Interval-Valued Erdélyi–Kober Fractional Integral Operators: Hölder, Minkowski, and Hermite–Hadamard Type Inequalities with Applications

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-343
Published: Oct 15, 2026
Cite as:
Mohammad Akram, Haitham Qawaqneh, Waqar Afzal, Mujahid Abbas, Muhammad Tariq, Ernesto Urenda-Cázares, Jorge E. Macías-Díaz, Hijaz Ahmad, Interval-Valued Erdélyi–Kober Fractional Integral Operators: Hölder, Minkowski, and Hermite–Hadamard Type Inequalities with Applications, Int. J. Anal. Appl., 24 (2026), 343.

Main Article Content

Mohammad Akram, Haitham Qawaqneh, Waqar Afzal, Mujahid Abbas, Muhammad Tariq, Ernesto Urenda-Cázares, Jorge E. Macías-Díaz, Hijaz Ahmad

Abstract

In this article, we introduce a novel interval-valued Erdélyi–Kober fractional integral operator. Using this operator, we establish several new variants of Hermite–Hadamard, midpoint, Hölder, and Minkowski type inequalities for interval-valued convex mappings. To illustrate the validity and sharpness of the obtained results, we provide nontrivial examples supported by tables and detailed remarks. We also show that a number of known results in the existing literature arise as special cases of our main inequalities. Moreover, the derived Hermite–Hadamard-type bounds are expressed in a new form involving special functions that explicitly involve the Gamma and Beta functions. As an application, we demonstrate the usefulness of our results in estimating the error of numerical quadrature rules, particularly the trapezoidal rule, for interval-valued functions.

Article Details

References

  1. D. Abu Judeh, Applications of Conformable Fractional Pareto Probability Distribution, Int. J. Adv. Soft Comput. Appl. 14 (2022), 116–124. https://doi.org/10.15849/ijasca.220720.08.
  2. H. Ahmad, H. Qawaqneh, W. Afzal, M. Abbas, L. Akın, et al., Error Estimates for Numerical Quadrature Rules via New Fractional Integral Inequalities Based on the Prabhakar Fractional Operator for Harmonic Mappings with Computational and Graphical Validation, Int. J. Anal. Appl. 24 (2026), 242. https://doi.org/10.28924/2291-8639-24-2026-242.
  3. W. Afzal, Regularity of Parabolic Ornstein–Uhlenbeck Equations via Boundedness of Fractional Muckenhoupt-Type Weighted Singular Operators in Variable Herz Spaces, J. Pseudo-Differ. Oper. Appl. 17 (2026), 2. https://doi.org/10.1007/s11868-025-00752-0.
  4. 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, J. Math. 2022 (2022), 3830324. https://doi.org/10.1155/2022/3830324.
  5. M.N. Aftab, H.A. Zinadah, A.A. Lupas, C.C. Lee, New (p,q)-Symmetric Hermite Hadamard-Type Inequalities in Quantum Calculus, Int. J. Appl. Math. Optim. Artif. Intell. 1 (2026), 1–10.
  6. M.H. Alqahtani, A.M. Abd El-latif, Separation Axioms via Novel Operators in the Frame of Topological Spaces and Applications, AIMS Math. 9 (2024), 14213–14227. https://doi.org/10.3934/math.2024690.
  7. H. Qawaqneh, H. Aydi, Fixed Points of Contraction Mappings Involving a Simulation Function and Applications, J. Math. Anal. 16 (2025), 1–13. https://doi.org/10.54379/jma-2025-4-1.
  8. 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 Math. 9 (2024), 16061–16092. https://doi.org/10.3934/math.2024778.
  9. D. Khan, I. Khan, J.E.N. Valdes, W.M. Abdelfattah, M. Azeem, Midpoint Type Inequality for Multiplicatively Strong Convex Function via Multiplicative Calculus, Int. J. Appl. Math. Optim. Artif. Intell. 1 (2026), 31–39.
  10. M.Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite–Hadamard's Inequalities for Fractional Integrals and Related Fractional Inequalities, Math. Comput. Model. 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048.
  11. X. Hai, S. Wang, Hermite-Hadamard Type Inequalities Based on the Erdélyi-Kober Fractional Integrals, AIMS Math. 6 (2021), 11494–11507. https://doi.org/10.3934/math.2021666.
