New fractional Hermite–Hadamard and Fejér inequalities involving special functions with applications to numerical quadrature rules

Authors

  • Haitham Qawaqneh
    Department of Basic Sciences, Al-Zaytoonah University of Jordan, Amman 11733, Jordan
  • Waqar Afzal
    Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan;
    Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan;
    International Center for Interdisciplinary Research in Sciences, University of Lahore, Lahore 54792, Pakistan
  • Mujahid Abbas
    Department of Mechanical Engineering Science, University of Johannesburg, Johannesburg 2092, South Africa
  • Muhammad Tariq
    Mathematics Research Center, Near East University, Near East Boulevard, Nicosia, Mersin 99138, Turkey
  • Mehreen Shehzadi Khan
    Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia
  • Siegfried Macías
    Department of Mathematics and Physics, Autonomous University of Aguascalientes, Aguascalientes 20100, Mexico
  • Jorge E. Macías-Díaz
    Department of Mathematics and Physics, Autonomous University of Aguascalientes, Aguascalientes 20100, Mexico
  • Hijaz Ahmad
    Irfan Suat Günsel Operational Research Institute, Near East University, Nicosia/TRNC, Mersin 10, 99138, Turkey;
    Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, South Korea
  • Waleed Mohammed Abdelfattah
    College of Engineering, University of Business and Technology, Jeddah 23435, Saudi Arabia;
    Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, P.O. 44519, Egypt

Keywords:

Gamma and Beta functions, Interval-valued Katugampola fractional integrals, Hermite–Hadamard and Fejér inequalities, Interval-valued convex mappings, Numerical quadrature

Abstract

The principal objective of this article is to establish new and refined variants of Hermite--Hadamard- and Fej'{e}r-type integral inequalities in the interval-valued setting by exploiting the auxiliary transformed mapping mathcal{G}(s) := mathcal{F}(s^{1/rho}). This transformation preserves convexity and facilitates the systematic application of fractional integral techniques over a suitably rescaled domain through the Katugampola fractional operator. The resulting bounds are expressed explicitly in terms of the Gamma function, which enriches the analytical structure of the inequalities and yields sharper and more precise estimates than existing results. The validity and applicability of the theoretical findings are demonstrated through nontrivial illustrative examples and detailed remarks, and several previously known inequalities are recovered as limiting cases under suitable parameter configurations. As a further application, the derived inequalities are used to construct inclusion-type error estimates for numerical quadrature rules, with particular emphasis on the trapezoidal rule applied to interval-valued functions in the Katugampola fractional framework.

Dimensions

[1] M. L. Cartwright, ``Jacques Hadamard, 1865--1963'', Biographical Memoirs of Fellows of the Royal Society 11 (1965) 75. https://doi.org/10.1098/rsbm.1965.0005.

[2] M. Z. Sarikaya, E. Set, H. Yaldiz & N. Bac{s}ak, ``Hermite--Hadamard's inequalities for fractional integrals and related fractional inequalities'', Mathematical and Computer Modelling 57 (2013) 2403. https://doi.org/10.1016/j.mcm.2011.12.048.

[3] S. Varov{s}anec, ``On h-convexity'', Journal of Mathematical Analysis and Applications 326 (2007) 303. https://doi.org/10.1016/j.jmaa.2006.02.086.

[4] M. A. Noor, F. Qi & M. U. Awan, ``Some Hermite--Hadamard type inequalities for log-h-convex functions'', Analysis 33 (2013) 367. https://doi.org/10.1524/anly.2013.1223.

[5] W. Afzal, M. Abbas, J. E. Mac'ias-D'iaz & S. Treanc{t}u{a}, ``Some h-Godunova--Levin function inequalities using center radius (cr) order relation'', Fractal and Fractional 6 (2022) 518. https://doi.org/10.3390/fractalfract6090518.

[6] W. Afzal, E. Y. Prosviryakov, S. M. El-Deeb & Y. Almalki, ``Some new estimates of Hermite--Hadamard, Ostrowski and Jensen-inclusions for h-convex stochastic process via interval-valued functions'', Symmetry 15 (2023) 831. https://doi.org/10.3390/sym15040831.

