Startseite Estimations of Levinson-type inequalities using novel 3-convex Green functions with Taylor’s formula
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Estimations of Levinson-type inequalities using novel 3-convex Green functions with Taylor’s formula

  • Awais Rasheed EMAIL logo , Khuram Ali Khan , Josip Pečarić und Đilda Pečarić
Veröffentlicht/Copyright: 14. November 2024
Journal of Applied Analysis
Aus der Zeitschrift Journal of Applied Analysis

Abstract

The goal of this study is to derive the generalized Levinson-type inequalities in the form of Taylor representation for higher order convex functions. This study uses Taylor’s formula and several novel types of 3-convex Green functions, to establish the novel identities associated with Bullen-type inequalities for higher order convex functions. Moreover, various Levinson-type inequalities are proved for positive real weights using Green functions and Taylor’s formula.

Acknowledgements

The authors wish to thank the editor for possible publication.

References

[1] M. Adeel, K. A. Khan, Đ. Pečarić and J. Pečarić, Generalization of the Levinson inequality with applications to information theory, J. Inequal. Appl. 2019 (2019), Paper No. 230. 10.1186/s13660-019-2186-4Suche in Google Scholar

[2] M. Adeel, K. A. Khan, Đ. Pečarić and J. Pečarić, Levinson type inequalities for higher order convex functions via Abel–Gontscharoff interpolation, Adv. Difference Equ. 2019 (2019), Paper No. 430. 10.1186/s13662-019-2360-5Suche in Google Scholar

[3] M. Adeel, K. A. Khan, Đ. Pečarić and J. Pečarić, Estimation of f-divergence and Shannon entropy by Levinson type inequalities via new Green’s functions and Lidstone polynomial, Adv. Difference Equ. 2020 (2020), Paper No. 27. 10.1186/s13662-020-2501-xSuche in Google Scholar

[4] M. Adeel, K. A. Khan, Đ. Pečarić and J. Pečarić, Estimation of f-divergence and Shannon entropy by using Levinson type inequalities for higher order convex functions via Hermite interpolating polynomial, J. Inequal. Appl. 2020 (2020), Paper No. 137. 10.1186/s13660-020-02403-ySuche in Google Scholar

[5] M. Adeel, K. A. Khan, Đ. Pečarić and J. Pečarić, Levinson-type inequalities via new Green functions and Montgomery identity, Open Math. 18 (2020), no. 1, 632–652. 10.1515/math-2020-0163Suche in Google Scholar

[6] M. Adeel, K. A. Khan, Đ. Pečarić and J. Pečarić, Estimation of f-divergence and Shannon entropy by Levinson type inequalities via Lidstone interpolating polynomial, Trans. A. Razmadze Math. Inst. 175 (2021), no. 1, 1–11. 10.1007/s00010-021-00845-3Suche in Google Scholar

[7] M. Adeel, K. A. Khan, Đ. Pečarić and J. Pečarić, Estimation of 𝔣 -divergence and Shannon entropy by Bullen type inequalities via Fink’s identity, Filomat 36 (2022), no. 2, 527–538. 10.2298/FIL2202527ASuche in Google Scholar

[8] R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Math. Appl. 262, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-011-2026-5Suche in Google Scholar

[9] I. Ansari, K. A. Khan, A. Nosheen, Đ. Pečarić and J. Pečarić, Estimation of divergences on time scales via the Green function and Fink’s identity, Adv. Difference Equ. 2021 (2021), Paper No. 394. 10.1186/s13662-021-03550-2Suche in Google Scholar

[10] M. Bilal, K. A. Khan, A. Nosheen and J. Pečarić, Bounds of some divergence measures on time scales via Abel–Gontscharoff interpolation, Math. Slovaca 74 (2024), no. 2, 417–436. 10.1515/ms-2024-0032Suche in Google Scholar

