New developments in convex analysis: Harmonically trigonometric $ p $-coordinated convex functions and related inequalities

Sabila Ali; Muhammad Samraiz; Salma Trabelsi; Hajer Zaway; Sabila Ali; Muhammad Samraiz; Salma Trabelsi; Hajer Zaway

doi:10.3934/math.20251066

AIMS Mathematics

2025, Volume 10, Issue 10: 23984-24015. doi: 10.3934/math.20251066

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New developments in convex analysis: Harmonically trigonometric $ p $-coordinated convex functions and related inequalities

1.
Department of Mathematics, University of Sargodha P.O. Box 40100, Sargodha, Pakistan; Email: sabila21imran@gmail.com, muhammad.samraiz@uos.edu.pk, msamraizuos@gmail.com
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia; Email: satrabelsi@kfu.edu.sa

Received: 04 August 2025 Revised: 29 September 2025 Accepted: 14 October 2025 Published: 21 October 2025
MSC : 26A51, 26D15

In this paper, we introduce two novel classes of functions termed the harmonically trigonometric $ p $-convex functions on $ \Delta = [\tau, \upsilon]\times[\varphi, \chi] $ and harmonically trigonometric $ p $-coordinated convex functions. We discuss relations between these two newly introduced classes of convex functions in two variables and validate the theoretical findings with visual 3D graphs that illustrate the relation landscapes and transition regimes between the function classes. We then study several special or limiting cases (e.g. $ p = 1 $, $ p = -1 $, pure harmonic trigonometric and pure trigonometric), showing that our general formulations leads to new novel convexities. Hermite-Hadamard, Fejér-Hermite-Hadamard, and related type integral inequalities, with bounds including hypergeometric functions, are presented via a novel coordinated class of convex functions.
Citation: Sabila Ali, Muhammad Samraiz, Salma Trabelsi, Hajer Zaway. New developments in convex analysis: Harmonically trigonometric $ p $-coordinated convex functions and related inequalities[J]. AIMS Mathematics, 2025, 10(10): 23984-24015. doi: 10.3934/math.20251066

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Abstract

In this paper, we introduce two novel classes of functions termed the harmonically trigonometric $ p $-convex functions on $ \Delta = [\tau, \upsilon]\times[\varphi, \chi] $ and harmonically trigonometric $ p $-coordinated convex functions. We discuss relations between these two newly introduced classes of convex functions in two variables and validate the theoretical findings with visual 3D graphs that illustrate the relation landscapes and transition regimes between the function classes. We then study several special or limiting cases (e.g. $ p = 1 $, $ p = -1 $, pure harmonic trigonometric and pure trigonometric), showing that our general formulations leads to new novel convexities. Hermite-Hadamard, Fejér-Hermite-Hadamard, and related type integral inequalities, with bounds including hypergeometric functions, are presented via a novel coordinated class of convex functions.

