Research Article | Open Access

Hao Li, Muhammad Shoaib Saleem, Ijaz Hussain, Muhammad Imran, "Strongly Reciprocally -Convex Functions and Some Inequalities", *Journal of Mathematics*, vol. 2020, Article ID 4957141, 9 pages, 2020. https://doi.org/10.1155/2020/4957141

# Strongly Reciprocally -Convex Functions and Some Inequalities

**Academic Editor:**Nan-Jing Huang

#### Abstract

In this paper, we generalize the concept of strong and reciprocal convexity. Some basic properties and results will be presented for the new class of strongly reciprocally -convex functions. Furthermore, we will discuss the Hermite–Hadamard-type, Jensen-type, and Fejér-type inequalities for the strongly reciprocally -convex functions.

#### 1. Introduction

The importance of convex functions and convex sets cannot be ignored, especially in nonlinear programing [1–5] and optimization theory [6], see, for instance, [7–14]. Generalization in the convexity is always appreciable. Also, many generalizations and extensions have been made in the theory of inequalities as well as in convexity. Several inequalities have been studied and established for the convexity of functions, and many generalizations, applications, and refinements take place, see [7, 9, 13, 15–18], for further study.

In the theory of inequalities, the famous inequality, Hermite–Hadamard inequality was established by Jaques Hadamard [19]. If is a convex function, thenholds for all with .

In [10], Lipot Fejér established the weighted version of the Hermite–Hadamard inequality.

If is a convex function, then the inequalityholds for all with and is integrable, nonnegative, and symmetric about .

For more details on the Fejér inequality, see [8, 9, 11, 20–22]. The main motivation of this article is based on [18].

Mathematically, Jensen-type inequality is stated as if is a convex function defined on , thenholds for all , and with .

This inequality has applications in probability and statistics.

The article is organized as follows: Section 2 is devoted to preliminaries and basic results, whereas in the last section, we will develop the main results for strongly reciprocally -convex functions.

#### 2. Preliminaries

This section concerns preliminaries and basic results for the strongly reciprocally -convex functions.

*Definition 1 (-convex set; see [23]). *An interval is called the -convex set if for all and , where or , , , and .

*Definition 2 (-convex function; see [24]). *A function is called -convex function iffor all and , where is the -convex set.

*Definition 3 (strongly convex function; see [14]). *Let be a positive number. A function is called a strongly convex function iffor all and .

*Definition 4 (strongly -convex function; see [25]). *Let be a positive number. A function is called strongly -convex function iffor all and .

*Definition 5 (harmonic convex function; see [22]). *Let be an interval. A function is harmonic convex iffor all and .

*Definition 6 (harmonic -convex function; see [26]). *A function is called a harmonic -convex function iffor all and .

*Definition 7. *(strongly reciprocally convex function; see [18]). Let and . A function is said to be strongly reciprocally convex with modulus on if the inequalityholds for all and .

Now, we are ready to introduce a new class of convexity named as strongly reciprocally -convex function.

*Definition 8 (strongly reciprocally -convex function). *A function is called strongly reciprocally -convex with modulus on if the inequalityholds, for all and .

*Remark 1. *(1)If we insert in inequality (10), then we retrace the strong and reciprocal convexity [18](2)If we insert in inequality (10), then we retrace the harmonic -convexity [26](3)If we insert and in inequality (10), then we retrace the harmonic convexity [22]The following proposition expresses the algebraic property of strongly reciprocally -convex functions.

Proposition 1. *Let be two strongly reciprocally -convex functions; then, the following statements hold:*(i)* is strongly reciprocally -convex*(ii)*For any , is strongly reciprocally -convex corresponding to *

*Proof. *(i)Choose ; then, by the definition of and , we obtain where .(ii)Let ; then, by definition, we obtainwhere and .

The next lemma establishes the connection between the strong and reciprocal -convexity and harmonic -convexity.