[7] M. Abbas, W. Afzal, T. Botmart & A. M. Galal, ``Jensen, Ostrowski and Hermite--Hadamard type inequalities for h-convex stochastic processes by means of Center-Radius order relation'', AIMS Mathematics 8 (2023) 16013. https://doi.org/10.3934/math.2023817.

[8] S. K. Sahoo, H. Alrweili, S. Treanc{t}u{a} & Z. A. Khan, ``New fractional integral inequalities pertaining to center-radius (cr)-ordered convex functions'', Fractal and Fractional 7 (2023) 81. https://doi.org/10.3390/fractalfract7010081.

[9] S. K. Sahoo, A. Kashuri, M. Aljuaid, S. Mishra & M. De La Sen, ``On Ostrowski--Mercer's type fractional inequalities for convex functions and applications'', Fractal and Fractional 7 (2023) 215. https://doi.org/10.3390/fractalfract7030215.

[10] A. F. Shah, S. Ozcan, M. Vivas-Cortez, M. S. Saleem & A. Kashuri, ``Fractional Hermite--Hadamard--Mercer inequalities for interval-valued convex stochastic processes with center--radius order and applications in entropy and information theory'', Fractal and Fractional 8 (2024) 408. https://doi.org/10.3390/fractalfract8070408.

[11] M. Tariq, S. K. Ntouyas, W. Afzal & J. Tariboon, ``Some new approaches of integral inequalities involving Raina and Mittag-Leffler function pertaining to Atangana-Baleanu fractional integral operator'', Journal of Mathematics and Computer Science 41 (2025) 244. https://doi.org/10.22436/jmcs.041.02.07.

[12] M. Tariq, W. Afzal, M. Nadeem, A. E. Mu~noz-Zavala & J. E. Mac'ias-D'iaz, ``Novel fractional Hermite--Hadamard and product-type inequalities via Raina function and preinvex mappings with entropy applications'', European Journal of Pure and Applied Mathematics 18 (2025) 6556. https://doi.org/10.29020/nybg.ejpam.v18i3.6556.

[13] M. Bin-Saad, J. Younis & M. A. Abd El Salam, ``Unification and extension of classical integral theorems via the generalized Mittag--Leffler function'', Integral Transforms and Special Functions 2026 (2026) 1. https://doi.org/10.1080/10652469.2026.2613779.

[14] M. Bin-Saad, J. Younis, M. S. Alatawi & A. N. Koam, ``On Hadamard and Fej'er--Hadamard inequalities for fractional integrals involving Mittag-Leffler-type function of arbitrary order'', Journal of Function Spaces 2026 (2026) 4817046. https://doi.org/10.1155/jofs/4817046.

[15] W. M. Abdelfattah, S. Bhatti, A. Asghar, M. Tariq, W. Afzal & H. Ahmad, ``Fractional integral inequalities associated with applications to special means'', International Journal of Mathematics and Computer Science 21 (2026) 289. https://doi.org/10.69793/ijmcs/02.2026/ahmad.

[16] W. M. Abdelfattah, S. Bhatti, A. Asghar, M. Tariq, W. Afzal & H. Ahmad, ``New fractional developments of Hermite--Hadamard type inequalities'', International Journal of Mathematics and Computer Science 21 (2026) 283. https://doi.org/10.69793/ijmcs/02.2026/hijaz.

[17] H. Cheng, D. Zhao & G. Zhao, ``Fractional quantum Hermite--Hadamard-type inequalities for interval-valued functions'', Fractals 31 (2023) 2350104. https://doi.org/10.1142/S0218348X23501049.

[18] W. Afzal, M. H. Alqahtani, M. Abbas & D. Breaz, ``Regularity and qualitative study of parabolic physical Ginzburg--Landau equations in variable exponent Herz spaces via fractional Bessel--Riesz operators'', Fractal and Fractional 9 (2025) 644. https://doi.org/10.3390/fractalfract9100644.

[19] W. Afzal, M. Abbas, M. Tariq, J. E. Mac'ias-D'iaz & H. Ahmad, ``A note on the boundedness of the multidimensional Katugampola operator in Campanato spaces'', Gulf Journal of Mathematics 23 (2026) 1. https://doi.org/10.56947/5e9waj50.

