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On the Approximation and Bounds of the Gini Mean Difference

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Abstract

A variety of mathematical inequalities are utilised to obtain approximation and bounds of the Gini mean difference. The Gini mean difference or the related index is a widely used measure of inequality in numerous areas such as health, finance and population attributes arenas. The paper provides a review of recent developments in the area with an emphasis on work with which the author has been involved.

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References

  1. Cerone, P.: On some generalisations of Steffensen’s inequality and related results. J. Ineq. Pure and Appl. Math. 2(3), Art. 28 (2001). [ONLINE http://jipam.vu.edu.au/v2n3/]

  2. Cerone, P.: On an identity for the Chebychev functional and some ramifications. J. Ineq. Pure and Appl. Math. 3(1), Art. 4 (2002). [ONLINE http://jipam.vu.edu.au/v3n1/]

  3. Cerone, P.: On Gini Mean Difference Bounds Via Generalised Iyengar Results, Submitted.

    Google Scholar 

  4. Cerone, P.: Bounding the Gini mean difference, Inequalities and Applications. In: Bandle, C., Losonczi, A., Pales, Zs., Plum, M. (eds.) International Series of Numerical Mathematics, vol. 157, pp. 77–89, Birkhüser Verlag, Basel (2008)

    Google Scholar 

  5. Cerone, P.: On Young’s Inequality and its Reverse for Bounding the Lorenz Curve and Gini mean. J. Math. Inequalities 3(3), 369–381 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cerone, P., Dragomir, S.S.: Bounds for the Gini mean difference via the Sonin identity. Comp. Math. Modelling 50, 599–609 (2005)

    MATH  MathSciNet  Google Scholar 

  7. Cerone, P., Dragomir, S.S.: Bounds for the Gini mean difference via the Korkine identity. J. Appl. Math. & Computing (Korea) 22(3), 305–315 (2006)

    Google Scholar 

  8. Cerone, P., Dragomir, S.S.: Bounds for the Gini mean difference of continuous distributions defined on finite intervals (II). Comput. Math. Appl. 52(10–11) 1555–1562 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cerone, P., Dragomir, S.S.: Bounds for the Gini mean difference of continuous distributions defined on finite intervals (I). Appl. Math. Lett. 20, 782–789 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cerone, P., Dragomir, S.S.: A survey on bounds for the Gini Mean Difference. In: Barnett, N.S., Dragomir, S.S. (eds.) Advances in Inequalities from Probability Theory and Statistics. pp. 81–111. Nova Science Publishers, New York (2008)

    Google Scholar 

  11. Cheng, X.-L., Sun, J.: A note on the perturbed trapezoid inequality. J. Inequal. Pure & Appl. Math. 3(2) (2002) Article. 29 [ONLINE http://jipam.vu.edu.au/article.php?sid=181]

  12. David, H.A., Gini’s mean difference rediscovered. Biometrika, 55, 573 (1968)

    MATH  Google Scholar 

  13. Gastwirth, J.L.: The estimation of the Lorentz curve and Gini index. Rev. Econom. Statist. 54 305–316 (1972)

    Article  MathSciNet  Google Scholar 

  14. Gini, C.: Variabilità e Metabilità, contributo allo studia della distribuzioni e relationi statistiche. Studi Economica-Gicenitrici dell’ Univ. di Coglani 3, 1–158 (1912) art 2

    Google Scholar 

  15. Giorgi, G.M.: Alcune considerazioni teoriche su di un vecchio ma per sempre attuale indice: il rapporto di concentrazione del Gini, Metron, XLII(3–4), 25–40 (1984)

    Google Scholar 

  16. Giorgi, G.M.: Bibliographic portrait of the Gini concentration ratio. Metron XLVIII(1–4), 103–221 (1990)

    Google Scholar 

  17. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge

    Google Scholar 

  18. Iyengar, K.S.K.: Note on an inequality. Math. Student 6, 75–76 (1938)

    MATH  Google Scholar 

  19. Karamata, J.: O prvom stavu srednjih vrednosti odredenih integrala. Glas Srpske Kraljevske Akademje CLIV, 119–144 (1933)

    Google Scholar 

  20. Karamata, J.: Sur certaines inégalités relatives aux quotients et à la différence de ∫ f g et ∫ f ⋅ ∫ g, Acad. Serbe Sci. Publ. Inst. Math. 2, 131–145 (1948)

    Google Scholar 

  21. Kendall, M., Stuart, A.: The Advanced Theory of Statistics, Distribution Theory, vol. 1, 4th edn. Charles Griffin & Comp. Ltd., London (1977)

    Google Scholar 

  22. Lupaş, A.: On two inequalities of Karamata. Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. 602–633, 119–123 (1978)

    Google Scholar 

  23. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, London (1993)

    Book  MATH  Google Scholar 

  24. Steffensen, J.F.: On certain inequalities between mean values and their application to actuarial problems. Skandinavisk Aktuarietidskrift 1, 82–97 (1918)

    Google Scholar 

  25. Witkowski, A.: On Young’s inequality. J. Ineq. Pure and Appl. Math. 7(5), Art. 164 (2007). [ONLINE http://jipam.vu.edu.au/article.php?sid=782]

  26. Young, W.H.: On classes of summable functions and their Fourier series. Proc. Roy. Soc. London (A), 87, 225–229 (1912)

    Google Scholar 

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Acknowledgements

Most of the work for this article was undertaken while at Victoria University, Melbourne Australia.

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Authors and Affiliations

  1. La Trobe University, Melbourne, Victoria, Australia

    Pietro Cerone

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  1. Pietro Cerone
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Correspondence to Pietro Cerone .

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Editors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Dedicated to Professor Hari M. Srivastava

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Cerone, P. (2014). On the Approximation and Bounds of the Gini Mean Difference. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_17

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