Superquadratic stochastic processes and their fractional perspective with applications in information theory

Mohsen Ayyash; Dawood Khan; Saad Ihsan Butt; Youngsoo Seol; Mohsen Ayyash; Dawood Khan; Saad Ihsan Butt; Youngsoo Seol

doi:10.3934/math.2025617

AIMS Mathematics

2025, Volume 10, Issue 6: 13695-13720. doi: 10.3934/math.2025617

Previous Article Next Article

Research article

Superquadratic stochastic processes and their fractional perspective with applications in information theory

1.
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
2.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
3.
Department of Mathematics, Dong-A University, Busan 49315, Korea

Received: 02 April 2025 Revised: 04 June 2025 Accepted: 09 June 2025 Published: 13 June 2025
MSC : 26D15, 26A51, 26A33, 26D10, 94A15, 94A17, 60G05

Superquadraticity is a generalization of convexity that yields more refined results compared to those obtained through convexity alone. In this work, we established, for the first time, a class of superquadratic stochastic processes and explored their fundamental properties. Based on these properties, we derived Jensen's and (Hermite-Hadamard) $ \mathbb{HH} $'s type inequalities, along with their fractional counterparts, in the context of mean-square stochastic (Riemann-Liouville) $ \mathbb{R.L} $ fractional integrals. The validity of our findings was supported by graphical illustrations using suitable examples. Furthermore, we extended the applicability of our results to information theory by introducing several stochastic divergence measures.
Citation: Mohsen Ayyash, Dawood Khan, Saad Ihsan Butt, Youngsoo Seol. Superquadratic stochastic processes and their fractional perspective with applications in information theory[J]. AIMS Mathematics, 2025, 10(6): 13695-13720. doi: 10.3934/math.2025617

Related Papers:

Abstract

Superquadraticity is a generalization of convexity that yields more refined results compared to those obtained through convexity alone. In this work, we established, for the first time, a class of superquadratic stochastic processes and explored their fundamental properties. Based on these properties, we derived Jensen's and (Hermite-Hadamard) $ \mathbb{HH} $'s type inequalities, along with their fractional counterparts, in the context of mean-square stochastic (Riemann-Liouville) $ \mathbb{R.L} $ fractional integrals. The validity of our findings was supported by graphical illustrations using suitable examples. Furthermore, we extended the applicability of our results to information theory by introducing several stochastic divergence measures.