Lemma 1. *Let be a function; is strongly reciprocally -convex iff the function , defined by , is harmonically -convex.*

*Proof. *Let be strongly reciprocally -convex; then, we haveThis shows that is a harmonic -convex function.

Conversely, if is harmonically -convex, thenThis implies that is a strongly reciprocally -convex function for all and .

#### 3. Main Results

In this section, Hermite–Hadamard-, Fejér-, and Jensen-type inequalities are investigated. The next theorem gives the generalization of the Hermite–Hadamard inequality for strongly reciprocally -convex functions.

Theorem 1. *(Hermite–Hadamard-type inequality). Let be an interval on the real line. If is a strongly reciprocally -convex function with modulus and , thenfor all with .*

*Proof. *We start by the definition; set in inequality (10), and we haveLet and , and by integrating w.r.t over [0, 1], the above inequality yieldsand then inequality (17) is reduced towhich is the left side of the inequality.

For the right side of inequality (15), set and in (10); we haveIntegrating w.r.t over [0, 1], the above inequality yieldsSincethen we obtainFrom (18) and (22), we get (15).

*Remark 2. *(1)For in (15), Hermite–Hadamard inequality for strongly reciprocally convex functions is obtained [18].(2)If we allow in inequalities (15), we obtain the Hermite–Hadamard-type inequalities for harmonically convex functions [22].For further details on Hermite–Hadamard inequities, see [27–30].

Theorem 2. *(Fejér-type inequality). Assume is a strongly reciprocally -convex function with modulus on ; then,holds for with and where is a nonnegative integrable function that satisfies*

*Proof. *Since is a strongly reciprocally -convex function, then by definition for in (10), we havefor all ; suppose and in the above inequality; then, we obtainSince is nonnegative and symmetric, we haveThe above inequality is integrated with respect to over [0, 1], and then putting , we obtainAfter simplification, the above inequality becomesFor the right-hand side of (23), set and in (10); we haveIntegrating with respect to over [0, 1] and then putting , we obtainAfter simplification, we haveFrom (32) and (27), we get (23).

*Remark 3. *If we set in (23), the Fejér-type inequality for strongly reciprocally convex functions is obtained.

Jensen-type inequality for the aforementioned inequality is described in the next theorem.

Theorem 3. *(Jensen-type inequality). If is a reciprocally strongly p-convex function with modulus , thenholds for all , with and .*

*Proof. *Fix and such that .

Put , and suppose a function of the formsupporting at , satisfying and . Then, for every , we haveMultiplying both sides by and summing up to *n*, we haveSince , we havewhich completes the proof.

*Remark 4. *In inequality (34), fixing and yields the Jensen-type inequality for the harmonic convex function [22]. See [31–34] for more details on Jensen-type inequalities.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Authors’ Contributions

Hao Li analyzed all results and proofread and revised the paper, Muhammad Shoaib Saleem proposed the problem and supervised the work, Ijaz Hussain proved the results, and Muhammad Imran wrote the whole paper.

#### Acknowledgments

This research was supported by the Higher Education Commission of Pakistan.