  12. B. Ahmad, A. Alsaedi, M. Kirane, B.T. Torebek, Hermite–Hadamard, Hermite–Hadamard–Fejér, Dragomir–Agarwal and Pachpatte Type Inequalities for Convex Functions via New Fractional Integrals, J. Comput. Appl. Math. 353 (2019), 120–129. https://doi.org/10.1016/j.cam.2018.12.030.
  13. T. Zhou, Z. Yuan, T. Du, On the Fractional Integral Inclusions Having Exponential Kernels for Interval-Valued Convex Functions, Math. Sci. 17 (2023), 107–120. https://doi.org/10.1007/s40096-021-00445-x.
  14. S.K. Sahoo, H. Ahmad, M. Tariq, B. Kodamasingh, H. Aydi, et al., Hermite–Hadamard Type Inequalities Involving k-Fractional Operator for (h̅,m)-Convex Functions, Symmetry 13 (2021), 1686. https://doi.org/10.3390/sym13091686.
  15. H. Budak, T. Tunç, M.Z. Sarikaya, Fractional Hermite-Hadamard-Type Inequalities for Interval-Valued Functions, Proc. Amer. Math. Soc. 148 (2020), 705–718. https://doi.org/10.1090/proc/14741.
  16. 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, J. Math. 2023 (2023), 8865992. https://doi.org/10.1155/2023/8865992.
  17. Z.A. Khan, W. Afzal, W. Nazeer, J.K.K. Asamoah, Some New Variants of Hermite–Hadamard and Fejér-Type Inequalities for Godunova–Levin Preinvex Class of Interval-Valued Functions, J. Math. 2024 (2024), 8814585. https://doi.org/10.1155/2024/8814585.
  18. O. Alghamdi, A. Al-Omari, M.H. Alqahtani, Novel Operators in the Frame of Primal Topological Spaces, AIMS Math. 9 (2024), 25792–25808. https://doi.org/10.3934/math.20241260.
  19. 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, Comput. Model. Eng. Sci. 143 (2025), 911–943. https://doi.org/10.32604/cmes.2025.062524.
  20. H. Qawaqneh, Fractional Analytic Solutions and Fixed Point Results with Some Applications, Adv. Fixed Point Theory 14 (2024), 1. https://doi.org/10.28919/afpt/8279.
  21. T. Kanan, M. Elbes, K. Abu Maria, M. Alia, Exploring the potential of IoT-based learning environments in education, Int. J. Adv. Soft Comput. Appl. 15 (2023), 166–178. https://doi.org/10.15849/IJASCA.230720.11.
  22. 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.
  23. 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.
  24. A. Adrees, W. Afzal, K. Shabbir, N.M. Dahshan, A.E. Abuzeid, et al., On Some Properties and Integral Inequalities for Modified (p,h)-Convex Stochastic Processes, Open J. Math. Sci. 10 (2026), 477–491. https://doi.org/10.30538/oms2026.0300.
  25. H. Ahmad, S. Bhatti, A. Asghar, M. Tariq, W. Afzal, et al., Hermite-Hadamard Type Inequality via Generalized Superquadratic Functions, Int. J. Math. Comput. Sci. 21 (2026), 307–313. https://doi.org/10.69793/ijmcs/02.2026/hsamwew.
  26. M. Tariq, S.K. Ntouyas, A.A. Shaikh, A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators, Mathematics 11 (2023), 1953. https://doi.org/10.3390/math11081953.
  27. H. Ahmad, M. Tariq, A. Asghar, W. Afzal, M. Aphane, et al., Some New Notions of Mathematical Integral Inequalities: Theory and Applications, Int. J. Anal. Appl. 24 (2026), 175. https://doi.org/10.28924/2291-8639-24-2026-175.
  28. W. Afzal, D. Breaz, M. Abbas, L.-I. Cotîrlă, Z.A. Khan, et al., Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér 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.
  29. M.A. Ali, H. Budak, G. Murtaza, Y.M. Chu, Post-Quantum Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions, J. Inequal. Appl. 2021 (2021), 84. https://doi.org/10.1186/s13660-021-02619-6.
  30. H. Kalsoom, M.A. Ali, M. Idrees, P. Agarwal, M. Arif, New Post Quantum Analogues of Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions, Math. Probl. Eng. 2021 (2021), 5529650. https://doi.org/10.1155/2021/5529650.
