DOI QR코드

DOI QR Code

THE LOG-CONVEXITY OF ANOTHER CLASS OF ONE-PARAMETER MEANS AND ITS APPLICATIONS

  • Yang, Zhen-Hang (System Division Zhejiang Province Electric Power Test and Research Institute)
  • Received : 2009.12.11
  • Published : 2012.01.31

Abstract

In this paper, the log-convexity of another class one-parameter mean is investigated. As applications, some new upper and lower bounds of logarithmic mean, new estimations for identric mean and new inequalities for power-exponential mean and exponential-geometric mean are first given.

Keywords

References

  1. H. Alzer, Ungleichungen fur Mittelwerte, Arch. Math. 47 (1986), no. 5, 422-426. https://doi.org/10.1007/BF01189983
  2. H. Alzer, Two inequalities for means, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no.1, 11-16.
  3. H. Alzer, Aufgabe 987, Elem. Math. 43 (1988), 93.
  4. H. Alzer, Uer eine einparametrige familie von Mitlewerten, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber 1987 (1988), 23-29.
  5. H. Alzer, Uer eine einparametrige familie von Mitlewerten. II, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber 1988 (1989), 23-29.
  6. H. Alzer, On Stolarsky's mean value family, Int. J. Math. Educ. Sci. Technol. 20 (1989), no. 1, 186-189.
  7. J. L. Brenner, A unified treatment and extension of some means of classical analysis I. Comparison theorems, J. Combin. Inform. System Sci. 3 (1978), no. 4, 175-199.
  8. F. Burk, By all means, Amer. Math. Monthly 92 (1985), no. 1, 50. https://doi.org/10.1080/00029890.1985.11971533
  9. B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 17 (1966), 32-39. https://doi.org/10.1090/S0002-9939-1966-0188497-6
  10. B. C. Carlson, The logarithmic mean, Amer. Math. Monthly 79 (1972), 615-618. https://doi.org/10.2307/2317088
  11. W.-S. Cheung and F. Qi, Logarithmic convexity of the one-parameter mean values, Taiwanese J. Math. 11 (2007), no. 1, 231-237. https://doi.org/10.11650/twjm/1500404648
  12. E. L. Dodd, Some generalizations of the logarithmic mean and of similar means of two variates which become indeterminate when the two variates are equal, Ann. Math. Stat. 12 (1941), 422-428. https://doi.org/10.1214/aoms/1177731680
  13. C. Gini, Diuna formula comprensiva delle media, Metron 13 (1938), 3-22.
  14. P. A. Hasto, A monotonicity property of ratios of symmetric homogeneous means, J. Inequal. Pure Appl. Math. 3 (2002), no. 5, Article 71, 23 pp.
  15. G. Jia and J.-D. Cao, A new upper bound of the logarithmic mean, J. Inequal. Pure Appl. Math. 4 (2003), no. 4, Article 80, 4 pp.
  16. O. Kouba, New bounds for the identric mean of two arguments, J. Inequal. Pure Appl. Math. 9 (2008), no. 3, Article 71, 6 pp.
  17. E. B. Leach and M. Sholander, Extended mean values, Amer. Math. Monthly 85 (1978), no. 2, 84-90. https://doi.org/10.2307/2321783
  18. E. B. Leach and M. Sholander, Extended mean values II, J. Math. Anal. Appl. 92 (1983), no. 1, 207-223. https://doi.org/10.1016/0022-247X(83)90280-9
  19. T. P. Lin, The power mean and the logarithmic mean, Amer. Math. Monthly 81 (1974), 879-883. https://doi.org/10.2307/2319447
  20. E. Neuman and J. Sandor, On certain means of two arguments and their extensions, Int. J. Math. Math. Sci. 2003 (2003), no. 16, 981-993. https://doi.org/10.1155/S0161171203208103
  21. E. Neuman and J. Sandor, Inequalities involving Stolarsky and Gini means, Math. Pannon. 14 (2003), no. 1, 29-44.
  22. E. Neuman, A generalization of an inequality of Jia and Cau, J. Inequal. Pure Appl. Math. 5 (2004), no. 1, Article 15, 4 pp.
  23. B. Ostle and H. L. Terwilliger, A comparison of two means, Proc. Montana Acad. Sci. 17 (1957), 69-70.
  24. Zs. Pales, Inequalities for sums of powers, J. Math. Anal. Appl. 131 (1988), no. 1, 265-270. https://doi.org/10.1016/0022-247X(88)90204-1
  25. Zs. Pales, Inequalities for differences of powers, J. Math. Anal. Appl. 131 (1988), no. 1, 271-281. https://doi.org/10.1016/0022-247X(88)90205-3
  26. A. O. Pittenger, Inequalities between arithmetic and logarithmic means, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 678-715 (1980), 15-18.
  27. F. Qi, Logarithmic convexity of extended mean values, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1787-1796. https://doi.org/10.1090/S0002-9939-01-06275-X
  28. F. Qi, On a two-parameter family of nonhomogeneous mean values, Tamkang J. Math. 29 (1998), no. 2, 155-163.
  29. J. Sandor, On the identric and logarithmic means, Aequationes Math. 40 (1990), no. 2-3, 261-270. https://doi.org/10.1007/BF02112299
  30. J. Sandor, On certain identities for means, Studia Univ. Babes-Bolyai, Math. 37 (1993), no. 4, 7-14.
  31. J. Sandor, On refinements of certain inequalities for means, Arch. Math. (Brno) 31 (1995), no. 4, 279-282.
  32. J. Sandor and T. Trif, Some new inequalities for means of two arguments, Int. J. Math. Math. Sci. 25 (2001), no. 8, 525-532. https://doi.org/10.1155/S0161171201003064
  33. K. B. Stolarsky, Generalizations of the logarithmic mean, Math. Mag. 48 (1975), 87-92. https://doi.org/10.2307/2689825
  34. K. B. Stolarsky, The power and generalized logarithmic means, Amer. Math. Monthly 87 (1980), no. 7, 545-548. https://doi.org/10.2307/2321420
  35. T. Trif, Note on certain inequalities for means in two variables, J. Inequal. Pure Appl. Math. 6 (2005), no. 2, Article 43, 5 pp.
  36. R. Yang and D. Cao, Generalizations of the logarithmic mean, J. Ningbo Univ. 2 (1989), no. 1, 105-108.
  37. Zh.-H. Yang, Simple discriminances of convexity of homogeneous functions and applications, Gaodeng Shuxue Yanjiu (Study in College Mathematics) 4 (2004), no. 7, 14-19.
  38. Zh.-H. Yang, On the monotonicity and log-convexity for one-parameter homogeneous functions, RGMIA Res. Rep. Coll. 8 (2005), no. 2, Art. 14.
  39. Zh.-H. Yang, Some identities for means and applications, RGMIA Res. Rep. Coll. 8 (2005), no. 3, Art. 17.
  40. Zh.-H. Yang, On the homogeneous functions with two parameters and its monotonicity, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 101, 11 pp.
  41. Zh.-H. Yang, On the log-convexity of two-parameter homogeneous functions, Math. Inequal. Appl. 10 (2007), no. 3, 499-516.
  42. Zh.-H. Yang, On the monotonicity and log-convexity of a four-parameter homogeneous mean, J. Ine. Appl. 2008 (2008), