, we investigate the following new (H–H) type inequality for –poly exp convex functions:

Remark 24. (i)In case of , then as a result, we obtain Theorem 4.1 in [17](ii)In case of and , then as a result, we obtain Theorem 3.1 in [16](iii)In case of and , then as a result, we obtain Corollary 1in [17]

In this section, in order to prove our main results regarding on some Hermite–Hadamard type inequalities for –poly –harmonic exp convex function, we need the following lemmas:

Lemma 25. Let be differentiable function on the of . If , then where and

Proof. Let Using integration by parts

Lemma 26 (see [23]). Let be differentiable function on the of . If , then where and

Theorem 27. Let be differentiable function on the of . If and is an -poly –harmonic exp convex function on , , then where

Proof. Using Lemma 25, properties of modulus, power mean inequality, and –poly –harmonic exp convexity of the , we have which completes the proof.

Corollary 28. Under the assumptions of Theorem 27 with , we have the following new result:

Corollary 29. Under the assumptions of Theorem 27 with , we have the following new result: where

Theorem 30. Let be differentiable function on the of . If and is an –poly –harmonic exp convex function on , , , then, where

Proof. Using Lemma 25, properties of modulus, Hölder’s inequality, and –poly –harmonic exp convexity of the , we have which completes the proof.

Corollary 31. Under the assumptions of Theorem 30 with , we have the following new result:

Corollary 32. Under the assumptions of Theorem 30 with , we have the following new result: where

Theorem 33. Let be differentiable function on the of . If and is an –poly –harmonic exp convex function on , then where

Proof. Using Lemma 26, properties of modulus, power mean inequality, and –poly -harmonic exp convexity of the we have which completes the proof.

Corollary 34. Under the assumptions of Theorem 33 with and we have the following new result:

Theorem 35. Let be differentiable function on the of . If and is an –poly –harmonic exp convex function on and then where

Proof. Using Lemma 26, properties of modulus, Hölder’s inequality, and –poly –harmonic exp convexity of the we have which completes the proof.

Corollary 36. Under the assumptions of Theorem 35 with and we have the following new result:

6. Applications

In this section, we recall the following special means of two positive number with : (1)The arithmetic mean(2)The geometric mean(3)The harmonic mean(4)The logarithmic mean

These means have a lot of applications in areas and different types of numerical approximations. However, the following simple relationship are known in the literature:

Proposition 37. Let and . Then we get the following inequality

Proof. Taking for in Theorem 21, then, inequality (59) is easily captured.

Proposition 38. Let and . Then, we get the following inequality:

Proof. Taking for in Theorem 21, then, inequality (60) is easily captured.

Proposition 39. Let and . Then, we get the following inequality:

Proof. Taking for in Theorem 21, then, inequality (61) is easily captured.

Proposition 40. Let and . Then, we get the following inequality:

Proof. Taking for in Theorem 21, then, inequality (62) is easily captured.

Proposition 41. Let and . Then, we get the following inequality:

Proof. Taking for in Theorem 21, then, inequality (63) is easily captured.

Proposition 42. Let . Then, we get the following inequality:

Proof. Taking for in Theorem 21, then, inequality (64) is easily captured.

Proposition 43. Let . Then, we get the following inequality:

Proof. Taking for in Theorem 21, then, inequality (65) is easily captured.

Remark 44. The above discussed means, namely, arithmetic, geometric, harmonic, and logarithmic are well known in literature because these means have remarkable applications in machine learning, probability, statistics, and numerical approximation [24]. But, in the future, we will try to find the applications of the He Chengtian mean (also called as He Chengtian average), which was introduced by the first time a famous ancient Chinese mathematician He Chengtian [25]. The He Chengtian average was extended to solve nonlinear oscillators and it is called as He’s max–min approach (also called as He’s max–min method), which was further developed into a frequency–amplitude formulation for nonlinear oscillators [26, 27].

7. Conclusion

We have introduced and investigated some algebraic properties of a new class of functions, namely, –poly –harmonic exp convex. We showed that our new introduced class of function have some nice properties. We proved that our new introduced class is very larger with respect to the known class of functions, like –polynomial convex and –polynomial harmonically convex. A new version of Hermite–Hadamard type inequality and an integral identity for the differentiable function are obtained. It is high time to find the applications of these inequalities along with efficient numerical methods. We believe that our new class of functions will have a very deep research in this fascinating field of inequalities and also in pure and applied sciences. The interesting techniques and wonderful ideas of this paper can be extended on the coordinates along with fractional calculus. In the future, our goal is that we will continue our research work in this direction furthermore.

Data Availability

Data will be provided on request to the first author.

Conflicts of Interest

The authors declare that there are no conflicts of interest associated with this publication.