References

[1]	S. S. Dragomir, J. Pečarić, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335–341.
[2]	P. Agarwal, S. S. Dragomir, M. Jleli, B. Samet, Advances in mathematical inequalities and applications, Singapore: Birkhäuser, 2018. https://doi.org/10.1007/978-981-13-3013-1
[3]	Y. Qin, Integral and discrete inequalities and their applications, Switzerland: Birkhäuser Cham, 2016. https://doi.org/10.1007/978-3-319-33304-5
[4]	A. P. Jayaraj, K. N. Gounder, J. Rajagopal, Optimizing signal smoothing using HERS algorithm and time fractional diffusion equation, Expert Syst. Appl., 238 (2024), 122250. https://doi.org/10.1016/j.eswa.2023.122250 doi: 10.1016/j.eswa.2023.122250
[5]	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 doi: 10.1016/j.mcm.2011.12.048
[6]	J. F. Han, P. O. Mohammed, H. D. Zeng, Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function, Open Math., 18 (2020), 794–806. https://doi.org/10.1515/math-2020-0038 doi: 10.1515/math-2020-0038
[7]	İ. Mumcu, E. Set, A. O. Akdemir, F. Jarad, New extensions of Hermite-Hadamard inequalities via generalized proportional fractional integral, Numer. Meth. Part. D. E., 40 (2024), e22767. https://doi.org/10.1002/num.22767 doi: 10.1002/num.22767
[8]	V. Stojiljković, R. Ramaswamy, F. Alshammari, O. A. Ashour, M. L. H. Alghazwani, S. Radenović, Hermite-Hadamard type inequalities involving $(k-p)$-fractional operator for various types of convex functions, Fractal Fract., 6 (2022), 376. https://doi.org/10.3390/fractalfract6070376 doi: 10.3390/fractalfract6070376
[9]	F. Zafar, S. Mehmood, A. Asiri, Weighted Hermite-Hadamard inequalities for r-times differentiable prein vex functions for $k$-fractional integrals, Preprints, 2023. https://doi.org/10.20944/preprints202208.0322.v1
[10]	H. Chen, U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274–1291. https://doi.org/10.1016/j.jmaa.2016.09.018 doi: 10.1016/j.jmaa.2016.09.018
[11]	S. S. Dragomir, Hermite-Hadamard type inequalities for generalized Riemann-Liouville fractional integrals of h-convex functions, Math. Method. Appl. Sci., 44 (2021), 2364–2380. https://doi.org/10.1002/mma.5893 doi: 10.1002/mma.5893
[12]	K. Liu, J. R. Wang, D. O'Regan, On the Hermite-Hadamard type inequality for $\psi$-Riemann-Liouville fractional integrals via convex functions, J. Inequal. Appl., 2019 (2019), 27. https://doi.org/10.1186/s13660-019-1982-1 doi: 10.1186/s13660-019-1982-1
[13]	M. Bohner, A. Kashuri, P. O. Mohammed, J. E. N. Valdés, Hermite-Hadamard-type inequalities for conformable integrals, Hacet. J. Math. Stat., 51 (2022), 775–786. https://doi.org/10.15672/hujms.946069 doi: 10.15672/hujms.946069
[14]	M. Merad, B. Meftah, H. Boulares, A. Moumen, M. Bouye, Fractional Simpson-like inequalities with parameter for differential s-tgs-convex functions, Fractal Fract., 7 (2023), 772. https://doi.org/10.3390/fractalfract7110772 doi: 10.3390/fractalfract7110772
[15]	F. Hezenci, Fractional inequalities of corrected Euler-Maclaurin-type for twice-differentiable functions, Comput. Appl. Math., 42 (2023), 92. https://doi.org/10.1007/s40314-023-02235-8 doi: 10.1007/s40314-023-02235-8
[16]	D. Zhao, M. A. Ali, H. Budak, Z. Y. He, Some Bullen-type inequalities for generalized fractional integrals, Fractals, 31 (2023), 2340060. https://doi.org/10.1142/s0218348x23400601 doi: 10.1142/s0218348x23400601
[17]	G. Rahman, M. V. Cortez, $ {\underset{\scriptscriptstyle\centerdot}{\rm C}}$. Yildiz, M. Samraiz, S. Mubeen, M. F. Yassen, On the generalization of Ostrowski-type integral inequalities via fractional integral operators with application to error bounds, Fractal Fract., 7 (2023), 683. https://doi.org/10.3390/fractalfract7090683 doi: 10.3390/fractalfract7090683
[18]	W. Saleh, A. Lakhdari, A. Kili$ {\underset{\scriptscriptstyle\centerdot}{\rm c}}$man, A. Frioui, B. Meftah, Some new fractional Hermite-Hadamard type inequalities for functions with co-ordinated extended $(s, m)$-prequasiinvex mixed partial derivatives, Alex. Eng. J., 72 (2023), 261–267. https://doi.org/10.1016/j.aej.2023.03.080. doi: 10.1016/j.aej.2023.03.080
