AccScience Publishing / IJOCTA / Volume 14 / Issue 3 / DOI: 10.11121/ijocta.1482
RESEARCH ARTICLE

Further refinements and inequalities of Fejer's type via GA-convexity

Muhammad Amer Latif1 H¨useyin Budak2 Artion Kashuri3*
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1 Basic Sciences Unit, Preparatory Year, King Faisal University, Hofuf 31982, Al-Hasa, Saudi Arabia
2 Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce 81620, T¨urkiye
3 Department of Mathematical Engineering, Polytechnic University of Tirana, Tirana 1001, Albania
IJOCTA 2024, 14(3), 229–248; https://doi.org/10.11121/ijocta.1482
Received: 3 November 2023 | Accepted: 28 April 2024 | Published online: 12 July 2024
© 2024 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract

In this study, we introduce some new mappings in connection with Hermite-Hadamard and Fejer type integral inequalities which have been proved using the GA-convex functions. As a consequence, we obtain certain new inequalities of the Fejer type that provide refinements of the Hermite-Hadamard and Fejer type integral inequalities that have already been obtained.

Keywords
Hermite-Hadamard inequality
convex function
GA-convex function
Fejer inequality
Conflict of interest
The authors declare they have no competing interests.
References

[1] Hermite, C. (1893). Sur deux limites d’une int´egrale d´e finie. Mathesis, 3, 82.

[2] Hadamard, J. (1893). E(´)tudesurles propri´et´es des fonctions enti´eres en particulier d’une function consid´er´e par Riemann. ournal de Math´ematiques Pures et Appliqu´ees, 9, 171–215.

[3] Fej´er, L. (1906). U(¨)ber die Fourierreihen, II. Math.Naturwiss. Anz Ungar. Akad. Wiss. , 24, 369–390.

[4] Ardic, M. A., Akdemir, A. O. & Set, E. (2016). New Ostrowski like inequalities for GG-convex and GA-convex functions. Mathematical Inequal- ities & Applications, 19(4), 1159–1168. https: //doi.org/10.7153/mia-19-85

[5] Ardic, M. A., Akdemir, A. O. & Yildiz, K. (2018). On some new inequalities via GG-convexity and GA-convexity. Filomat, 32(16), 5707–5717. https://doi.org/10.2298/FIL1816707A

[6] Dragomir, S. S., Latif, M. A. & Momoniat, E.(2019). Fej´er type integral inequalities related with geometrically-arithmetically convex func- tions with applications. Acta et Commentationes Universitatis Tartuensis de Mathematica, 23(1), 51–64. https://doi.org/10.12697/ACUTM.201 9.23.05

[7] Dragomir, S. S. (2018). Some new inequali- ties of Hermite-Hadamard type for GA-convex functions. Annales Universitatis Mariae Curie- Sklodowska Lublin-Polonia, 72(1), 55–68. https://doi.org/10.17951/a.2018.72.1.55-68

[8] Dragomir, S. S. (2018). Inequalities of Hermite- Hadamard type for GA-convex functions. Annales Mathematicae Silesianae, 32, 145–168. https: //doi.org/10.2478/amsil-2018-0001

[9] Dragomir, S. S. (2018). Inequalities of Jensen type for GA-convex functions. Nonlinear Functional Analysis and Applications, 23(2), 275-304.

[10] Dragomir, S. S., Cho, Y. J. & Kim, S. S.(2000). Inequalities of Hadamard’s type for Lips- chitzian mappings and their applications. Journal of Mathematical Analysis and Applications, 245, 489–501. https://doi.org/10.1006/jmaa.200 0.6769

[11] Dragomir, S. S., Milosevic, D. S. & Sandor, J. (1993). On some refinements of Hadamard’s inequalities and applications. Publikacije Elek- trotehniˇckog Fakulteta. Serija Matematika, 4, 3– 10.

[12] Dragomir, S. S. (1992). On Hadamard’s inequal- ity for convex functions. Mathematica Balkanica, 6, 215–222.

