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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 46 Issue: 4

On converses of some comparison inequalities for homogeneous means

Zhen-hang Yang
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Yayın Tarihi
2017-08-01
DOI
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In this paper, the necessary and suffcient conditions for the converses of comparison inequalities for Stolarsky means and for Gini means to hold are proved, and the necessary and suffcient conditions for some companion inequalities for bivariate means to hold are given, which unify, generalize and improve known results.

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Amaç & Hipotez:In this paper, the necessary and suffcient conditions for the converses of comparison inequalities for Stolarsky means and for Gini means to hold are proved, and the necessary and suffcient conditions for some companion inequalities for bivariate means to hold are given, which unify, generalize and improve known results.
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Temel Bulgular & Veri:In this paper, the necessary and suffcient conditions for the converses of comparison inequalities for Stolarsky means and for Gini means to hold are proved, and the necessary and suffcient conditions for some companion inequalities for bivariate means to hold are given, which unify, generalize and improve known results.
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Sonuç & Katkı:In this paper, the necessary and suffcient conditions for the converses of comparison inequalities for Stolarsky means and for Gini means to hold are proved, and the necessary and suffcient conditions for some companion inequalities for bivariate means to hold are given, which unify, generalize and improve known results.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Stolarsky means · Gini means · Inequality

Abstract

In this paper, the necessary and suffcient conditions for the converses of comparison inequalities for Stolarsky means and for Gini means to hold are proved, and the necessary and suffcient conditions for some companion inequalities for bivariate means to hold are given, which unify, generalize and improve known results.

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Bu Makaleye Atıf Yap

Yang, Z. (2017). On converses of some comparison inequalities for homogeneous means. Hacettepe Journal of Mathematics and Statistics, Volume: 46 Issue: 4. https://ajindex.com/articles/1403020

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Kaynakça (43)

