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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 4

Issues

Refinements of the majorization-type inequalities via green and fink identities and related results

Sadia Khalid
  • Department of Mathematics COMSATS University Islamabad Lahore Campus, Islamabad, Pakistan
  • Faculty of Textile Technology University of Zagreb Prilaz baruna Filipovića 28a, 10000, Zagreb, Croatia
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/ Josip Pečarić / Ana Vukelić
Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0144

Abstract

In this work, the Green’s function of order two is used together with Fink’s approach in Ostrowski’s inequality to represent the difference between the sides of the Sherman’s inequality. Čebyšev, Grüss and Ostrowski-type inequalities are used to obtain several bounds of the presented Sherman-type inequality. Further, we construct a new family of exponentially convex functions and Cauchy-type means by looking to the linear functionals associated with the obtained inequalities.

MSC 2010: Primary 26A51; 26D15

Keywords: logarithmic convexity; divided difference; n-convex function; majorization; Sherman’s inequality; Green’s function; Fink’s identity; Čebyšev functional; Grüss-type inequality; Ostrowski-type inequality; exponential convexity

References

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    Cerone, P.—Dragomir, S. S.: Some new Ostrowski-type bounds for the Čebyšev functional and applications, J. Math. Inequal. 8 (2014), 159–170.Google Scholar

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    Fink, A. M.: Bounds of the deviation of a function from its avereges, Czechoslovak Math. J. 42 (1992), 289–310.Google Scholar

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    Khalid, S.—Pečarić, J.—Vukelić, A.: Refinements of the majorization theorems via Fink identity and related results, J. Classical. Anal. 7 (2015), 129–154.Google Scholar

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    Khan, M. A.—Latif, N.—Perić, I.—Pečarić, J.: On majorization for matrices, Math. Balkanica (N.S.) 27 (2013), Fasc 1–2, 3–19.Google Scholar

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    Pečarić, J.: On some inequalities for functions with nondecreasing increments, J. Math. Anal. Appl. 98 (1984), 188–197.CrossrefGoogle Scholar

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    Pečarić, J.-Perić, J.: Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform. 39 (2012), 65–75.Google Scholar

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    Pečarić, J.—Praljak, M.: Hermite interpolation and inequalities involving weighted averages of n-convex functions, Math. Inequal. Appl. 19 (2016), 1169–1180.Web of ScienceGoogle Scholar

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    Pečarić, J.—Proschan, F.—Tong, Y. L.: Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, New York, 1992.Google Scholar

About the article

skhalid@ttf.hr skhalid@ciitlahore.edu.pk


Received: 2016-09-19

Accepted: 2017-04-11

Published Online: 2018-08-06

Published in Print: 2018-08-28


Communicated by Ján Borsík


Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 773–788, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0144.

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