Some Refinements of Integral Inequalities over Triangular Fuzzy Co-Domain

Muhammad Bilal Khan; Loredana Ciurdariu

doi:10.31181/sor31202626

Authors

Muhammad Bilal Khan Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov 500036, Romania Author https://orcid.org/0000-0001-7450-8067
Loredana Ciurdariu Departembt If Mathematica, Politehnica University of Timișoara,300006, Timisoara, Romania Author https://orcid.org/0000-0002-1234-7939

DOI:

https://doi.org/10.31181/sor31202626

Keywords:

Lp space over triangular fuzzy number, Fuzzy domain, Triangular Hölder-like inequality, Triangular Minkowski’s-like inequality, Triangular Beckenbach’s-like inequality

Abstract

Integral inequalities, in general, serve as powerful tools for various applications. Specifically, when an integral operator is used as a predictive tool, an integral inequality can play a key role in defining, quantifying, and analyzing such processes. Real-valued functions over fuzzy domain, also referred to as real-valued functions, offer a valuable approach for incorporating uncertainty into prediction models. In this paper, using a straightforward proof method over newly defined triangular fuzzy space, we established several new refinements for integral forms of classical Hölder’s and newly defined triangular Hölder’s-like inequality. Numerous existing inequalities linked with triangular Hölder's-like inequality over fuzzy domain can be improved through the newly obtained ones, as illustrated through applications like triangular Hölder’s power-mean-like integral inequality, triangular Cauchy-Schwarz-like inequality, triangular Minkowski’s-like inequality, and triangular Beckenbach’s-like inequality over fuzzy domain. Our main results Additionally, our outcomes represent significant progressions in the field of mathematics.

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References

Todinov, M. (2021). Interpretation of algebraic inequalities: Practical engineering optimisation and generating new knowledge. CRC Press. https://doi.org/10.1201/9781003184470

Qin, Y. (2016). Integral and discrete inequalities and their applications. Birkhäuser. https://doi.org/10.1007/978-3-319-32921-8

Khan, M. B., Santos-García, G., Noor, M. A., & Soliman, M. S. (2022). Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities. Chaos, Solitons & Fractals, 164, 112692. https://doi.org/10.1016/j.chaos.2022.112692

Khan, M. B., Catas, A., Aloraini, N., & Soliman, M. S. (2023). Some certain fuzzy fractional inequalities for up and down ℏ-pre-invex via fuzzy-number valued mappings. Fractal and Fractional, 7(2), 171. https://doi.org/10.3390/fractalfract7020171

Zhang, T., Deng, F., & Shi, P. (2023). Non-fragile finite-time stabilization for discrete mean-field stochastic systems. IEEE Transactions on Automatic Control. https://doi.org/10.1109/TAC.2023.3266789

Jiang, X., Wang, Y., Zhao, D., & Shi, L. (2024). Online Pareto optimal control of mean-field stochastic multi-player systems using policy iteration. Science China Information Sciences, 67(4), 1–17. https://doi.org/10.1007/s11432-023-3821-9

Jia, G., Luo, J., Cui, C., Kou, R., Tian, Y., & Schubert, M. (2023). Valley quantum interference modulated by hyperbolic shear polaritons. Physical Review B, 109(15), 155417. https://doi.org/10.1103/PhysRevB.109.155417

Ullah, N., Khan, M. B., Aloraini, N., & Treanțǎ, S. (2023). Some new estimates of fixed point results under multi-valued mappings in G-metric spaces with application. Symmetry, 15(3), 517. https://doi.org/10.3390/sym15030517

Tian, F., Liu, Z., Zhou, J., Chen, L., & Feng, X. T. (2024). Quantifying post-peak behavior of rocks with type-I, type-II, and mixed fractures by developing a quasi-state-based peridynamics. Rock Mechanics and Rock Engineering, 57(1), 1–37. https://doi.org/10.1007/s00603-023-03688-3

Guo, S., Zuo, X., Wu, W., Yang, X., Zhang, J., Li, Y., ... & Zhu, S. (2024). Mitigation of tropospheric delay induced errors in TS-InSAR ground deformation monitoring. International Journal of Digital Earth, 17(1), 2316107. https://doi.org/10.1080/17538947.2024.2316107

