Abstract
Grüss inequality is subject of interest for many authors due to its effectiveness in predicting bounds in several quadrature problems. In the present article, we give weighted treatment of the discrete Čebyšev and Grüss type inequalities pertaining two n-tuples of real numbers in which the bounding constants are mobilised with bounding sequences of real numbers. As an application estimations of discrete Ostrowski type inequalities are provided. Finally, by practicing obtained results along with Jensen’s difference, a wide range of estimations are formalised by considering Jensen-Grüss differences.
Communicated by Marek Balcerzak
References
[1] Aglić Aljinović, A.—Pečarić, J.: Discrete weighted Montgomery identity and discrete Ostrowski type inequalities, Comput. Math. Appl. 48 (2004), 731–745.10.1016/j.camwa.2004.03.004Search in Google Scholar
[2] Andrica, D.—Badea, C.: Grüss’ inequality for positive linear functionals, Period. Math. Hungar. 19(2) (1988), 155–167.10.1007/BF01848061Search in Google Scholar
[3] Butt, S. I.—Bakula, M. K.—Pečarić, J.: Steffensen-Grüss inequality, J. Math. Inequal. 15(2) (2021), 799–810.10.7153/jmi-2021-15-56Search in Google Scholar
[4] Butt, S. I.—Pečarić, Đ.—Pečarić, J.: Several Jensen-Gruss inequalities with applications in information theory, Ukr. Math. J. 74(12) (2023), 1654–1672.10.37863/umzh.v74i12.6554Search in Google Scholar
[5] Butt, S. I.—Bakula, M. K.—Pečarić, Đ.—Pečarić, J.: Jensen-Grüss inequality and its applications for the Zipf-Mandelbrot law, Math. Methods Appl. Sci. 44(2) (2021), 1664–1673.10.1002/mma.6869Search in Google Scholar
[6] Butt, S. I.—Pečarić, J.—Perić, I.—Praljak, M.: Multidimensional reversed Hardy type inequalities for monotone functions, Asian-Eur. J. Math. 7(4) (2014), Art. ID 1450055.10.1142/S1793557114500557Search in Google Scholar
[7] Butt, S. I.—Pečarić, J.—Spužević, S. T.: Generalized Čebyšev and Grüss type results in weighted Lebesgue spaces, Mathematics 11(7) (2023), Art. No. 1756.10.3390/math11071756Search in Google Scholar
[8] Cerone, P.—Dragomir, S. S.: New inequalities for the Čebyšev functional involving two n-tuples of real numbers and applications, RGMIA research report collection 5(3) (2002).Search in Google Scholar
[9] Cerone, P.—Dragomir, S. S.: A refinement of Grüss inequality and applications, RGMI Research Report Collection 5(2) (2000), Art. No. 14.Search in Google Scholar
[10] Grüss, G.: Über das maximum des absoluten Betrages von
[11] Horvath, L.: Grüss type and related integral inequalities in probability spaces, Aequationes Math. 93 (2019), 743–756.10.1007/s00010-018-0612-1Search in Google Scholar
[12] Izumino, S.—Pečarić, J.: Some extensions Of Grüss inequality and its applications, Nihonkai Math. J. 13(2) (2002), 159–166.Search in Google Scholar
[13] Izumino, S.—Pečarić, J.—Tepeš, B.: Some extensions of Grüss inequality, Math. J. Toyama Univ. 26 (2003), 61–73.Search in Google Scholar
[14] Izumino, S.—Pečarić, J.—Tepeš, B.: A Grüss-type inequality and its applications, J. Inequal. Appl. 2005 (2005), Art. ID 918685.10.1155/JIA.2005.277Search in Google Scholar
[15] Li, X.—Mohapatra, R. N.—Rodriguez, R. S.: Grüss-type inequalities, J. Math. Anal. Appl. 267 (2002), 434–443.10.1006/jmaa.2001.7565Search in Google Scholar
[16] Mitrinović, D. S.—Pečarić, J.—Fink, A. M.: Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Boston, London, 1993.10.1007/978-94-017-1043-5Search in Google Scholar
[17] Mitrinović, D. S.—Pečarić, J. E.—Fink, A. M.: Inequalities for Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.Search in Google Scholar
[18] Niezgoda, M.: A new inequality of Ostrowski-Grüss type and Applications to some Numerical Quadrature rules, Comput. Math. Appl. 58 (2009), 589–596.10.1016/j.camwa.2009.03.089Search in Google Scholar
[19] Pečarić, J.—Proschan, F.—Tong, Y. L.: Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.Search in Google Scholar
[20] Pečarić, J.—Tepeš, B.: Improvements of some inequalities for moments of guessing function, Math. Inequal. Appl. 8(1) (2005), 53–62.10.7153/mia-08-05Search in Google Scholar
[21] Qin, Y.: Integral and Discrete Inequalities and their Applications, Birkhauser, Springer International Publishing Switzerland, 2016.Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
- Euclidean operator radius inequalities of d-tuple operators and operator matrices
- Equable parallelograms on the Eisenstein lattice
- On certain star versions of the Hurewicz property using ideals
- Relative versions of star-Menger property
- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
- A new family of copulas based on probability generating functions
- Induced mappings on the hyperspace of totally disconnected sets