[19]	Y. X. Zhou, T. S. Du, The Simpson-type integral inequalities involving twice local fractional differentiable generalized $(s, P)$-convexity and their applications, Fractals, 31 (2023), 2350038. https://doi.org/10.1142/s0218348x2350038x doi: 10.1142/s0218348x2350038x
[20]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/tsci160111018a doi: 10.2298/tsci160111018a
[21]	K. Nikodem, On convex stochastic processes, Aequationes Math., 2 (1998), 427–446. https://dx.doi.org/10.1007/BF02190513 doi: 10.1007/BF02190513
[22]	A. Skowronski, On some properties of j-convex stochastic processes, Aequationes Math., 2 (1992), 249–258. https://dx.doi.org/10.1007/BF01830983 doi: 10.1007/BF01830983
[23]	D. Kotrys, Hermite-Hadamard inequality for convex stochastic processes, Aequationes Math., 83 (2012), 143–151. https://doi.org/10.1007/s00010-011-0090-1 doi: 10.1007/s00010-011-0090-1
[24]	N. Okur, I. Isçan, E. Y. Dizdar, Hermite-Hadamard type inequalities for $p$-convex stochastic processes, Int. J. Optimiz. Contro., 9 (2019), 148–153. https://doi.org/10.11121/ijocta.01.2019.00602 doi: 10.11121/ijocta.01.2019.00602
[25]	E. Set, M. Tomar, S. Maden, Hermite-Hadamard type inequalities for s-convex stochastic processes in the second sense, Turk. J. Anal. Number Theor., 2 (2014), 202–207. https://doi.org/10.12691/tjant-2-6-3 doi: 10.12691/tjant-2-6-3
[26]	H. Budak, M. Z. Sarikaya, A new Hermite-Hadamard inequality for h-convex stochastic processes, RGMIA Res. Rep. Collect., 7 (2019), 356-363. https://doi.org/10.20852/ntmsci.2019.376 doi: 10.20852/ntmsci.2019.376
[27]	D. Barraez, L. Gonzalez, N. Merentes, On h-convex stochastic processes, Math. Aeterna, 5 (2015), 571–581.
[28]	H. Budak, M. Z. Sarikaya, On generalized stochastic fractional integrals and related inequalities, Theor. Appl., 5 (2018), 471–481. https://doi.org/10.15559/18-vmsta117 doi: 10.15559/18-vmsta117
[29]	W. Afzal, M. Abbas, S. M. Eldin, Z. A. Khan, Some well known inequalities for $(h_{1}, h_{2})$-convex stochastic process via interval set inclusion relation, AIMS Math., 8 (2023), 19913–19932. https://doi.org/10.3934/math.20231015 doi: 10.3934/math.20231015
[30]	M. Tunç, Ostrowski-type inequalities via $h$-convex functions with applications to special means, J. Inequal. Appl., 1 (2013), 1–10. https://doi.org/10.1186/1029-242x-2013-326 doi: 10.1186/1029-242x-2013-326
[31]	D. Zhao, T. An, G. Ye, W. Liu, New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 1 (2018), 1–14. https://doi.org/10.1186/s13660-018-1896-3 doi: 10.1186/s13660-018-1896-3
[32]	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 doi: 10.3934/math.2023366
[33]	J. E. H. Hernandez, On $(m, h_{1}, h_{2})$-G-convex dominated stochastic processes, Kragujev. J. Math., 46 (2022), 215–227. https://doi.org/10.46793/kgjmat2202.215h doi: 10.46793/kgjmat2202.215h
[34]	M. J. V. Cortez, On $(m, h_{1}, h_{2})$-convex stochastic processes using fractional integral operator, Appl. Math. Inform. Sci., 12 (2018), 45–53. https://doi.org/10.18576/amis/120104 doi: 10.18576/amis/120104
[35]	F. M. Hafiz, The fractional calculus for some stochastic processes, Stoch. Anal. Appl., 22 (2004), 507–523. https://doi.org/10.1081/sap-120028609 doi: 10.1081/sap-120028609
[36]	D. Kotrys, Remarks on Jensen, Hermite-Hadamard and Fejer inequalities for strongly convex stochastic processes, Math. Aeterna, 5 (2015), 104.
[37]	H. Agahi, A. Babakhani, On fractional stochastic inequalities related to Hermite-Hadamard and Jensen types for convex stochastic processes, Aequationes Math., 90 (2016), 1035–1043. https://doi.org/10.1007/s00010-016-0425-z doi: 10.1007/s00010-016-0425-z
[38]	H. Fu, M. S. Saleem, W. Nazeer, M. Ghafoor, P. Li, On Hermite-Hadamard type inequalities for $n$-polynomial convex stochastic processes, AIMS Math., 6 (2021), 6322–6339. https://doi.org/10.3934/math.2021371 doi: 10.3934/math.2021371
