Journal of Applied Science and Engineering

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Saba Naser MajeedThis email address is being protected from spambots. You need JavaScript enabled to view it.

Department of Mathematics, College of Education for Pure Science Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq


 

Received: January 11, 2024
Accepted: April 9, 2024
Publication Date: May 28, 2024

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.


Download Citation: ||https://doi.org/10.6180/jase.202504_28(4).0005  


In this paper, a new class of generalized convex function called the class of exponentially E-preinvex functions is introduced by combining the classes of exponentially E-convex functions and exponentially preinvex functions. Some important properties and relations are proved, especially those which relate exponentially E-preinvex functions with different γ-level sets and different epigraphs associated with these functions. Also, some optimality properties for nonlinear optimization problems involving exponentially E-preinvex functions are established.


Keywords: E-invex set, preinvex function, exponentially preinvex function, exponentially E-preinvex function.


  1. [1] R. T. Rockafellar. Convex analysis. 18. Princeton uni versity press, 1970.
  2. [2] I. Ekeland and R. Temam. Convex analysis and vari ational problems. Society for Industrial and Applied Mathematics, 1999.
  3. [3] C.Zalinescu. Convex analysis in general vector spaces. World scientific, 2002.
  4. [4] S.N.Majeed,(2023)“StrongFenchelDualityForEvenly Convex Optimization Problems" Journal of Applied Science and Engineering 26(12): 1723–1729. DOI: 10.6180/jase.202312_26(12).0004.
  5. [5] C.Niculescu and L.-E. Persson. Convex functions and their applications. 23. Springer, 2006.
  6. [6] G.Cristescu and L. Lupsa. Non-connected convexities and applications. 68. Springer Science & Business Me dia, 2013.
  7. [7] M. A. Hanson, (1981) “On sufficiency of the KuhnTucker conditions" Journal of Mathematical Analysis and Applications 80(2): 545–550. DOI: 10.1016/0022-247X(81)90123-2.
  8. [8] R.Nehse,(1979) “Problems in connection with the Hahn Banach-Theorem" Proc. of the Summer-school" Non linear Analysis", Berlin:
  9. [9] B. Craven, (1981) “Invex functions and constrained local minima" Bulletin of the Australian Math ematical society 24(3): 357–366. DOI: 10.1017/S0004972700004895.
  10. [10] B. Craven. Duality for generalized convex fractional pro grams. Academic Press New York, 1981, 473–489.
  11. [11] A. Ben-Israel and B. Mond, (1986) “What is invex ity?" The ANZIAM Journal 28(1): 1–9. DOI: 10.1017/S0334270000005142.
  12. [12] M.HansonandB.Mond,(1987)“Convextransformable programming problems and invexity" Journal of Infor mation and Optimization Sciences 8(2): 201–207. DOI: 10.1080/02522667.1987.10698886.
  13. [13] T. Weir and B. Mond, (1988) “Pre-invex functions in multiple objective optimization" Journal of Mathemat ical Analysis and applications 136(1): 29–38. DOI: 10.1016/0022-247X(88)90113-8.
  14. [14] S. Mohan and S. Neogy, (1995) “On invex sets and preinvex functions" Journal of Mathematical Analy sis and Applications 189(3): 901–908. DOI: 10.1006/jmaa.1995.1057.
  15. [15] M.A.NoorandK.I. Noor, (2006) “Some characteriza tions of strongly preinvex functions" Journal of Mathe matical Analysis and Applications 316(2): 697–706. DOI: 10.1016/j.jmaa.2005.05.014.
  16. [16] M.A.Noor, (2007) “On Hadamard integral inequalities involving two log-preinvex functions" J. Inequal. Pure Appl. Math 8(3): 1–14.
  17. [17] S.-l. Chen, N.-J. Huang, D. O’Regan, et al., (2014) “Geodesic B-preinvex functions and multiobjective opti mization problems on Riemannian manifolds" Journal of Applied Mathematics 2014: DOI: 10.1155/2014/ 524698.
