Geometric Interpretation and Some Analytical Properties of First Type S-Convex Function
Abstract
There are at least two types of s-convex functions. This article will only discuss the properties related to the first type of s-convex function, consist of the geometric meaning, monotonicity, continuity, inclusion properties, and Jensen's inequality. The novelties of this research are the geometric meaning of first type s-convex function, the continuity at 0, inclusion properties of two s-convex function classes on an interval, and necessary and sufficient condition for Jensen's type equality. All the properties will be proved analitically.
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References
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I. M. R. Pinheiro, {\em Second note on the definition of $S_1$-convexity} , Adv. Pure Math., vol. 05, no. 03, pp. 127–130, (2015).
M. R. Pinheiro, {\em Exploring the concept of s-convexity}, Aequationes mathematicae, vol. 74, pp. 201-209, 12, (2007).
Y. P. D. Ole, S. A. Hazmy, and A. A. Masta, {\em Geometric interpretation of s-convex function on real numbers}, in IConMAA 2022: Analysis, Uncertainty, and Optimization, AIP Publishing, (2024).
Y. Sayyari, H. Barsam, and L. Ciurdariu, {\em A new refinement of jensen-type inequality with respect to uniformly convex functions with applications in information theory}, Journal of Mathematical Inequalities, pp. 1311-1322, (2023).
Y. Sayyari and H. Barsam, {\em Jensen-mercer inequality for uniformly convex functions with some applications}, Afrika Matematika, vol. 34, p. 38, 9,(2023).
F. P. Mohebbi, M. Hassani, M. E. Omidvar, H. R. Moradi, and S. Furuichi, {\em Further jensen–mercer’s type inequalities for convex functions}, Journal of Mathematical Inequalities, pp. 719–737, (2024).
H. R. Moradi, M. E. Omidvar, M. A. Khan, and K. Nikodem, “Around jensen’s inequality for strongly convex functions,” Aequationes mathematicae, vol. 92, pp. 25–37, 2, (2018).
I. Nikoufar and D. Saeedi, {\em An operator version of the jensen inequality for s-convex functions}, Complex Analysis and Operator Theory, vol. 15, p. 92, 7, (2021).
M. A. Khan, S. I. Bradanovi´c, and H. A. Mahmoud, {\em New improvements of the jensen–mercer inequality for strongly convex functions with applications}, Axioms, vol. 13, p. 553, 8, (2024).
S. Simic and B. Almohsen, {\em Some generalizations of jensen’s inequality}, Contemporary Mathematics, 12, (2020).
M. A. Khan, M. Hanif, Z. A. H. Khan, K. Ahmad, and Y.-M. Chu, {\em Association of jensen’s inequality for s-convex function with csisz´ar divergence}, Journal of Inequalities and Applications, vol. 2019, p. 162, 12, (2019).
I. M. R. Pinheiro, {\em Jensen’s inequality in detail and s-convex functions}, International Journal of Mathematical Analysis, vol. 3, pp. 95–98, (2009).
R. K. Raina, P. Sharma, and J. Sokol, {\em A class of strongly close-to-convex functions}, Bol. Soc. Parana. Mat. (3), vol. 38, pp. 9–24, May (2019).
Dasep, S. Fatimah, C. Kustiawan, E. Sumiaty, and A. A. Masta, {\em An inclusion properties of generalized orlicz sequence spaces}, in IConMAA 2022:: Analysis, Uncertainty, and Optimization, AIP Publishing, (2024).
R. Dermawan, S. Fatimah, S. A. Hazmy, A. A. Masta, and C. Kustiawan, {\em Generalization of young’s function}, in AIP Conference Proceedings, vol. 2734, p. 080005, AIP Publishing, (2023).
R. Dermawan, A. A. Masta, E. Sumiaty, S. Fatimah, C. Kustiawan, and S. A. Hazmy, {\em An inclusion properties of generalized orlicz space}, in IConMAA 2022: Analysis, Uncertainty, and Optimization, AIP Publishing, (2024).
S. Fatimah, R. Dermawan, S. A. Hazmy, and A. A. Masta, {\em Generalized orlicz spaces}, in AIP Conference Proceedings, vol. 2734, p. 080004, AIP Publishing, (2023).
C. Kustiawan, A. A. Masta, D. Dasep, E. Sumiaty, S. Fatimah, and S. A. Hazmy, {\em Generalized orlicz sequence spaces}, Barekeng J. Ilmu Mat. dan Terap., vol. 17, pp. 0427–0438, Apr. (2023).
A. A. Masta, H. Gunawan, and W. Setya-Budhi, {\em On inclusion properties of two versions of orlicz–morrey spaces}, Mediterranean Journal of Mathematics, vol. 14, p. 228, 12, (2017).
S. Fatimah, S. A. Hazmy, and A. A. Masta, “Necessary condition for boundedness of stein-weiss operator on orlicz spaces, {\em Journal of Engineering Science and Technology}, vol. 16, pp. 3792–3800, (2021).
S. Fatimah, A. A. Masta, S. A. Hazmy, C. Kustiawan, and I. Rukmana, {\em Discrete orlicz-morrey spaces and their inclusion properties}, Journal of Engineering Science and Technology, vol. 16, pp. 2018–2027, (2021).
S. A. Hazmy, S. Fatimah, and M. A. A. Masta, {\em Some notes about boundedness of fractional integral on orlicz spaces}, in AIP Conference Proceedings, vol. 2734, p. 080002, AIP Publishing, 2023.l. 16, pp. 2018–2027, (2021).
Published
2025-08-13
Section
Advances in Nonlinear Analysis and Applications
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