[20] W. Afzal, M. Abbas, J. E. Mac'ias-D'iaz, M. Z. Meetei, M. S. Khan & A. Gallegos, ``Gradient descent and twice differentiable Simpson-type inequalities via K-Riemann-Liouville fractional operators in function spaces'', European Journal of Pure and Applied Mathematics 18 (2025) 5790. https://doi.org/10.29020/nybg.ejpam.v18i1.5790.

[21] O. Alghamdi, A. Al-Omari & M. H. Alqahtani, ``Novel operators in the frame of primal topological spaces'', AIMS Mathematics 9 (2024) 25792. https://doi.org/10.3934/math.20241260.

[22] A. M. Abd El-latif, M. H. Alqahtani & F. A. Gharib, ``Strictly wider class of soft sets via supra soft $delta$-Closure operator'', International Journal of Analysis and Applications 22 (2024) 47. https://doi.org/10.28924/2291-8639-22-2024-47.

[23] M. H. Alqahtani & A. M. Abd El-latif, ``Separation axioms via novel operators in the frame of topological spaces and applications'', AIMS Mathematics 9 (2024) 14213. https://doi.org/10.3934/math.2024690.

[24] A. M. Abd El-latif & M. H. Alqahtani, ``Novel categories of supra soft continuous maps via new soft operators'', AIMS Mathematics 9 (2024) 7449. https://doi.org/10.3934/math.2024361.

[25] D. Judeh & M. Abu Hammad, ``Applications of conformable fractional Pareto probability distribution'', International Journal of Advances in Soft Computing and its Applications 14 (2022) 115.

[26] R. Hatamleh, D. A. M. Mahmoud, H. Qawaqneh, H. Y. Saleh, A. M. Abd El-latif, E. Almuhur, A. Hussain, A. Mehmood & C. L. Armada, ``Complex triple-valued neutrosophic soft sets and their topological framework with AI-driven signal-template analysis'', Journal of the Nigerian Society of Physical Sciences 8 (2026) 3261. https://doi.org/10.46481/jnsps.2026.3261.

[27] S. Y. Musa, B. A. Asaad, H. Alohali, Z. A. Ameen & M. H. Alqahtani, ``Fuzzy n-bipolar soft sets for multi-criteria decision-making: theory and application'', Computer Modeling in Engineering & Sciences 143 (2025) 911. https://doi.org/10.32604/cmes.2025.062524.

[28] Z. A. Ameen & M. H. Alqahtani, ``Congruence representations via soft ideals in soft topological spaces'', Axioms 12 (2023) 1015. https://doi.org/10.3390/axioms12111015.

[29] Z. A. Ameen, M. H. Alqahtani & O. F. Alghamdi, ``Lower density soft operators and density soft topologies'', Heliyon 10 (2024) e35280. https://doi.org/10.1016/j.heliyon.2024.e35280.

[30] A. M. Abd El-latif, A. A. Azzam, R. Abu-Gdairi, M. Aldawood & M. H. Alqahtani, ``New versions of maps and connected spaces via supra soft sd-operators'', PLoS ONE 19 (2024) e0304042. https://doi.org/10.1371/journal.pone.0304042.

[31] O. Almutairi & A. Ki lic{c}man, ``Some integral inequalities for h-Godunova--Levin preinvexity'', Symmetry 11 (2019) 1500. https://doi.org/10.3390/sym11121500.

[32] S. Ali, R. S. Ali, M. Vivas-Cortez, S. Mubeen, G. Rahman & K. S. Nisar, ``Some fractional integral inequalities via h-Godunova--Levin preinvex functions'', AIMS Mathematics 7 (2022) 13832. https://doi.org/10.3934/math.2022763.

[33] R. S. Ali, S. Mubeen, S. Ali, G. Rahman, J. Younis & A. Ali, ``Generalized Hermite--Hadamard integral inequalities for h-Godunova--Levin functions'', Journal of Function Spaces 2022 (2022) 9113745. https://doi.org/10.1155/2022/9113745.

[34] G. Farid, U. N. Katugampola & M. Usman, ``Ostrowski-fractional integral inequalities for s-Godunova--Levin functions via Katugampola fractional integrals'', Open Journal of Mathematical Sciences 1 (2017) 97. https://doi.org/10.30538/oms2017.0010.