#### References

- G. Farid, “A unified integral operator and further its consequences,”
*Open Journal of Mathematical Analysis*, vol. 4, no. 1, pp. 1–7, 2020. View at: Publisher Site | Google Scholar - S. Mehmood, G. Farid, and G. Farid, “Fractional integrals inequalities for exponentially \(m\)-convex functions,”
*Open Journal of Mathematical Sciences*, vol. 4, no. 1, pp. 78–85, 2020. View at: Publisher Site | Google Scholar - S. Mehmood, G. Farid, G. Farid, K. A. Khan, and M. Yussouf, “New Hadamard and Fejér-Hadamard fractional inequalities for exponentially m-convex function,”
*Engineering and Applied Science Letters*, vol. 3, no. 1, pp. 45–55, 2020. View at: Publisher Site | Google Scholar - S. Mehmood, G. Farid, K. A. Khan, and M. Yussouf, “New fractional Hadamard and Fejr-Hadamard inequalities associated with exponentially (h, m)-convex functions,”
*Engineering and Applied Science Letters*, vol. 3, pp. 9–18, 2020. View at: Google Scholar - M. A. Khan, S. Z. Ullah, and Y. M. Chu, “The concept of coordinate strongly convex functions and related inequalities,”
*Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matematicas*, vol. 113, no. 3, pp. 2235–2251, 2019. View at: Google Scholar - M. A. Khan, S. Z. Ullah, and Y. Chu, “Generalization of favards and berwalds inequalities for strongly convex functions,”
*Communications in Mathematics and Applications*, vol. 10, no. 4, pp. 693–705, 2019. View at: Publisher Site | Google Scholar - S. M. Kang, G. Farid, W. Nazeer, and B. Tariq, “Hadamard and FejrHadamard inequalities for extended generalized fractional integrals involving special functions,”
*Journal of Inequalities and Applications*, vol. 2018, no. 1, p. 119, 2018. View at: Google Scholar - S. S. Dragomir, “Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces,”
*Proyecciones (Antofagasta)*, vol. 34, no. 4, pp. 323–341, 2015. View at: Publisher Site | Google Scholar - Y. C. Kwun, G. Farid, S. Ullah, W. Nazeer, K. Mahreen, and S. M. Kang, “Inequalities for a unified integral operator and associated results in fractional calculus,”
*IEEE Access*, vol. 7, pp. 126283–126292, 2019. View at: Publisher Site | Google Scholar - L. Fejér, “Uberdie fourierreihen, II,”
*Math. Naturwise. Anz Ungar. Akad., Wiss*, vol. 24, pp. 369–390, 1906. View at: Google Scholar - Y. C. Kwun, M. S. Saleem, M. Ghafoor, W. Nazeer, and S. M. Kang, “HermiteHadamard-type inequalities for functions whose derivatives are ?-convex via fractional integrals,”
*Journal of Inequalities and Applications*, vol. 2019, no. 1, p. 44, 2019. View at: Google Scholar - G. H. Hardy, J. E. Littlewood, and G. Polya,
*Inequalities*, University Press, Cambridge, UK, 1934. - H. Bai, M. S. Saleem, W. Nazeer, M. S. Zahoor, and T. Zhao, “Hermite-Hadamard-and jensen-type inequalities for interval h1, h2 nonconvex function,”
*Journal of Mathematics*, vol. 2020, Article ID 3945384, 6 pages, 2020. View at: Publisher Site | Google Scholar - J. Zhao, S. I. Butt, J. Nasir, Z. Wang, I. Tlili, and Y. Liu, “Hermite-jensen-mercer type inequalities for caputo fractional derivatives,”
*Journal of Function Spaces*, vol. 2020, Article ID 7061549, 11 pages, 2020. View at: Publisher Site | Google Scholar - G. Hong, G. Farid, W. Nazeer et al., “Boundedness of fractional integral operators containing mittag-leffler function via exponentially-convex functions,”
*Journal of Mathematics*, vol. 2020, Article ID 3584105, 7 pages, 2020. View at: Publisher Site | Google Scholar - Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, and S. M. Kang, “Generalized riemann-liouville $k$ -fractional integrals associated with ostrowski type inequalities and error bounds of Hadamard inequalities,”
*IEEE Access*, vol. 6, pp. 64946–64953, 2018. View at: Publisher Site | Google Scholar - B. T. Polyak,