  31. H. Ahmad, S. Bhatti, A.A. Siyal, M. Tariq, W. Afzal, et al., k-Fractional Hermite-Hadamard Inequality Over Superquadraticity in the Frame of Interval Analysis, Int. J. Math. Comput. Sci. 21 (2026), 621–628. https://doi.org/10.69793/ijmcs/03.2026/abstabeek.
  32. 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.
  33. J. Tariboon, M.A. Ali, H. Budak, S.K. Ntouyas, Hermite–Hadamard Inclusions for Co-Ordinated Interval-Valued Functions via Post-Quantum Calculus, Symmetry 13 (2021), 1216. https://doi.org/10.3390/sym13071216.
  34. 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, Int. J. Comput. Intell. Syst. 15 (2022), 8. https://doi.org/10.1007/s44196-021-00061-6.
  35. Z.A. Khan, W. Afzal, An Estimation of Different Kinds of Integral Inequalities for a Generalized Class of Godunova–Levin Convex and Preinvex Functions via Pseudo and Standard Order Relations, J. Funct. Spaces 2025 (2025), 3942793. https://doi.org/10.1155/jofs/3942793.
  36. Y. Tan, D. Zhao, M.Z. Sarikaya, Hermite–Hadamard-Type Inequalities for ħ-Preinvex Interval-Valued Functions via Fractional Integral, Int. J. Comput. Intell. Syst. 16 (2023), 120. https://doi.org/10.1007/s44196-023-00300-y.
  37. W.M. Abdelfattah, S. Bhatti, A. Asghar, M. Tariq, W. Afzal, et al., Fractional Integral Inequalities Associated with Applications to Special Means, Int. J. Math. Comput. Sci. 21 (2026), 289–294. https://doi.org/10.69793/ijmcs/02.2026/ahmad.
  38. H. Ahmad, H. Qawaqneh, W. Afzal, K. Shabbir, M.Z. Meetei, et al., Boundedness Properties of Multidimensional Fractional Hadamard-Type Operators on Campanato Spaces, Int. J. Anal. Appl. 24 (2026), 200. https://doi.org/10.28924/2291-8639-24-2026-200.
  39. Z.A. Ameen, M.H. Alqahtani, Some Classes of Soft Functions Defined by Soft Open Sets Modulo Soft Sets of the First Category, Mathematics 11 (2023), 4368. https://doi.org/10.3390/math11204368.
  40. 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 Intell. Syst. 8 (2022), 413–427. https://doi.org/10.1007/s40747-021-00379-w.
  41. 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.
  42. W. Afzal, E.Y. Prosviryakov, S.M. El-Deeb, Y. Almalki, Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions, Symmetry 15 (2023), 831. https://doi.org/10.3390/sym15040831.
  43. O. Rholam, M. Barmaki, Hermite-Hadamard Inequalities Type for Log-h-Convex Interval-Valued Stochastic Processes, J. Interdiscip. Math. 27 (2024), 1405–1420. https://doi.org/10.47974/JIM-1861.
  44. H. Ahmad, W. Afzal, M. Abbas, M. Tariq, W.M. Abdelfattah, On Boundedness, Continuity, and Hyers–Ulam Stability of Interval-Valued Stochastic Processes in a Probabilistic Framework, Stat. Optim. Inf. Comput. 16 (2026), 2185–2199. https://doi.org/10.19139/soic-2310-5070-3884.
  45. A. Fahad, Y. Qian, Z. Ali, A. Younus, On Generalization of Hermite–Hadamard–Mercer Inequalities for Interval-Valued Functions with Generalized Geometric–Arithmetic Convexity, Int. J. Geom. Methods Mod. Phys. 23 (2026), 2440026. https://doi.org/10.1142/S0219887824400267.
  46. 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 Fract. 9 (2025), 578. https://doi.org/10.3390/fractalfract9090578.
  47. W. Afzal, S.M. Eldin, W. Nazeer, A.M. Galal, Some Integral Inequalities for Harmonical cr-h-Godunova-Levin Stochastic Processes, AIMS Math. 8 (2023), 13473–13491. https://doi.org/10.3934/math.2023683.
  48. M. Dilshad, F.M. Alamrani, A. Alamer, E. Alshaban, M.G. Alshehri, Viscosity-Type Inertial Iterative Methods for Variational Inclusion and Fixed Point Problems, AIMS Math. 9 (2024), 18553–18573. https://doi.org/10.3934/math.2024903.
  49. R.G. Artes Jr., W. Afzal, S. Asjad, M. Tariq, H. Ahmad, Wolff Potential and Birkhoff's Ergodic Theorem in Grand Variable Herz Spaces, Gulf J. Math. 23 (2026). https://doi.org/10.56947/51vxdc48.