Acknowledgments

The authors received financial support from the Taif University Researches Supporting Project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia

References

S. S. Dragomir and V. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, Victoria, RGMIA Monographs, 2000.
K. Mehrez and P. Agarwal, “New Hermite–Hadamard type integral inequalities for convex functions and their applications,” Journal of Computational and Applied Mathematics, vol. 350, pp. 274–285, 2019.
View at: Publisher Site | Google Scholar
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and new inequalities in analysis. Mathematics and its Applications (East European Series), vol. 61, Kluwer Academic Publishers Group, Dordrecht, 1993.
Y. Khurshid, M. Adil Khan, and Y.-M. Chu, “Conformable integral inequalities of the Hermite-Hadamard type in terms of GG- and GA-convexities,” Journal of Function Spaces, vol. 2019, Article ID 6926107, 8 pages, 2019.
View at: Publisher Site | Google Scholar
C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer, New York, 2006.
S. Varošanec, “On h–convexity,” Journal of Mathematical Analysis and Applications, vol. 326, pp. 303–311, 2007.
View at: Publisher Site | Google Scholar
T. Toplu, M. Kadakal, and İ. ̇İşcan, “On n-polynomial convexity and some related inequalities,” AIMS Mathematics, vol. 5, no. 2, pp. 1304–1318, 2020.
View at: Google Scholar
H. N. Shi and J. Zhang, “Some new judgement theorems of Schur geometric and schur harmonic convexities for a class of symmetric function,” Journal of Inequalities and Applications, vol. 2013, Article ID 527, 2013.
View at: Publisher Site | Google Scholar
G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Generalized convexity and inequalities,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1294–1308, 2007.
View at: Publisher Site | Google Scholar
M. A. Noor, K. I. Noor, and S. Iftikhar, “Harmite–Hadamard inequalities for harmonic nonconvex function,” MAGNT Research Report, vol. 4, no. 1, pp. 24–40, 2016.
View at: Google Scholar
M. U. Awan, N. Akhtar, S. Iftikhar, M. A. Noor, and Y.-M. Chu, “New Hermite–Hadamard type inequalities for n-polynomial harmonically convex functions,” Journal of Inequalities and Applications, vol. 2020, Article ID 125, 2020.
View at: Publisher Site | Google Scholar
S. I. Butt, A. Kashuri, M. Tariq, J. Nasir, A. Aslam, and W. Gao, “n–polynomial exponential type p–convex function with some related inequalities and their applications,” Heliyon, vol. 6, article e05420, 2020.
View at: Publisher Site | Google Scholar
S. I. Butt, A. Kashuri, M. Tariq, J. Nasir, A. Aslam, and W. Gao, “Hermite–Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications,” Advances in Difference Equations, vol. 2020, Article ID 508, 2020.
View at: Publisher Site | Google Scholar
S. S. Dragomir and I. Gomm, “Some Hermite–Hadamard's inequality functions whose exponentials are convex,” Babes Bolyai Math., vol. 60, pp. 527–534, 2015.
View at: Google Scholar
M. U. Awan, N. Akhtar, S. Iftikhar, M. A. Noor, and Y. -M. Chu, “Hermite–Hadamard type inequalities for exponentially convex functions,” Applied Mathematics & Information Sciences, vol. 12, no. 2, pp. 405–409, 2018.
View at: Google Scholar
M. Kadakal and İ. İşcan, “Exponential type convexity and some related inequalities,” Journal of Inequalities and Applications, vol. 2020, Article ID 82, 2020.
View at: Publisher Site | Google Scholar
W. Geo, A. Kashuri, S. I. Butt, M. Tariq, A. Aslam, and M. Nadeem, “New inequalities via n–polynomial harmoniaclly exponential type convex functions,” AIMS Mathematics, vol. 5, no. 6, pp. 6856–6873, 2020.
View at: Google Scholar
G. Alirezaei and R. Mahar, “On exponentially concave functions and their impact in information theory,” in 2018 Information Theory and Applications Workshop (ITA), pp. 1–10, San Diego, CA, USA, 2018.
View at: Publisher Site | Google Scholar
S. Pal and T. K. L. Wong, “Exponentially concave functions and a new information geometry,” The Annals of Probability, vol. 46, no. 2, pp. 1070–1113, 2018.
View at: Publisher Site | Google Scholar
İ. ̇İşcan, “Hermite-Hadamard type inequalities for harmonically convex functions,” Hacettepe Journal of Mathematics, vol. 43, no. 6, pp. 935–942, 2014.
View at: Publisher Site | Google Scholar
M. A. Noor, K. I. Noor, and S. Iftikhar, “Integral inequalities for diffrential p–harmonic convex function,” Filomet, vol. 31, no. 20, pp. 6575–6584, 2017.
View at: Google Scholar
M. A. Noor, K. I. Noor, M. U. Awan, and S. Costache, “Some integral inequalities for harmonically h-convex functions,” UPB Scientific Bulletin, Series A, vol. 77, no. 1, pp. 5–16, 2015.
View at: Google Scholar
M. A. Noor, K. I. Noor, and S. Iftikhar, “Newton inequalities for p-harmonic convex function,” Honam Mathematical Journal, vol. 40, no. 2, pp. 239–250, 2018.
View at: Google Scholar
R. J. Dalpatadu, The Arithmetic–Geometric–Harmonic Mean, JSM Math. Stat., 2014.
J. H. He, “He Chengtian's inequality and its applications,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 887–891, 2004.
View at: Google Scholar
J. H. He, “Max-min approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 207–210, 2008.
View at: Publisher Site | Google Scholar
J. H. He, “An improved amplitude-frequency formulation for nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 211-212, 2008.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Saad Ihsan Butt et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Journal of Function Spaces

Developments in Geometric Function Theory

Hermite–Hadamard Type Inequalities via Generalized Harmonic Exponential Convexity and Applications

Abstract

1. Introduction

2. Preliminaries

3. Generalized Exponential Type Convex Functions and Its Properties

4. (H–H) Type Inequality via Generalized Exponential Type Convexity

5. Refinements of (H–H) Type Inequality via Generalized Exponential Type Convexity

6. Applications

7. Conclusion

Data Availability

Conflicts of Interest

Acknowledgments

References

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