  1. Alzer, H. Aufgabe 987, Elem. Math. 43, 93, 1988. (German)
  2. Brenner, J. L. A unied treatment and extension of some means of classical analysisI: Comparison theorems, J. Combin. Inform. System Sci. 3, 175-199,1978.
  3. Burk, F. By all means, Amer. Math. Monthly 92, 50, 1985.
  4. Carlson, B. C. The logarithmic mean, Amer. Math. Monthly 79, 615-618, 1972.
  5. Czinder P. and Zs. Páles, Some comparison inequalities for Gini and Stolarsky means, Math. Inequal. Appl. 9(4), 607616, 2006.
  6. Dodd, E. L. Some generalizations of the logarithmic mean and of similar means of two variates which become indeterminate when the two variates are equal, Ann. Math. Statist. 12, 422-428, 1971.
  7. Du, H.-X. Some inequalities for bivariate means, Commun. Korean Math. Soc. 24(4), 553559, 2009; available online at http://icms.kaist.ac.kr/mathnet/thesis_file/08_C09-025.pdf:
  8. Gini, C. Diuna formula comprensiva delle media, Metron 13, 322, 1938.
  9. Peter A. Hästö, A Monotonicity property of ratios of symmetric homogeneous means, J. Inequal. Pure Appl. Math. 3(5), Art. 71, 2002
  10. Jia G. and Cao, J.-D. A new upper bound of the logarithmic mean, J. Inequal. Pure Appl. Math. 4(4), Art. 80, 2003;
  11. Leach E. B. and Sholander, M. C. Extended mean values, Amer. Math. Monthly 85, 84-90, 1978.
  12. Leach E. B. and Sholander, M. C. Extended mean values II, J. Math. Anal. Appl. 92, 207-223, 1983.
  13. Lin, T.-P. The power mean and the logarithmic mean, Amer. Math. Monthly 81, 879883, 1974.
  14. Losonczi, L. On the Comparison of Cauchy Mean Values, J. Inequal. Appl. 7(1), 11-24, 2002.
  15. Neuman E. and Sándor, J. On certain means of two arguments and their extensions, Intern. J. Math. Math. Sci. 2003(16), 981-983, 2003.
  16. Neuman E. and Sándor, J. Inequalities involving Stolarsky and Gini means, Math. Pannonica 14(1), 29-44, 2003.
  17. Neuman E. and Páles, Zs. On comparison of Stolarsky and Gini means, J. Math. Anal. Appl. 278(2), 274-284, 2003.
  18. Neuman, E. A generalization of an inequality of Jia and Cau, J. Inequal. Pure Appl. Math. 5(1), Art. 15, 2004; available online at http://jipam.vu.edu.au/images/010_04_JIPAM/010_04.pdf:
  19. Neuman E. and Sándor, J. Companion inequalities for certain bivariate means, Appl. Anal. Discrete Math. 3(1), 46-51, 2009; available online at http://www.doiserbia.nb.rs/img/doi/1452-8630/2009/1452-86300901046N.pdf.
  20. Páles, Zs. Inequalities for sums of powers, J. Math. Anal. Appl. 131, 265-270, 1988.
  21. Páles, Zs. Inequalities for differences of powers, J. Math. Anal. Appl. 131, 271-281, 1988.
  22. Páles, Zs. On comparison of homogeneous means, Annales Univ. Sci. 32, 261-266, 1989
  23. Páles, Zs. Comparison of two variables homogeneous means, Inter. Ser. Num. Math. 103, 59-70, 1992.
  24. Pittinger, A. O. Inequalities between arithmetic and logarithmic means, Univ. Beogard Publ. Elektr. Fak. Ser. Mat. Fiz, 680, 15-18, 1980.
  25. Sándor, J. A note on some inequalities for means, Arch. Math. (Basel) 56(5), 471-473, 1991.
  26. Sándor, J. On certain identities for means, Studia Univ. Babes-Bolyai, Math. 38(4), 7-14, 1993.
  27. Sándor, J. On certain inequalities for means, J. Math. Anal. Appl. 189, 602606, 1995.
  28. Sándor, J. On certain inequalities for means, II, J. Math. Anal. Appl. 199(1996), no. 2, 629-635.
  29. Sándor J. and Rasa, I. Inequalities for certain means in two arguments, Nieuw Arch. Wisk. 15, 5155, 1997.
  30. Stolarsky, K. B. Generalizations of the Logarithmic Mean, Math. Mag. 48, 87-92, 1975.
  31. Stolarsky, K. B. The power and generalized logarithmic means, Amer. Math. Monthly 87, 545548, 1980,.
  32. Witkowski, A. Comparison theorem for generalization of Stolarsky means, RGMIA Res. Rep. Coll. 8(1), Art. 6, 2005; available online at http://rgmia.vu.edu.au/v8n1.html.
  33. Witkowski, A. Comparison theorem for two-parameter means, Math. Inequal. Appl., 12(1), 11-20, 2009.
  34. Yang, Zh.-H. Some identities for means and applications, RGMIA Res. Rep. Coll. 8(3), Art. 17, 2005; available online at http://rgmia.vu.edu.au/v8n3.html.
  35. Yang, Zh.-H. ON the homogeneous functions with two parameters and its monotonicity, J. Inequal. Pure Appl. Math. 6(4), Art. 101, 2005; available online at http://jipam.vu.edu.au/images/155_05_JIPAM/155_05.pdf:
  36. Yang, Zh.-H. ON the log-convexity of two-parameter homogeneous functions, Math. Inequal. Appl. 10(3), 499-516, 2007.
  37. Yang, Zh.-H. On the monotonicity and log-convexity of a four-parameter homogeneous mean, J. Inequal. Appl. 2008, Art. ID 149286, 2008; available online at http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/149286.
  38. Yang, Zh.-H. Some monotonictiy results for the ratio of two-parameter symmetric homogeneous functions, Int. J. Math. Math. Sci. 2009, Art. ID 591382, 12 pages, 2009. doi:10.1155/2009/591382; available online at http://www.hindawi.com/journals/ijmms/2009/591382.html.
  39. Yang, Zh.-H. Log-convexity of ratio of the two-parameter symmetric homogeneous functions and an application, J. Inequal. Spec. Func. 1(1), 16-29, 2010; available online at http://www.ilirias.com
  40. Yang, Zh.-H. The log-convexity of another class of one-parameter means and its applications, Bull. Korean Math. Soc. 49(1), 33-47, 2012; available online at http://dx.doi.org/10.4134/BKMS.2012.49.1.033.
  41. Yang, Zh.-H. New sharp bounds for identric mean in terms of logarithmic mean and arithmetic mean, J. Math. Inequal. 6(4), 533543, 2012, doi:10.7153/jmi-06-5; available online at http://files.ele-math.com/articles/jmi-06-51.pdf.
  42. Yang, Zh.-H. The monotonicity results for the ratio of certain mixed means and their applications, Int. J. Math. Math. Sci. 2012, Art. ID 540710, 13 pages, 2012, doi:10.1155/2012/540710; available online at http://www.hindawi.com/journals/ijmms/2012/540710/
  43. Stolarsky, K. B. The power and generalized logarithmic means, Amer. Math. Monthly 87, 545548, 1980,.