References

[1]	C. Niculescu, L. E. Persson, Convex functions and their applications, New York: Springer, 2006, 7–64. https://doi.org/10.1007/0-387-31077-0_2
[2]	B. G. Pachpatte, On some integral inequalities involving convex functions, Rgmia Res. Rep. Coll., 3 (2000), 487–492.
[3]	S. Ali, R. S. Ali, M. Vivas-Cortez, S. Mubeen, G. Rahman, K. S. Nisar, Some fractional integral inequalities via $h$-Godunova–Levin preinvex function, AIMS Math., 7 (2022), 13832–13844. https://doi.org/10.3934/math.2022763 doi: 10.3934/math.2022763
[4]	S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X doi: 10.1016/S0893-9659(98)00086-X
[5]	M. Samraiz, M. Malik, S. Naheed, A. O. Akdemir, Error estimates of Hermite‐Hadamard type inequalities with respect to a monotonically increasing function, Math. Meth. Appl. Sci., 46 (2023), 14527–14546. https://doi.org/10.1002/mma.9334 doi: 10.1002/mma.9334
[6]	J. Hadamard, Ètude sur les propriètès des fonctions entières et en particulier d'une fonction considèrèe par Riemann, J. Math. Pure. Appl., 58 (1983), 171–215.
[7]	S. Ali, S. Mubeen, R. S. Ali, G. Rahman, A. Morsy, K. S. Nisar, M. Zakarya, Dynamical significance of generalized fractional integral inequalities via convexity, AIMS Math., 6 (2021), 9705–9730. https://doi.org/10.3934/math.2021565 doi: 10.3934/math.2021565
[8]	M. Samraiz, T. Atta, H. A. Nabwey, S. Naheed, S. Etemad, On a new $\alpha$-convexity with respect to a parameter: applications on the means and fractional inequalities, Fractals, 32 (2024), 2440035. https://doi.org/10.1142/S0218348X24400358 doi: 10.1142/S0218348X24400358
[9]	A. Almutairi, A. Kiliçman, New fractional inequalities of midpoint type via s-convexity and their application, J. Inequal. Appl., 2019 (2019), 1–19. https://doi.org/10.1186/s13660-019-2215-3 doi: 10.1186/s13660-019-2215-3
[10]	İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935–942.
[11]	K. S. Zhang, J. P. Wan, $p$-Convex functions and their properties, Pure Appl. Math., 23 (2007), 130–133.
[12]	H. Kadakal, Hermite-Hadamard type inequalities for trigonometrically convex functions, Sci. Stud. Res. Ser. Math. Inform., 28 (2018), 19–28.
[13]	S. Ali, M. Samraiz, S. Naheed, M. Vivas-Cortez, Advancements in harmonic convexity and its role in modern mathematical analysis, J. Math., 1 (2025), 8847839. https://doi.org/10.1155/jom/8847839 doi: 10.1155/jom/8847839
[14]	S. Ali, M. Samraiz, S. Naheed, M. Vivas-Cortez, Y. Elmasryc, A refined class of harmonically convex functions and Bullen-Mercer-type inequalities, J. Math. Comput. Sci., 39 (2025), 500–521. https://doi.org/10.22436/jmcs.039.04.06 doi: 10.22436/jmcs.039.04.06
[15]	S. S. Dragomir, On the Hadamard's inequlality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math, 5 (2001), 775–788. https://doi.org/10.11650/twjm/1500574995 doi: 10.11650/twjm/1500574995
[16]	A. Almutairi, A. Kiliçman, New refinements of the Hadamard inequality on coordinated convex function, J. Inequal. Appl., 2019 (2019), 1–9. https://doi.org/10.1186/s13660-019-2143-2 doi: 10.1186/s13660-019-2143-2
[17]	M. A. Noor, K. I. Noor, M. U. Awan, Integral inequalities for coordinated harmonically convex functions, Complex Var. Elliptic Equ., 60 (2015), 776–786. https://doi.org/10.1080/17476933.2014.976814 doi: 10.1080/17476933.2014.976814
[18]	E. Set, İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions on the co-ordinates, arXiv preprint, 2014. https://doi.org/10.48550/arXiv.1404.6397
[19]	M. Samraiz, F. Nawaz, S. Iqbal, T. Abdeljawad, G. Rahman, K. S. Nisar, Certain mean-type fractional integral inequalities via different convexities with applications, J. Inequal. Appl., 2020 (2020), 1–19. https://doi.org/10.1186/s13660-020-02474-x doi: 10.1186/s13660-020-02474-x
[20]	S. S. Dragomir, Lebesgue integral inequalities of Jensen type for $\lambda $-convex functions, Armen. J. Math., 10 (2018), 1–19. https://doi.org/10.52737/18291163-2018.10.8-1-19 doi: 10.52737/18291163-2018.10.8-1-19
[21]	M. Tunç, On some new inequalities for convex functions, Turkish J. Math., 36 (2012), 245–251.
[22]	M. A. Noor, K. I. Noor, S. Iftikhar, Hermite-Hadamard inequalities for harmonic nonconvex functions, MAGNT Res. Rep., 4 (2016), 24–40.
[23]	M. R. Delavar, S. M. Aslani, M. D. L. Sen, Hermite-Hadamard-Fejér inequality related to generalized convex functions via fractional integrals, J. Math., 2018 (2018). https://doi.org/10.1155/2018/5864091
[24]	M. Muddassar, M. I. Bhatti, M. Iqbal, Some new s-Hermite-Hadamard type inequalities for differentiable functions and their applications, Proc. Pakistan Acad. Sci., 49 (2012), 9–17. https://doi.org/10.1186/1029-242X-2012-267 doi: 10.1186/1029-242X-2012-267
[25]	M. Samraiz, M. Hammad, S. Naheed, A comprehensive study of refined Hermite-Hadamard inequalities and their applications, Filomat, 39 (2025), 683–696. https://doi.org/10.2298/FIL2502683S doi: 10.2298/FIL2502683S
[26]	S. I. Butt, M. Umar, D. Khan, Y. Seol, S. Tipurić-Spužević, Hermite-Hadamard-type inequalities for harmonically convex functions via proportional Caputo-Hybrid operators with applications, Fractal Fract., 9 (2025), 77. https://doi.org/10.3390/fractalfract9020077 doi: 10.3390/fractalfract9020077
[27]	S. S. Dragomir, Two mappings in connection to Hadamard's inequalities, J. Math. Anal. Appl., 167 (1992), 49–56. https://doi.org/10.1016/0022-247X(92)90233-4 doi: 10.1016/0022-247X(92)90233-4
[28]	M. A. Noor, K. I. Noor, M. U. Awan, S. Khan, Hermite-Hadamard inequalities for s-Godunova-Levin preinvex functions, J. Adv. Math. Stud., 7 (2014), 12–19.
[29]	R. S. Ali, S. Mubeen, S. Ali, G. Rahman, J. Younis, A. Ali, Generalized Hermite-Hadamard-type integral inequalities for-Godunova-Levin functions, J. Funct. Spaces, 2022 (2022). https://doi.org/10.1155/2022/9113745
[30]	L. Fejèr, Über die fourierreihen, Ⅱ, Math. Naturwiss. Anz Ungar. Akad. Wiss., 24 (1906), 369–390.

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