[13] Kashuri, A. & Liko, R. (2019). Some new Hermite-Hadamard type inequalities and their applications. Studia Scientiarum Mathemati- carum Hungarica, 56(1), 103–142. https://do i.org/10.1556/012.2019.56.1.1418

[14] Kashuri, A., Sahoo, S. K., Mohammed, P. O., Sarairah, E. A. & Hamed, Y. S. (2023). Some new Hermite-Hadamard type inequalities pertain- ing to fractional integrals with an exponential ker- nel for subadditive functions. Symmetry, 15, 748. https://doi.org/10.3390/sym15030748

[15] Dragomir, S. S. (2000). On Hadamard’s inequal- ity for the convex mappings defined on a ball in the space and applications. Mathematical In- equalities & Applications, 3(2), 177–187. https: //doi.org/10.7153/mia-03-21

[16] Dragomir, S. S. (1996). On some integral inequal- ities for convex functions. Zb.-Rad. (Kragujevac) , 18, 21–25.

[17] Dragomir, S. S. & Agarwal, R. P. (1998). Two new mappings associated with Hadamard’s inequal- ities for convex functions. Applied Mathematics Letters, 11(3), 33–38. https://doi.org/10.101 6/S0893-9659(98)00030-5

[18] Dragomir, S. S. (1992). Two mappings in connec- tion to Hadamard’s inequalities. Journal of Math- ematical Analysis and Applications, 167, 49–56. https://doi.org/10.1016/0022-247X(92)902 33-4

[19] Kunt, M. & I(˙)¸scan, I(˙) . (2018). Fractional Hermite-Hadamard-Fej´er type inequalities for GA-convex functions. Turkish Journal of Inequalities, 2, 1– 20.

[20] I(˙)¸scan, I(˙) . (2014). Hermite-Hadamard type inequal-ities for GA-s-convex functions. Le Matematiche, 19, 129–146.

[21] Latif, M. A., Kalsoom, H., Khan, Z. A., & Al-moneef, A. A. (2022). Refinement mappings related to Hermite-Hadamard type inequalities for GA-convex function. Mathematics, 10, 1398. https://doi.org/10.3390/math10091398

[22] Latif, M. A. (2014). New Hermite-Hadamard type integral inequalities for GA-convex functions with applications. Analysis, 34, 379–389. https://do i.org/10.1515/anly-2012-1235

[23] Latif, M. A., Dragomir, S. S. & Momo- niat, E. (2017). Some estimates on the Hermite-Hadamard inequality through ge- ometrically quasi-convex functions. Mis- colc Mathematical Notes, 18(2), 933–946. https://doi.org/10.18514/MMN.2017.1819

[24] Latif, M. A. (2015). Hermite-Hadamard type in- equalities for GA-convex functions on the co- ordinates with applications. Proceedings of the Pakistan Academy of Sciences, 52(4), 367–379.

[25] Latif, M. A., Dragomir, S. S. & Momoniat, E. (2018). Some Fej´er type integral inequalities for geometrically-arithmetically-convex functions with applications. Filomat, 32(6), 2193–2206. ht tps://doi.org/10.2298/FIL1806193L

[26] Latif, M. A. (2022). Weighted Hermite-Hadamard type inequalities for differentiable GA-convex and geometrically quasi-convex mappings. Rocky Mountain Journal of Mathematics, 51(6),1899–1908. https://doi.org/10.1216/rmj.2021.5 1.1899

[27] Latif, M. A. Fej´er type inequalities for GA-convex functions and related results. (Submitted)

[28] Latif, M. A. Fej´er type inequalities and GA- convex functions. (Submitted)

[29] Latif, M. A. Some companions of Fej´er type in- equalities using GA-convex functions. (Submit- ted)

[30] Niculescu, C. P. (2000). Convexity according to the geometric mean. Mathematical Inequalities and Applications, 3, 155–167. https://doi.or g/10.7153/mia-03-19

[31] Noor, M. A., Noor, K. I. & Awan, M. U. (2014). Some inequalities for geometrically-arithmetically h-convex functions. Creative Mathematics and In- formatics, 23(1), 91–98. https://doi.org/10.3 7193/CMI.2014.01.14

[32] Obeidat, S. & Latif, M. A. (2018). Weighted ver- sion of Hermite-Hadamard type inequalities for geometrically quasi-convex functions and their applications. Journal of Inequalities and Applica- tions, 2018, Article 307. https://doi.org/10.1 186/s13660-018-1904-7

[33] Qi, F. & Xi, B. Y. (2014). Some Hermite- Hadamard type inequalities for geometrically quasi-convex functions. Indian Academy of Sci- ences Proceedings - Mathematical Sciences, 124(3), 333–342. https://doi.org/10.1007/ s12044-014-0182-7

