Advancements in Convex Analysis Through Inverse Cosine Function with Applications

Document Type : Research Paper

Authors

Department of Mathematics, University of Sargodha P.O. Box 40100, Sargodha, Pakistan.

Abstract
In this article, we introduce a new class of convex functions called $\alpha$-inverse cosine convex functions ($\alpha$-ICCF), which extends the traditional classes. We analyze various algebraic and geometric properties by illustrating the graphs of several significant $\alpha$-ICCF via visual representations. Utilizing this novel class, we derive the Hermite-Hadamard (HH) inequality and certain refinements for functions whose first derivative in absolute value is $\alpha$-ICCF. The primary tools employed in deriving the main results include Hölder's inequality, Hölder-Iscan inequality and power-mean integral inequality. Our findings demonstrate that the approximations obtained using Hölder-Iscan and the improved power-mean integral inequality are superior to those derived from other methods. In particular, when $\alpha=1$, the derived results will coincide with those of classical ICCF. This innovative concept of $\alpha$-inverse cosine convexity opens new avenues for research, encouraging further exploration of such convexity classes.

Keywords

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1. M. Agueh, Sharp Gagliardo-Nirenberg inequalities and mass transport theory, J. Dynam. Differ. Equ., 18 (4) (2006), pp. 1069-1093.
2. M.U. Awan, M.A. Noor and K.I. Noor, Hermite-Hadamard inequalities for exponentially convex functions, Appl. Math. Inf. Sci., 12 (2) (2018), pp. 405-409.
3. A. Bakht and M. Anwar, Hermite-Hadamard and Ostrowski type inequalities via α-exponential type convex functions with applications, AIMS Math., 9 (4) (2024), pp. 9519-9535.
4. S.P. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, 2004.
5. S.S. Dragomir, An Ostrowski like inequality for convex functions and applications, Rev. Mat. Complut., 16 (2) (2003), pp. 373-382.
6. S.S. Dragomir, Refinements of the Hermite-Hadamard integral inequality for log-convex functions, RGMIA Res. Rep. Coll., 3 (2) (2000), pp. 98-115.
7. S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, App. Math. Lett., 11 (5) (1998), pp. 91-95.
8. M.J. Farrell, The convexity assumption in the theory of competitive markets, J. Pol. Econ., 67 (4) (1959), pp. 377-391.
9. İ. İşcan, New refinements for integral and sum forms of Hölder inequality, J. Inequal. Appl., 2019 (1) (2019), pp. 1-11.
10. J. A. Jiddah, M.S. Shagari, M. Noorwali, A. Aloqaily and N. Mlaiki, Hybrid fixed point theorems of graphic contractions with applications, Heliyon., 10 (10) (2024), pp. 1-13.
11. H.U. Kadakal, Hermite-Hadamard type inequalities for trigonometrically convex functions, Sci. Stud. Res. Ser. Math. Inform., 28 (2) (2018), pp. 19-28.
12. H. Kadakal, Harmonic trigonometrically convexity, Filomat., 37 (23) (2023), pp. 8029-8038.
13. M. Kadakal, İ.İşcan, P. Agarwal and M. Jleli, Exponential trigonometric convex functions and Hermite-Hadamard type inequalities, Math. Slovaca., 71 (1) (2021), pp. 43-56.
14. M. Kadakal, İ.İşcan, H. Kadakal and K. Bekar, On improvements of some integral inequalities, Honam Math. J., 43 (3) (2021), pp. 441-452.
15. M.B. Khan, P.O. Mohammed, M.A. Noor and Y.S. Hamed, New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities, Symmetry., 13 (4) (2021).
16. A.J. Kurdila and M. Zabarankin, Convex Functional Analysis, Springer Science & Business Media, New York, (2006).
17. L.H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc., 55 (1949), pp. 961-962.
18. A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities: Theory of Majorization and its Applications, Springer, New York, (2011).
19. M.A. Noor and K.I. Noor., On exponentially convex functions, J. Orisa. Math. Soc., 38 (1) (2019), pp. 33-51.
20. J.A. Oguntuase, L-E. Persson and A. Cižmešija, Multidimensional Hardy-type inequalities via convexity, Bull. Aust. Math. Soc., 77 (2) (2008), pp. 245-260.
21. S. Özcan, M. Kadakal, I. Iscan and H. Kadakal, Generalized strongly n-polynomial convex functions and related inequalities, Bound. Value Probl., 2024 (1) (2024), pp. 1-24.
22. J.E. Pecaric, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, (1992).
23. T. Rasham, A. Mustafa, A. Mukheimer, M. Nazam and W. ShatanawiNovel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications, Demonstr. Math., 57 (1) (2024), 20230161.
24. A. Salim, C. Derbazi, J. Alzabut and A. Küçükaslan, Existence and κ-Mittag-Leffler-Ulam-Hyers stability results for implicit coupled (κ, ϑ)-fractional differential systems, Arab J. Basic Appl. Sci., 31 (1) (2024), pp. 225-241.
25. M. Samraiz, A. Imran and S. Naheed, Inverse cosine convex functions: algebraic, geometric and analytic perspetives, Submitted, (2024).
26. M. Samraiz, K. Saeed, S. Naheed, G. Rahman and K. Nonlaopon, On inequalities of Hermite-Hadamard type via n-polynomial exponential type s-convex functions, AIMS Math., 7 (8) (2022), pp. 14282-14298.
27. M. Samraiz, T. Atta, S. Naheed, T. Abdeljawad and M. T. Ghaffar, A novel class of integral inequalities with graphical approach and diverse applications, Math. Comput. Model. Dyn. Syst., 30 (1) (2024), pp. 156-178.
28. G. Scutari, D. Palomar, F. Facchinei and J. Pang, Convex optimization, game theory and variational inequality theory, IEEE Signal Process. Mag., 27 (3) (2010), pp. 35-49.
29. T. Sears, Generalized Maximum Entropy, Convexity and Machine Learning, (2010).
30. S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326 (1) (2007), pp. 303-311.
31. M. Vivas-Cortez, M. Samraiz, M. T. Ghaffar, S. Naheed, G. Rahman and Y. Elmasry, Exploration of Hermite-Hadamard-type integral inequalities for twice differentiable h-convex functions, Fractal and Fract., 7 (7) (2023).
32. G. Zabandan, A new refinement of the Hermite-Hadamard inequality for convex functions, J. Inequal. Pure Appl. Math., 10 (2) (2009).