  18. [18] S. N. Majeed, (2019) “Fuzzy preinvexity via ranking value functions with applications to fuzzy optimization problems" Journal of Interdisciplinary Mathemat ics 22(8): 1485–1494. DOI: 10.1080/09720502.2019.1706846.
  19. [19] M. B. Khan, M. A. Noor, L. Abdullah, and Y.-M. Chu,(2021)“Somenewclassesofpreinvex fuzzy-interval valued functions and inequalities" International Jour nal of Computational Intelligence Systems 14(1): 1403–1418. DOI: 10.2991/ijcis.d.210409.001.
  20. [20] M.Khan,G.Santos-García,M.Noor,andM.Soliman, (2022) “New class of preinvex fuzzy mappings and related inequalities" Mathematics 10: 3753. DOI: 10.3390/math10203753.
  21. [21] A. A. Shaikh, A. Iqbal, and C. K. Mondal, (2018) “Some results on φ-convex functions and geodesic φ convex functions" Differential Geometry- Dynamical System 20: 159–169.
  22. [22] A. A. Shaikh, C. K. Mondal, and I. Ahmad, (2019) “Non-existence of certain type of convex functions on a Riemannian manifold with a pole" Journal of Geometry and Physics 140: 104–110. DOI: 10.1016/j.geomphys.2019.02.006.
  23. [23] A.A.Shaikh, R. P. Agarwal, and C. K. Mondal, (2020) “Geodesic sandwich theorem with an application" Mathe matical Inequalities and Applications 23: 161–167. DOI: 10.7153/mia-2020-23-13.
  24. [24] E. Youness, (1999) “E-convex sets, E-convex functions, and E-convex programming" Journal of Optimization Theory and Applications 102: 439–450. DOI: 10.1023/A:1021792726715.
  25. [25] E. A. Youness et al., (2001) “Stability in E-convex pro gramming" International Journal of Mathematics and Mathematical Sciences 26: 643–648. DOI: 10.1155/S0161171201006317.
  26. [26] E. A. Youness, (2001) “Optimality criteria in E-convex programming" Chaos, Solitons & Fractals 12(9): 1737 1745. DOI: 10.1016/S0960-0779(00)00036-9.
  27. [27] A. Megahed, H. Gomma, E. Youness, and A. El Banna, (2013) “A study on the duality of E-convex non linear programming problem" International Journal of Mathematical Analysis 7(4): 175–185. DOI: 10.12988/ijma.2013.13015.
  28. [28] A. E.-M. A. Megahed, H. G. Gomma, E. A. Youness, andA.-Z.H.El-Banna,(2013)“Optimality conditions of E-convex programming for an E-differentiable function" Journal of Inequalities and Applications 2013: 1–11. DOI: 10.1186/1029-242X-2013-246.
  29. [29] X. Chen, (2002) “Some properties of semi-E-convex func tions" Journal of Mathematical Analysis and Appli cations 275(1): 251–262. DOI: 10.1016/S0022-247X(02)00325-6.
  30. [30] M. Soleimani-Damaneh, (2011) “E-convexity and its generalizations" International Journal of Computer Mathematics 88(16): 3335–3349. DOI: 10.1080/00207160.2011.589899.
  31. [31] N. Abdulaleem. “E-invexity and generalized E invexity in E-differentiable multiobjective program ming”. In: ITM Web of Conferences. 24. EDP Sciences. 2019, 01002. DOI: 10.1051/itmconf/20192401002.
  32. [32] S. Majeed and A. Enad. “On semi strongly (E, F) convex functions and semi strongly (E, F)-convex optimization problems”. In: Journal of Physics: Confer ence Series. 1879. 2. IOP Publishing. 2021, 022110. DOI: 10.1088/1742-6596/1879/2/022110.
  33. [33] W. Saleh, (2022) “Hermite–Hadamard type inequality for (E, F)-convex functions and geodesic (E, F)-convex functions" RAIRO-Operations Research 56(6): 4181–4189. DOI: 10.1051/ro/2022185.
  34. [34] M. Elbrolosy, (2022) “Semi-(E, F)-convexity in com plex programming problems" AIMS Mathematics 7(6): 11119–11131. DOI: 10.3934/math.2022621.
  35. [35] R.E. HazimandS.N.Majeed, (2023) “Some properties of semi (p, e)-convexity" Journal of Interdisciplinary Mathematics 26(4): 629–635. DOI: 10.47974/JIM 1481.