[35] Z. A. Khan, W. Afzal, W. Nazeer & J. K. Asamoah, ``Some new variants of Hermite--Hadamard and Fej'er-type inequalities for Godunova--Levin preinvex class of interval-valued functions'', Journal of Mathematics 2024 (2024) 8814585. https://doi.org/10.1155/2024/8814585.

[36] M. W. Akram, S. Iqbal, A. Fahad & Y. Wang, ``Hermite--Hadamard-type inequalities for h-Godunova--Levin convex fuzzy interval-Valued functions via Riemann--Liouville fractional q-integrals'', Fractal and Fractional 9 (2025) 578. https://doi.org/10.3390/fractalfract9090578.

[37] Y. Almalki, W. Afzal, K. Shabbir, D. Breaz, L. I. Cot^irlu{a} & A. Albaity, ``New structural properties and Hermite--Hadamard inequalities for Godunova--Levin mappings via a novel analytical approach'', Research in Mathematics 12 (2025) 2574096. https://doi.org/10.1080/27684830.2025.2574096.

[38] E. R. Nwaeze, M. A. Khan & Y. M. Chu, ``Fractional inclusions of the Hermite--Hadamard type for m-polynomial convex interval-valued functions'', Advances in Difference Equations 2020 (2020) 507. https://doi.org/10.1186/s13662-020-02977-3.

[39] D. Zhao, G. Zhao, G. Ye, W. Liu & S. S. Dragomir, ``On Hermite--Hadamard-type inequalities for coordinated h-convex interval-valued functions'', Mathematics 9 (2021) 2352. https://doi.org/10.3390/math9192352.

[40] H. M. Srivastava, S. K. Sahoo, P. O. Mohammed, D. Baleanu & B. Kodamasingh, ``Hermite--Hadamard type inequalities for interval-valued preinvex functions via fractional integral operators'', International Journal of Computational Intelligence Systems 15 (2022) 8. https://doi.org/10.1007/s44196-021-00061-6.

[41] Y. Tan, D. Zhao & M. Z. Sarikaya, ``Hermite--Hadamard-type inequalities for $bar{h}$-preinvex Interval-Valued functions via fractional integral'', International Journal of Computational Intelligence Systems 16 (2023) 120. https://doi.org/10.1007/s44196-023-00300-y.

[42] K. K. Lai, S. K. Mishra, J. Bisht & M. Hassan, ``Hermite--Hadamard type inclusions for interval-valued coordinated preinvex functions'', Symmetry 14 (2022) 771. https://doi.org/10.3390/sym14040771.

[43] M. A. Ali, H. Budak, G. Murtaza & Y. M. Chu, ``Post-quantum Hermite--Hadamard type inequalities for interval-valued convex functions'', Journal of Inequalities and Applications 2021 (2021) 84. https://doi.org/10.1186/s13660-021-02619-6.

[44] F. Faisal, Z. Haider, U. Ghani, H. Ali, M. Amjad, H. Ahmad & I. Alraddadi, ``Derivation of Hermite--Hadamard-type inequalities via quasi-preinvex functions and strongly preinvex functions'', International Journal of Geometric Methods in Modern Physics 23 (2026) 2650048. https://doi.org/10.1142/S0219887826500489.

[45] P. Liu, M. B. Khan, M. A. Noor & K. I. Noor, ``New Hermite--Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense'', Complex & Intelligent Systems 8 (2022) 413. https://doi.org/10.1007/s40747-021-00379-w.

[46] D. Khan, S. I. Butt, A. Fahad, Y. Wang & B. B. Mohsin, ``Analysis of superquadratic fuzzy interval-valued functions and their integral inequalities'', AIMS Mathematics 10 (2025) 551. https://doi.org/10.3934/math.2025025.

[47] H. Ahmad, S. Bhatti, A. Asghar, M. Tariq, W. Afzal, E. Hincal & W. M. Abdelfattah, ``Hermite--Hadamard type inequality via generalized superquadratic functions'', International Journal of Mathematics and Computer Science 21 (2026) 307. https://www.researchgate.net/publication/404146436_Hermite-Hadamard_Type_Inequality_via_Generalized_Superquadratic_Functions.