  50. B. Bin-Mohsin, M.Z. Javed, M.U. Awan, B. Meftah, A. Kashuri, Fractional Reverse Inequalities Involving Generic Interval-Valued Convex Functions and Applications, Fractal Fract. 8 (2024), 587. https://doi.org/10.3390/fractalfract8100587.
  51. 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 (α,m)-Convex Mappings, J. Math. 2026 (2026), 8858239. https://doi.org/10.1155/jom/8858239.
  52. M. Bohner, L. Nguyen, B. Schneider, T. Truong, Inequalities for Interval-Valued Riemann Diamond-Alpha Integrals, J. Inequal. Appl. 2023 (2023), 86. https://doi.org/10.1186/s13660-023-02993-3.
  53. D. Khan, S.I. Butt, A. Fahad, Y. Wang, B.B. Mohsin, Analysis of Superquadratic Fuzzy Interval Valued Function and Its Integral Inequalities, AIMS Math. 10 (2025), 551–583. https://doi.org/10.3934/math.2025025.
  54. W. Afzal, A. Alb Lupaş, K. Shabbir, Hermite–Hadamard and Jensen-Type Inequalities for Harmonical (h₁,h₂)-Godunova–Levin Interval-Valued Functions, Mathematics 10 (2022), 2970. https://doi.org/10.3390/math10162970.
  55. H. Kalsoom, B. Almohsen, New Fractional-Type Inequalities for Generalized n-Polynomial Convexity in Interval-Valued Functions, Fractals 33 (2025), 2550060. https://doi.org/10.1142/S0218348X25500604.
  56. A.F. Shah, S. Özcan, M. Vivas-Cortez, M.S. Saleem, A. Kashuri, Fractional Hermite–Hadamard–Mercer-Type Inequalities for Interval-Valued Convex Stochastic Processes with Center-Radius Order and Their Related Applications in Entropy and Information Theory, Fractal Fract. 8 (2024), 408. https://doi.org/10.3390/fractalfract8070408.
  57. O. Almutairi, A. Kılıçman, Some Integral Inequalities for h-Godunova-Levin Preinvexity, Symmetry 11 (2019), 1500. https://doi.org/10.3390/sym11121500.
  58. W. Afzal, M. Abbas, J.E. Macías-Díaz, S. Treanţă, Some H-Godunova–Levin Function Inequalities Using Center Radius (CR) Order Relation, Fractal Fract. 6 (2022), 518. https://doi.org/10.3390/fractalfract6090518.
  59. M.A. Noor, K.I. Noor, M.U. Awan, S. Khan, Fractional Hermite-Hadamard Inequalities for Some New Classes of Godunova-Levin Functions, Appl. Math. Inf. Sci. 8 (2014), 2865–2872. https://doi.org/10.12785/amis/080623.
  60. H. Agahi, H. Román-Flores, A. Flores-Franulič, General Barnes–Godunova–Levin Type Inequalities for Sugeno Integral, Inf. Sci. 181 (2011), 1072–1079. https://doi.org/10.1016/j.ins.2010.11.029.
  61. H. Qawaqneh, W. Afzal, M. Abbas, M. Tariq, M.S. Khan, et al., New Fractional Hermite–Hadamard and Fejér Inequalities Involving Special Functions with Applications to Numerical Quadrature Rules, J. Niger. Soc. Phys. Sci. 8 (2026), 3543. https://doi.org/10.46481/jnsps.2026.3543.
  62. E. Tolsted, An Elementary Derivation of the Cauchy, Hölder, and Minkowski Inequalities from Young's Inequality, Math. Mag. 37 (1964), 2–12. https://doi.org/10.2307/2688239.
  63. A. Iusem, C. Isnard, D. Butnariu, A Mixed Hölder and Minkowski Inequality, Proc. Amer. Math. Soc. 127 (1999), 2405–2415. https://doi.org/10.1090/S0002-9939-99-04800-5.
  64. M. Akman, J. Lewis, O. Saari, A. Vogel, The Brunn-Minkowski Inequality and a Minkowski Problem for A-Harmonic Green's Function, Adv. Calc. Var. 14 (2021), 247–302. https://doi.org/10.1515/acv-2018-0064.