*Introduction to Optimization*, Optimization Software. Inc., Publications Division, New York, NY, USA, 1987. - M. A. Khan, F. Alam, and S. Z. Ullah, “Majorization type inequalities for strongly convex functions,”
*Turkish Journal of Inequalities*, vol. 3, no. 2, pp. 62–78, 2019. View at: Google Scholar - J. Hadamard, “Étude sur les propriétés des fonctions entiéres et en particulier d’une fonction considérée par Riemann,”
*Journal de Mathématiques Pures et Appliquées*, vol. 9, pp. 171–216, 1893. View at: Google Scholar - F. Chen and S. Wu, “Fejér and Hermite-Hadamard type inequalities for harmonically convex functions,”
*Journal of Applied Mathematics*, vol. 2014, Article ID 386806, 6 pages, 2014. View at: Publisher Site | Google Scholar - S. S. Dragomir and R. P. Agarwal, “Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula,”
*Applied Mathematics Letters*, vol. 11, no. 5, pp. 91–95, 1998. View at: Publisher Site | Google Scholar - S. Zaheer Ullah, M. A. Khan, Z. A. Khan, and Y. M. Chu, “Integral majorization type inequalities for the functions in the sense of strong convexity,”
*Journal of Function Spaces*, vol. 2019, Article ID 9487823, 11 pages, 2019. View at: Publisher Site | Google Scholar - A. Canino, “On p-convex sets and geodesics,”
*Journal of Differential Equations*, vol. 75, no. 1, pp. 118–157, 1988. View at: Publisher Site | Google Scholar - K. S. Zhang and J. P. Wan, “
*p*-Convex functions and their properties,”*Pure and Applied Mathematics*, vol. 23, no. 1, pp. 130–133, 2007. View at: Google Scholar - S. Maden, S. Turhan, and Í. Íşcan, “Hermite-Hadamard inequality for strongly
*p*-convex function,”*Turkish Journal of Mathematics and Computer Science*, vol. 10, pp. 184–189, 2018. View at: Google Scholar - I. Işcan, S. Numan, and K. Bekar, “Hermite-Hadamard and Simpson type inequalities for differentiable harmonically p-functions,”
*Journal of Advances in Mathematics and Computer Science*, vol. 4, no. 14, pp. 1908–1920, 2014. View at: Google Scholar - Y. Wu, F. Qi, and D. W. Niu, “Integral inequalities of Hermite-Hadamard type for the product of strongly logarithmically convex and other convex functions,”
*Maejo International Journal of Science and Technology*, vol. 9, no. 3, 2015. View at: Google Scholar - M. V. Cortez, “Relative strongly h-convex functions and integral inequalities,”
*Applied Mathematics & Information Sciences Letters*, vol. 4, no. 2, pp. 39–45, 2016. View at: Publisher Site | Google Scholar - M. A. Noor, K. I. Noor, M. A. Ashraf, M. U. Awan, and B. Bashir, “Hermite-Hadamard inequalities for h -convex functions,”
*Nonlinear Analysis Forum*, vol. 18, pp. 65–76, 2013. View at: Google Scholar - B. Y. Xi and F. Qi, “Some inequalities of Hermite-Hadamard type for h-convex functions,”
*Advances in Inequalities and Applications*, vol. 2, no. 1, pp. 1–15, 2013. View at: Google Scholar - A. Iqbal, M. A. Khan, S. Ullah, Y. M. Chu, and A. Kashuri, “Hermite-Hadamard type inequalities pertaining conformable fractional integrals and their applications,”
*AIP Advances*, vol. 8, no. 7, Article ID 75101, 2018. View at: Publisher Site | Google Scholar - M. A. Khan, A. Iqbal, M. Suleman, and Y. M. Chu, “HermiteHadamard type inequalities for fractional integrals via Greens function,”
*Journal of Inequalities and Applications*, vol. 2018, no. 1, p. 161, 2018. View at: Google Scholar - M. Adil Khan, M. Anwar, J. Jaksetic, and J. Pecaric, “On some improvements of the Jensen inequality with some applications,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 323615, pp. 1–15, 2009. View at: Publisher Site | Google Scholar - Y. Q. Song, M. Adil Khan, S. Zaheer Ullah, and Y. M. Chu, “Integral inequalities involving strongly convex functions,”
*Journal of Function Spaces*, vol. 2018, Article ID 6595921, 8 pages, 2018. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2020 Hao Li 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.