[48] A. Kashuri, S. K. Sahoo, P. O. Mohammed, E. Al-Sarairah & N. Chorfi, ``Novel inequalities for subadditive functions via tempered fractional integrals and their numerical investigations'', AIMS Mathematics 9 (2024) 13195. https://doi.org/10.3934/math.2024643.

[49] M. A. Hern'andez Cifre, D. Iglesias & J. Yepes Nicol'as, ``On a discrete Brunn--Minkowski type inequality'', SIAM Journal on Discrete Mathematics 32 (2018) 1840. https://doi.org/10.1137/18M1166067.

[50] M. Akman, J. Lewis, O. Saari & A. Vogel, ``The Brunn--Minkowski inequality and a Minkowski problem for a-harmonic Green's function'', Advances in Calculus of Variations 14 (2021) 247. https://doi.org/10.1515/acv-2018-0064.

[51] S. Verma & P. Viswanathan, ``A note on Katugampola fractional calculus and fractal dimensions'', Applied Mathematics and Computation 339 (2018) 220. https://doi.org/10.1016/j.amc.2018.07.035.

[52] W. Afzal, K. Shabbir, M. Arshad, J. K. K. Asamoah & A. M. Galal, ``Some novel estimates of integral inequalities for a generalized class of harmonical convex mappings by means of Center-Radius order relation'', Journal of Mathematics 2023 (2023) 8865992. https://doi.org/10.1155/2023/8865992.

[53] W. Afzal, M. Abbas, J. E. Mac'ias-D'iaz, A. Gallegos & Y. Almalki, ``Boundedness and Sobolev-type estimates for the exponentially damped Riesz potential with applications to the regularity theory of elliptic PDEs'', Fractal and Fractional 9 (2025) 458. https://doi.org/10.3390/fractalfract9070458.

[54] A. Adrees, W. Afzal, M. S. Khan, H. Sultana, O. O. Y. Karrar, L. Aliyarukunju, J. E. Mac'{i}as-D'{i}az, A. Rom'{a}n-Loera, D. Breaz & L. I. Cot^{i}rlu{a}, ``Novel unified variants, properties, and applications of Ostrowski, Jensen, and Hermite--Hadamard inequalities for generalized ($eta$, $p$, $h$)-Convex stochastic processes'', International Journal of Analysis and Applications 24 (2026) 164. https://doi.org/10.28924/2291-8639-24-2026-164.

[55] K. Issa, R. A. Bello & M. H. Sulaiman, ``Approximate solution of time-fractional non-linear parabolic equations arising in Mathematical physics'', African Scientific Reports 3 (2024) 176. https://doi.org/10.46481/asr.2024.3.2.176.

[56] V. O. Atabo & S. O. Adee, ``A new special 15-step block method for solving general fourth order ordinary differential equations'', Journal of the Nigerian Society of Physical Sciences 3 (2021) 308. https://doi.org/10.46481/jnsps.2021.337.

[57] A. Adrees, W. Afzal, K. Shabbir, N. M. Dahshan, A. E. Abuzeid, R. A. Tahir, M. Tariq & H. Ahmad, ``On some properties and integral inequalities for modified (p,h)-Convex stochastic processes'', Open Journal of Mathematical Sciences 10 (2026) 477. https://doi.org/10.30538/oms2026.0300.

[58] T. Saeed & S. Treanc{t}u{a}, ``New classes of interval-valued variational problems and inequalities'', Results in Control and Optimization 13 (2023) 100324. https://doi.org/10.1016/j.rico.2023.100324.

[59] R. A. Bello, A. O. Lasode, R. O. Ayinla & A. D. Adeshola, ``Quasi-subordination for a subclass of non-Bazilevic functions connected with a q-derivative operator'', African Scientific Reports 4 (2025) 326. https://doi.org/10.46481/asr.2025.4.3.326.

[60] Z. A. Ameen & M. H. Alqahtani, ``Baire category soft sets and their symmetric local properties'', Symmetry 15 (2023) 1810. https://doi.org/10.3390/sym15101810.

[61] H. Qawaqneh, J. Manafian, M. Alharthi & Y. Alrashedi, ``Stability analysis, modulation instability, and beta-time fractional exact soliton solutions to the van der Waals equation'', Mathematics 12 (2024) 2257. https://doi.org/10.3390/math12142257.