  36. [36] N.Abdulaleem, (2023) “Optimality and duality for E differentiable multiobjective programming problems in volving E-type functions" Journal of Industrial and Management Optimization 19(2): 1513–1527. DOI: 10.3934/jimo.2022004.
  37. [37] N.Abdulaleem, (2023) “E-univex sets, E-univex func tions and E-differentiable E-univex programming" Re sults in Mathematics 78(1): 3. DOI: 10.1007/s00025-022-01775-5.
  38. [38] N. Abdulaleem, (2023) “Optimality conditions for a class of E-differentiable vector optimization problems with interval-valued objective functions under E-invexity" International Journal of Computer Mathematics 100(7): 1601–1624. DOI: 10.1080/00207160.2023.2207389.
  39. [39] C.Fulga and V. Preda, (2009) “Nonlinear programming with E-preinvex and local E-preinvex functions" Euro pean Journal of Operational Research 192(3): 737 743. DOI: 10.1016/j.ejor.2007.11.056.
  40. [40] Z.Luoetal., (2011) “Some properties of semi-E-preinvex maps in Banach spaces" Nonlinear Analysis: Real World Applications 12(2): 1243–1249. DOI: 10.1016/j.nonrwa.2010.09.019.
  41. [41] S. N. Majeed, (2019) “Strongly and Semi Strongly E_h b-Vex Functions: Applications to Optimization Problems" Iraqi Journal of Science: 2022–2029. DOI: 10.24996/ijs.2019.60.9.16.
  42. [42] N.Abdulaleem, (2021) “EB-invexity in E-differentiable mathematical programming" Results in Control and Optimization 4: 100046. DOI: 10.1016/j.rico.2021.100046.
  43. [43] A.A.Shaikh, C. K.Mondal, andR.P. Agarwal, (2021) “Some properties of geodesic (α, E)-preinvex functions on Riemannian manifolds" Khayyam Journal of Mathe matics 7(2): 201–210. DOI: 10.22034/KJM.2020.220226.1717.
  44. [44] A. Iqbal and P. Kumar, (2023) “Geodesic E-prequasi invex function and its applications to non-linear program ming problems" Numerical Algebra, Control and Op timization 13(1): 1–10. DOI: 10.3934/naco.2021040.
  45. [45] G. Alirezaei and R. Mathar. “On exponentially con cave functions and their impact in information the ory”. In: 2018 Information Theory and Applications Workshop (ITA). IEEE. 2018, 1–10. DOI: 10.1109/ITA.2018.8503202.
  46. [46] S.PalandT.-K.L.Wong,(2018)“Exponentially concave functions and a new information geometry" The Annals of probability 46(2): 1070–1113.
  47. [47] M.A.NoorandK.I.Noor, (2019) “On exponentially convex functions" JOURNAL OF ORISSA MATHE MATICALSOCIETY975:2323.
  48. [48] M.NoorandK.Noor, (2019) “Strongly exponentially convex functions and their properties" Journal of Ad vanced Mathematical Studies 12(2): 177–185.
  49. [49] M.NoorandK.Noor, (2019) “Some properties of differ entiable exponential convex functions" preprint, Math, Deparment, COMSATS university, Islamabad, Pak istan:
  50. [50] M. A. Noor and K. I. Noor, (2019) “New classes of strongly exponentially preinvex functions" AIMS Math 4(6): 1554–1568. DOI: 10.3934/math.2019.6.1554.
  51. [51] M. A. Noor, K. I. Noor, and T. M. Rassias, (2021) “Relative strongly exponentially convex functions" Non linear Analysis and Global Optimization: 357–371. DOI: 10.1007/97.
  52. [52] J. Bisht, N. Sharma, S. K. Mishra, and A. Hamdi, (2023) “Some new integral inequalities for higher-order strongly exponentially convex functions" Journal of In equalities and Applications 2023(1): 41. DOI: 10. 1186/s13660-023-02952-y.
  53. [53] M.A.NoorandK.I. Noor, (2019) “Some properties of exponentially preinvex functions" Facta Universitatis, Series: Mathematics and Informatics: 941–955. DOI: 10.22190/FUMI1905941N. [54] N. Abdulaleem, (2024) “Exponentially E-convex vec tor optimization problems" Journal of Industrial and Management Optimization: 0–0. DOI: 10.3934/jimo.2023164.