[39]	S. Abramovich, G. Jameson, G. Sinnamon, Refining Jensen's inequality, B. Math. Soc. Sci. Math., 47 (2004), 3–14.
[40]	S. Abramovich, G. Jameson, G. Sinnamon, Inequalities for averages of convex and superquadratic functions, J. Inequal. Pure Appl. Math., 5 (2004), 1–14.
[41]	G. Li, F. Chen, Hermite-Hadamard type inequalities for superquadratic functions via fractional integrals, Abstr. Appl. Anal., 2014. https://doi.org/10.1155/2014/851271
[42]	M. W. Alomari and C. Chesneau, On h-superquadratic functions, Afr. Mat., 33 (2022), 41. https://doi.org/10.1007/s13370-022-00984-z doi: 10.1007/s13370-022-00984-z
[43]	D. Khan, S. I. Butt, Superquadraticity and its fractional perspective via center-radius cr-order relation, Chaos Soliton. Fract., 182 (2024), 114821. https://doi.org/10.1016/j.chaos.2024.114821 doi: 10.1016/j.chaos.2024.114821
[44]	S. I. Butt, D. Khan, Integral inequalities of $h$-superquadratic functions and their fractional perspective with applications, Math. Method. Appl. Sci., 2024, 1–30. https://doi.org/10.1002/mma.10418
[45]	D. Khan, S. I. Butt, Y. Seol, Analysis of $(P, m)$-superquadratic function and related fractional integral inequalities with applications, J. Inequal. Appl., 2024 (2024), 137. https://doi.org/10.1186/s13660-024-03218-x doi: 10.1186/s13660-024-03218-x
[46]	S. I. Butt, D. Khan, Superquadratic function and its applications in information theory via interval calculus, Chaos Soliton. Fract., 190 (2025), 115748. https://doi.org/10.1016/j.chaos.2024.115748 doi: 10.1016/j.chaos.2024.115748
[47]	S. I. Butt, D. Khan, S. Jain, G. I. Oros, P. Agarwal, S. Momani, Fractional integral inequalities for superquadratic functions via Atangana-Baleanu's operator with applications, Fractals, 2024. https://doi.org/10.1142/S0218348X25400687
[48]	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:10.3934/math.2025025 doi: 10.3934/math.2025025
[49]	S. Banić, J. Pečarić, S. Varošanec, Superquadratic functions and refinements of some classical inequalities, J. Korean Math. Soc., 45 (2008), 513–525. https://doi.org/10.4134/jkms.2008.45.2.513 doi: 10.4134/jkms.2008.45.2.513
[50]	D. Khan, S. I. Butt, Y. Seol, Analysis on multiplicatively $(P, m)$-superquadratic functions and related fractional inequalities with applications, Fractals, 33 (2025) 2450129. https://doi.org/10.1142/s0218348x24501299
[51]	D. Khan, S. I. Butt, Y. Seol, Properties and integral inequalities of P-superquadratic functions via multiplicative calculus with applications, Bound. Value Probl., 2024 (2024), 1–40. https://doi.org/10.1186/s13661-024-01978-5 doi: 10.1186/s13661-024-01978-5
[52]	S. I. Butt, D. Khan, Y. Seol, Fractal perspective of superquadratic functions with generalized probability estimations, PloS One, 20 (2025), e0313361. https://doi.org/10.1371/journal.pone.0313361 doi: 10.1371/journal.pone.0313361
[53]	J. Lin, Divergence measures based on the Shannon entropy, IEEE T. Inform. Theory., 37 (1991), 145–151. https://doi.org/10.1109/18.61115 doi: 10.1109/18.61115
[54]	H. Shioya, T. D, Te, A generalization of Lin divergence and the derivation of a new information divergence, Electron. Comm. Jap. 3, 78 (1995), 34–40. https://doi.org/10.1002/ecjc.4430780704 doi: 10.1002/ecjc.4430780704
[55]	I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Stud. Math. Hung., 2 (1967), 299–318. https://doi.org/10.2996/kmj/1138845386 doi: 10.2996/kmj/1138845386
[56]	A. Basu, H. Shioya, C. Park, Statistical inference: The minimum distance approach, in: Chapman & Hall/CRC Monographs on Statistics & Applied Probability, CRC Press, Boca Raton, FL, 2011.
[57]	H. Agahi, M. Yadollahzadeh, A generalization of HH $f$-divergence, J. Comput. Appl. Math., 343 (2018), 309–317. https://doi.org/10.1016/j.cam.2018.04.060 doi: 10.1016/j.cam.2018.04.060
[58]	H. Agahi, M. Yadollahzadeh, Some stochastic HH-divergences in information theory, Aequationes Math., 92 (2018), 1051–1059. https://doi.org/10.1007/s00010-018-0567-2 doi: 10.1007/s00010-018-0567-2

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)