[62] H. Qawaqneh, M. S. Noorani & H. Aydi, ``Some new characterizations and results for fuzzy contractions in fuzzy b-metric spaces and applications'', AIMS Mathematics 8 (2023) 6682. https://doi.org/10.3934/math.2023338.

[63] H. Qawaqneh & H. Aydi, ``Fixed points of contraction mappings involving a simulation function and applications'', Journal of Mathematical Analysis 16 (2025) 1. https://doi.org/10.54379/jma-2025-4-1.

[64] M. Arif, K. Ullah, A. Asghar, M. Tariq, E. Hincal, W. Afzal, H. Ahmad & W. M. Abdelfattah, ``Some iterative approximations of generalized nonexpansive operators in Banach spaces'', International Journal of Mathematics and Computer Science 21 (2026) 327. https://doi.org/10.69793/ijmcs/02.2026/auathaaa.

[65] T. Kanan, M. Elbes, K. Abu Maria & M. Alia, ``Exploring the potential of IoT-based learning environments in education'', International Journal of Advances in Soft Computing and its Applications 15 (2023) 167.

[66] F. Shi, G. Ye, D. Zhao & W. Liu, ``Some fractional Hermite--Hadamard type inequalities for interval-valued functions'', Mathematics 8 (2020) 534. https://doi.org/10.3390/math8040534.

[67] Z. A. Khan, W. Afzal, M. Abbas, J. Ro & A. A. Zaagan, ``Some well known inequalities on two dimensional convex mappings by means of pseudo L-R interval order relations via fractional integral operators having non-singular kernel'', AIMS Mathematics 9 (2024) 16061. https://doi.org/10.3934/math.2024778.

[68] H. Ahmad, M. Tariq, A. Asghar, W. Afzal, M. Aphane & R. Efendiev, ``Some new notions of mathematical integral inequalities: theory and applications'', International Journal of Analysis and Applications 24 (2026) 175. https://doi.org/10.28924/2291-8639-24-2026-175.

[69] X. Zhang, K. Shabbir, W. Afzal, H. Xiao & D. Lin, ``Hermite--Hadamard and Jensen-type inequalities via Riemann integral operator for a generalized class of Godunova--Levin functions'', Journal of Mathematics 2022 (2022) 3830324. https://doi.org/10.1155/2022/3830324.

[70] W. Afzal, D. Breaz, M. Abbas, L. I. Cot^{i}rlu{a}, Z. A. Khan & E. Rapeanu, ``Hyers--Ulam stability of 2D-Convex mappings and some related new Hermite--Hadamard, Pachpatte, and Fej'er type integral inequalities using novel fractional integral operators via totally interval-order relations with open problem'', Mathematics 12 (2024) 1238. https://doi.org/10.3390/math12081238.

[71] W. Liu, F. Shi, G. Ye & D. Zhao, ``Some inequalities for cr-log-h-convex functions'', Journal of Inequalities and Applications 2022 (2022) 160. https://doi.org/10.1186/s13660-022-02900-2.

[72] S. Firdous, W. Nazeer, W. Afzal, J. K. K. Asamoah & H. AlQadi, ``New variations and structural refinements of discrete weighted Jensen and Hermite--Hadamard inequalities using ($alpha$, $m$)-convex mappings'', Journal of Mathematics 2026 (2026) 8858239. https://doi.org/10.1155/jom/8858239.

Q21

Published

2026-08-13

How to Cite

New fractional Hermite–Hadamard and Fejér inequalities involving special functions with applications to numerical quadrature rules. (2026). Journal of the Nigerian Society of Physical Sciences, 8(4), 3543. https://doi.org/10.46481/jnsps.2026.3543

Issue

Volume 8, Issue 4, November 2026 (in progress)

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2026 Haitham Qawaqneh, Waqar Afzal, Mujahid Abbas, Muhammad Tariq, Mehreen Shehzadi Khan, Siegfried Macías, Jorge E. Macías-Díaz, Hijaz Ahmad, Waleed Mohammed Abdelfattah (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

New fractional Hermite–Hadamard and Fejér inequalities involving special functions with applications to numerical quadrature rules. (2026). Journal of the Nigerian Society of Physical Sciences, 8(4), 3543. https://doi.org/10.46481/jnsps.2026.3543

Similar Articles

21-30 of 167

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)