Generalized results on Levinson-type inequalities via Montgomery identity and Green functions

Awais Rasheed, Khuram Ali Khan, Josip Pečarić, Ðilda Pečarić

Abstract


The aim of this study is to obtain new generalizations of Levinson type inequalities for the class of n-convex(n ≥ 3) functions using new Green functions along with Montgomery identity. Some new estimations for novel functionals are derived via Bullen-type inequalities. Furthermore, generalized Levinson-type inequalities are established for positive real weights involving Montgomery identity and Green function.

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References


J. Pečarić, F. Proschan, Y.L. Tong, Convex functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.

N. Levinson, Generalization of an inequality of Kay Fan, Journal of Mathematical Analysis and Applications 6 (1969), 133-134.

T. Popoviciu, Sur une inegalité de N. Levinson, Mathematica (Cluj) 6 (1969), 301-306.

A. Aljinovič, J. Pečarić, A. Vukelič, On some Ostrowski type inequalities via Montgomery identity and Taylor's formula II, Tamkang J. Math. 36 (2005), no. 4, 279-301.

D.S. Mitrinovi_c, J. Pečarić, A.M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.

P.S. Bullen, An inequality of N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. (1973), 109-112.

J. Pečarić, On an inequality on N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. (1980), 71-74.

A.M. Mercer, 94.33 Short proofs of Jensen's and Levinson's Inequalities, The Mathematical Gazette 94 (2010), no. 531, 492-495.

N. Mehmood, R.P. Agarwal, S.I. Butt, J. Pečarić, New generalizations of Popoviciu type inequalities via new Green's functions and Montgomery identity, Journal of Inequalities and Applications 2017 (2017), 1-21.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Levinson-type inequalities via new Green functions and Montgomery identity, Open Mathematics 18 (2020), no. 1, 632-652.

A. Rasheed, K.A. Khan, J. Pečarić, Ð. Pečarić, Generalization of the Levinson inequality via new Green functions with applications to information theory, Journal of Inequalities and Applications 2023 (2023), no. 1, 124.

A. Rasheed, K.A. Khan, J. Pečarić, Ð. Pečarić,. Generalizations of Levinson-type inequalities via new Green functions and Hermite interpolating polynomial, Journal of inequalities and applications 2024 (2024), no. 1, 70.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Generalization of the Levinson inequality with applications to information theory, Journal of Inequalities and Applications 2019(2019), no. 1, 230.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Levinson type inequalities for higher order convex functions via Abel-Gontscharoff interpolation, Advances in Difference equations 2019 (2019), no. 1, 430.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Estimation of f-divergence and Shannon entropy by Levinson type inequalities via new green's functions and Lidstone polynomial, Advances in Difference equations 2020 (2020), no. 1, 27.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Estimation of f-divergence and Shannon entropy by using Levinson type Inequalities for higher order convex functions via Hermite interpolating polynomial, Journal of Inequalities and Applications 2020 (2020), no. 1, 137.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Estimation of f-divergence and Shannon entropy by Bullen type inequalities via Fink's identity, Filomat 36 (2022), no. 2, 527-538.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Entropy results for Levinson-type inequalities via Green functions and Hermite interpolating polynomial, Aequationes mathematicae 96 (2022), no. 1, 1-16.

M. Adeel, K.A. Khan, Ð. Pečarić, J. Pečarić, Estimation of f-divergence and shannon entropy by levinson type inequalities via lidstone interpolating polynomial, Transactions of A. Razmadze Mathematical Institute 175 (2021), no. 1, 1-11.

A. Rasheed, K.A. Khan, J. Pečarić, Ð. Pečarić, Estimations of Levinson-type inequalities using novel 3-convex Green functions with Taylor's formula, Journal of Applied Analysis 31 (2024), no. 2.

M. Bilal, K.A. Khan, A. Nosheen, J. Pečarić, Generalizations of Shannon type inequalities via diamond integrals on time scales, Journal of Inequalities and Applications 2023 (2023), no. 1, 24.

M. Bilal, K.A. Khan, A. Nosheen, J. Pečarić, Some inequalities related to Csiszár divergence via diamond integral on time scales, Journal of Inequalities and Applications 2023 (2023), no. 1, 55.




DOI: https://doi.org/10.52846/ami.v53i1.2132