Title: Majorization Inequalities via Peano's Representation of Hermite's Polynomial
Author(s): N. Latif, N. Siddique, J. Pecaric
Pages: 374-399
Cite as:
N. Latif, N. Siddique, J. Pecaric, Majorization Inequalities via Peano's Representation of Hermite's Polynomial, Int. J. Anal. Appl., 16 (3) (2018), 374-399.

Abstract


The Peano's representation of Hermite polynomial and new Green functions are used to construct the identities related to the generalization of majorization type inequalities in discrete as well as continuous case. $\check{C}$eby$\check{s}$ev functional is used to find the bounds for new generalized identities and to develop the Gr$\ddot{u}$ss and Ostrowski type inequalities. Further more, we present exponential convexity together with Cauchy means for linear functionals associated with the obtained inequalities and give some applications.

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References


  1. R. P. Agarwal, S. Iveli´c Bradanovi´c and J. Peˇcari´c, Generalizations of Sherman’s inequality by Lidstone’s interpolating polynomial, J. Inequal. Appl., 2016 (2016), Art. ID 6. Google Scholar

  2. M. Adil Khan, N. Latif and J. Peˇcari´c, Generalization of majorization theorem, J. Math. Inequal., 9(3) (2015), 847-872. Google Scholar

  3. R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer Academic Publishers, Dordrecht/ Boston/ London, 1993. Google Scholar

  4. P. R. Beesack, On the Greens function of an N-point boundary value problem, Pacific J. Math. 12 (1962), 801-812. Kluwer Academic Publishers, Dordrecht / Boston / London, 1993. Google Scholar

  5. S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1–66. Google Scholar

  6. P. Cerone and S. S. Dragomir, Some new Ostrowski-type bounds for theˇCebyˇsev functional and applications, J. Math. Inequal. 8(1) (2014), 159–170. Google Scholar

  7. P. J. Davis, Interpolation and Approximation, Blaisedell Publishing Co., Boston, 1961. Google Scholar

  8. L. Fuchs, A new proof of an inequality of Hardy-Littlewood-Polya, Mat. Tidsskr, B (1947), 53-54. Google Scholar

  9. J. Jakˇseti´c and J. Peˇcari´c, Exponential convexity method, J. Convex Anal. 20(2013), no. 1, 181-197. Google Scholar

  10. J. Jakˇseti´c, J. Peˇcari´c and A. Peruˇsi´c, Steffensen inequality, higher order convexity and exponential convexity, Rend. Circ. Mat. Palermo 63 (1) (2014), 109–127. Google Scholar

  11. N. Mahmood, R. P. Agarwal, S. I. Butt and J. Peˇcari´c, New Generalization of Popoviciu type inequalities via new Green functions and Montgomery identity, J. Inequal. Appl., 2017 (2017), Art. ID 108. Google Scholar

  12. A. W. Marshall, I. Olkin and Barry C. Arnold, Inequalities: Theory of Majorization and Its Applications (Second Edition), Springer Series in Statistics, New York 2011. Google Scholar

  13. J. Peˇcari´c, F. Proschan and Y. L. Tong, Convex functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992. Google Scholar

  14. J. Peˇcari´c and J. Peri´c, Improvements of the Giaccardi and the Petrovi´c inequality and related results, An. Univ. Craiova Ser. Mat. Inform., 39(1) (2012), 65–75. Google Scholar

  15. J. Peˇcari´c, On some inequalities for functions with nondecreasing increments, J. Math. Anal. Appl., 98 (1984), 188-197. Google Scholar

  16. A. Yu. Levin, Some problems bearing on the oscillation of solutions of linear differential equations, Soviet Math. Dokl., 4(1963), 121-124. Google Scholar


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