  65. M.A. Hernández Cifre, D. Iglesias, J.Y. Nicolás, On a Discrete Brunn-Minkowski Type Inequality, SIAM J. Discrete Math. 32 (2018), 1840–1856. https://doi.org/10.1137/18M1166067.
  66. V. Agostiniani, M. Fogagnolo, L. Mazzieri, Minkowski Inequalities via Nonlinear Potential Theory, Arch. Ration. Mech. Anal. 244 (2022), 51–85. https://doi.org/10.1007/s00205-022-01756-6.
  67. T. Zhou, T. Du, On the Reverse Minkowski's, Reverse Hölder's and Other Fractional Integral Inclusions Arising from Interval-Valued Mappings, IAENG Int. J. Appl. Math. 53 (2023), 1294–1307.
  68. 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, J. Math. Comput. Sci. 41 (2026), 244–263. https://doi.org/10.22436/jmcs.041.02.07.
  69. W. Afzal, T. Botmart, Some Novel Estimates of Jensen and Hermite-Hadamard Inequalities for h-Godunova-Levin Stochastic Processes, AIMS Math. 8 (2023), 7277–7291. https://doi.org/10.3934/math.2023366.
  70. 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.
  71. J. Nasir, M. Tariq, W. Afzal, H. Ahmad, M. Aphane, et al., Some New Notions of Fractional Mercer-Hadamard-Type Inequalities over Generalized Convexity with Applications to Q-Diagamma and Modified Bessel Functions, Results Nonlinear Anal. 9 (2026), 105–130.
  72. M. Akram, M. Dilshad, A.K. Rajpoot, F. Babu, R. Ahmad, et al., Modified Iterative Schemes for a Fixed Point Problem and a Split Variational Inclusion Problem, Mathematics 10 (2022), 2098. https://doi.org/10.3390/math10122098.
  73. A.M. Abd El-latif, M.H. Alqahtani, F.A. Gharib, Strictly Wider Class of Soft Sets via Supra Soft δ-Closure Operator, Int. J. Anal. Appl. 22 (2024), 47. https://doi.org/10.28924/2291-8639-22-2024-47.
  74. D. Filali, M. Dilshad, M. Akram, M.F. Khan, S.S. Irfan, Hybrid Inertial Viscosity-Type Forward-Backward Splitting Algorithms for Variational Inclusion Problems, AIMS Math. 10 (2025), 28829–28860. https://doi.org/10.3934/math.20251269.
  75. T. Du, T. Zhou, On the Fractional Double Integral Inclusion Relations Having Exponential Kernels via Interval-Valued Co-Ordinated Convex Mappings, Chaos Solitons Fractals 156 (2022), 111846. https://doi.org/10.1016/j.chaos.2022.111846.
  76. M. Tariq, T. Cagin, H. Budak, C. Cesarano, M. Aphane, Some Results within the Framework of Conformable Fractional Calculus, Int. J. Appl. Math. Optim. Artif. Intell. 1 (2026), 21–30.
  77. M. Tomar, P. Agarwal, S. Jain, G.V. Milovanović, Some Reverse Hölder Type Inequalities Involving (k,s)-Riemann-Liouville Fractional Integrals, in: T.S. Kalmenov, E.D. Nursultanov, M.V. Ruzhansky, M.A. Sadybekov, (eds) Functional Analysis in Interdisciplinary Applications, Springer Proceedings in Mathematics and Statistics, Springer, Cham, pp. 302–311, (2017). https://doi.org/10.1007/978-3-319-67053-9_29.
  78. F.O. Nwawuru, O.K. Narain, M. Dilshad, J.N. Ezeora, Splitting Method Involving Two-Step Inertial for Solving Inclusion and Fixed Point Problems with Applications, Fixed Point Theory Algorithms Sci. Eng. 2025 (2025), 8. https://doi.org/10.1186/s13663-025-00781-w.
  79. H. Ahmad, H. Qawaqneh, W. Afzal, M. Abbas, K. Shabbir, et al., Improved Hermite–Hadamard, Fejér, and Jensen-Type Inequalities for Generalized Logarithmic Godunova–Levin Mappings via Erdélyi–Kober Fractional Operators, Int. J. Anal. Appl. 24 (2026), 262. https://doi.org/10.28924/2291-8639-24-2026-262.
  80. H. Román-Flores, V. Ayala, A. Flores-Franulič, Milne Type Inequality and Interval Orders, Comput. Appl. Math. 40 (2021), 130. https://doi.org/10.1007/s40314-021-01500-y.