[34] Tseng, K. L., Hwang, S. R. & Dragomir, S. S.(2007). On some new inequalities of Hermite- Hadamard- Fej´er type involving convex functions. Demonstratio Mathematica, 40(1), 51–64. https: //doi.org/10.1515/dema-2007-0108

[35] Tseng, K. L., Hwang, S. R. & Dragomir, S. S.(2010). Fej´er-type inequalities (I). Journal of In- equalities and Applications, 2010, Article 531976. https://doi.org/10.1155/2010/531976

[36] Tseng, K. L., Hwang, S. R. & Dragomir, S. S.(2015). Some companions of Fej´er’s inequality for convex functions. Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas, 109, 645–656. https://doi.org/ 10.1007/s13398-014-0206-2

[37] Tseng, K. L., Hwang, S. R. & Dragomir, S. S.(2017). Fej´er-type Inequalities (II). Math. Slo- vaca, 67(1), 109–120. https://doi.org/10.1 515/ms-2016-0252

[38] Tseng, K. L., Hwang, S. R. & Dragomir, S. S.(2011). On some weighted integral inequalities for convex functions related Fej´er result. Filomat, 25(1), 195–218. https://doi.org/10.2298/FI L1101195T

[39] Tseng, K. L., Yang, G. S. & Hsu, K. C. (2009). On some inequalities for Hadamard’s type and applications. Taiwanese Journal of Mathematics, 13(6B), 1929–1948. https://doi.org/10.11650 /twjm/1500405649

[40] Xiang, R. (2015). Refinements of Hermite- Hadamard type inequalities for convex functions via fractional integrals. Journal of Applied Math- ematics & Informatics, 33(1-2), 119–125. https://doi.org/10.14317/jami.2015.119

[41] Yang, G. S. & Hong, M. C. (1997). A note on Hadamard’s inequality. Tamkang Journal of Mathematics, 28(1), 33–37. https://doi.org/ 10.5556/j.tkjm.28.1997.4331

[42] Yang, G. S. & Tseng, K. L. (1999). On certain in- tegral inequalities related to Hermite-Hadamard inequalities. Journal of Mathematical Analysis and Applications, 239, 180–187. https://doi. org/10.1006/jmaa.1999.6506

[43] Yang, G. S. & Tseng, K. L. (2001). Inequalities of Hadamard’s type for Lipschitzian mappings. Journal of Mathematical Analysis and Applica- tions, 260, 230–238. https://doi.org/10.100 6/jmaa.2000.7460

[44] Yang, G. S. & Tseng, K. L. (2002). On certain multiple integral inequalities related to Hermite- Hadamard inequalities. Utilitas Mathematica, 62, 131–142.

[45] Yang, G. S. & Tseng, K. L. (2003). Inequalities of Hermite-Hadamard-Fej´er type for convex func- tions and Lipschitzian functions. Taiwanese Jour- nal of Mathematics, 7(3), 433–440.

[46] Zhang, X. M., Chu, Y. M. & Zhang, X. H.(2010). The Hermite-Hadamard type inequality of GA-convex functions and its application. Jour- nal of Inequalities and Applications, 2010, Article 507560. https://doi.org/10.1155/2010/507560

[47] Kashuri, A. & Liko, R. (2020). Fractional trapez-ium type inequalities for twice differentiable preinvex functions and their applications. An In-ternational Journal of Optimization and Control:Theories & Applications (IJOCTA), 10(2), 226-236. https://doi.org/10.11121/ijocta.01.2 020.00795

[48] Kadakal, M. (2020). Some Hermite-Hadamard type inequalities for(P, m)-function and quasi m- convex functions. An International Journal of Optimization and Control: Theories & Applica- tions (IJOCTA), 10(1), 78–84. https://doi.or g/10.11121/ijocta.01.2020.00787

[49] Okur, N., I(´)scan, I(´). & Dizdar, E. Y. (2019).Hermite-Hadamard type inequalities for p-convex stochastic processes. An International Journal of Optimization and Control: Theories & Applica- tions (IJOCTA), 9(2), 148–153. https://doi.or g/10.11121/ijocta.01.2019.00602

[50] Kadakal, H. (2019). Some integral inequalities for multiplicatively geometrically P-functions. An International Journal of Optimization and Con- trol: Theories & Applications (IJOCTA) , 9(2), 216–222. https://doi.org/10.11121/ijoct a.01.2019.00738

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