[11] M. Bilal, K. A. Khan, A. Nosheen and J. Pečarić, Bounds of some divergence measures using Hermite polynomial via diamond integrals on time scales, Qual. Theory Dyn. Syst. 23 (2024), no. 2, Paper No. 54. 10.1007/s12346-023-00911-ySuche in Google Scholar

[12] P. S. Bullen, An inequality of N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. (1973), no. 412–460, 109–112. Suche in Google Scholar

[13] S. I. Butt, K. A. Khan and J. Pečarić, Generalization of Popoviciu inequality for higher order convex functions via Taylor polynomial, Acta Univ. Apulensis Math. Inform. 42 (2015), 181–200. 10.17114/j.aua.2015.42.12Suche in Google Scholar

[14] S. I. Butt, K. A. Khan and J. Pečarić, Popoviciu type inequalities via Hermite’s polynomial, Math. Inequal. Appl. 19 (2016), no. 4, 1309–1318. 10.7153/mia-19-96Suche in Google Scholar

[15] S. I. Butt, N. Mehmood and J. Pečarić, New generalizations of Popoviciu type inequalities via new Green functions and Fink’s identity, Trans. A. Razmadze Math. Inst. 171 (2017), no. 3, 293–303. 10.1016/j.trmi.2017.04.003Suche in Google Scholar

[16] S. I. Butt and J. Pečarić, Weighted Popoviciu type inequalities via generalized Montgomery identities, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 19(523) (2015), 69–89. Suche in Google Scholar

[17] S. I. Butt and J. Pečarić, Popoviciu’s Inequality for N-Convex Functions, LAP Lambert Academic, London, 2016. Suche in Google Scholar

[18] N. Latif, N. Siddique and J. Pečarić, Generalization of majorization theorem—II, J. Math. Inequal. 12 (2018), no. 3, 731–752. 10.7153/jmi-2018-12-56Suche in Google Scholar

[19] N. Levinson, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 (1964), 133–134. 10.1016/0022-247X(64)90089-7Suche in Google Scholar

[20] N. Mehmood, R. P. Agarwal, S. I. Butt and J. Pečarić, New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity, J. Inequal. Appl. 2017 (2017), Paper No. 108. 10.1186/s13660-017-1379-ySuche in Google Scholar PubMed PubMed Central

[21] A. M. Mercer, 94.33 short proofs of Jensen’s and Levinson’s inequalities, Math. Gaz. 94 (2010), no. 531, 492–495. 10.1017/S0025557200001820Suche in Google Scholar

[22] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Math. Appl. (East European Ser.) 61, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-017-1043-5Suche in Google Scholar

[23] T. Niaz, K. A. Khan, Đ. Pečarić and J. Pečarić, Estimation of different entropies via Taylor one point and Taylor two points interpolations using Jensen type functionals, Int. J. Anal. Appl. 17 (2019), no. 5, 686–710. Suche in Google Scholar

[24] J. E. Pečarić, On an inequality of N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. (1980), no. 678–715, 71–74. Suche in Google Scholar

[25] J. E. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Math. Sci. Eng. 187, Academic Press, Boston, 1992. Suche in Google Scholar

[26] T. Popoviciu, Sur une inegalite de N. Levinson, Math. (Cluj) 6 (1969), 301–306. Suche in Google Scholar

[27] A. Rasheed, K. A. Khan, J. Pečarić and Đ. Pečarić, Generalizations of Levinson type inequalities via new Green functions with applications to information theory, J. Inequal. Appl. 2023 (2023), Paper No. 124. 10.1186/s13660-023-03040-xSuche in Google Scholar

[28] A. Rasheed, K. A. Khan, J. Pečarić and Đ. Pečarić, Generalizations of Levinson-type inequalities via new Green functions and Hermite interpolating polynomial, J. Inequal. Appl. 2024 (2024), Paper No. 70. 10.1186/s13660-024-03146-wSuche in Google Scholar

Received: 2024-05-04
Revised: 2024-10-08
Accepted: 2024-10-09
Published Online: 2024-11-14

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 21.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jaa-2024-0073/html?lang=de
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