Guo, J., Liu, Y., Zou, Q., Ye, L., Zhu, S., & Zhang, H. (2023). Study on optimization and combination strategy of multiple daily runoff prediction models coupled with physical mechanism and LSTM. Journal of Hydrology, 624, 129969. https://doi.org/10.1016/j.jhydrol.2023.129969

Chang, X., Guo, J., Qin, H., Huang, J., Wang, X., & Ren, P. (2024). Single-objective and multi-objective flood interval forecasting considering interval fitting coefficients. Water Resources Management, 38(1), 1–20. https://doi.org/10.1007/s11269-023-03689-5

Lin, Q. (2019). Jensen inequality for superlinear expectations. Statistics & Probability Letters, 151, 79–83. https://doi.org/10.1016/j.spl.2019.03.012

White III, C. C., & Harrington, D. P. (1980). Application of Jensen’s inequality to adaptive suboptimal design. Journal of Optimization Theory and Applications, 32(1), 89–99. https://doi.org/10.1007/BF00934554

Mitrinović, D. S., Pečarić, J. E., & Fink, A. M. (1993). Classical and new inequalities in analysis. Kluwer Academic. https://doi.org/10.1007/978-94-017-1043-5

Mesiar, R., Li, J., & Pap, E. (2010). The Choquet integral as Lebesgue integral and related inequalities. Kybernetika, 46(6), 1098–1107. https://doi.org/10.1007/s00034-010-9215-3

Pap, E., & Štrboja, M. (2010). Generalization of the Jensen inequality for pseudo-integral. Information Sciences, 180(4), 543–548. https://doi.org/10.1016/j.ins.2009.11.012

Pečarić, J. E., & Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Elsevier Science.

Puri, M. L., & Ralescu, D. A. (1983). Differentials of fuzzy functions. Journal of Mathematical Analysis and Applications, 91(2), 552–558. https://doi.org/10.1016/0022-247X(83)90169-5

Román-Flores, H., Flores-Franulic, A., & Chalco-Cano, Y. (2007). A Jensen type inequality for fuzzy integrals. Information Sciences, 177(15), 3192–3201. https://doi.org/10.1016/j.ins.2007.02.007

Štrboja, M., Grbic, T., Štajner-Papuga, I., Grujic, G., & Medic, S. (2013). Jensen and Chebyshev inequalities for pseudo-integrals of set-valued functions. Fuzzy Sets and Systems, 222, 18–32. https://doi.org/10.1016/j.fss.2012.12.004

Wang, R. S. (2011). Some inequalities and convergence theorems for Choquet integral. Journal of Applied Mathematics and Computing, 35(1), 305–321. https://doi.org/10.1007/s12190-010-0389-4

Zhao, X., & Zhang, Q. (2011). Hölder type inequality and Jensen type inequality for Choquet integral. In Knowledge engineering and management: Proceedings of the sixth international conference on intelligent systems and knowledge engineering, Shanghai, China, Dec 2011 (ISKE2011) (pp. 1–10). Springer. https://doi.org/10.1007/978-3-642-25661-5_1

Costa, T. M., & Román-Flores, H. (2017). Some integral inequalities for fuzzy-interval-valued functions. Information Sciences, 420, 110–125. https://doi.org/10.1016/j.ins.2017.08.045

Khan, M. B., Mohammed, P. O., Noor, M. A., & Hamed, Y. S. (2021). New Hermite–Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities. Symmetry, 13(4), 673. https://doi.org/10.3390/sym13040673

Khan, M. B., Mohammed, P. O., Noor, M. A., & Abuahalnaja, K. (2021). Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions. Mathematical Biosciences and Engineering, 18(6), 6552–6580. https://doi.org/10.3934/mbe.2021325

Zhao, D. F., An, T. Q., Ye, G. J., & Liu, W. (2020). Chebyshev type inequalities for interval-valued functions. Fuzzy Sets and Systems, 396, 82–101. https://doi.org/10.1016/j.fss.2019.07.012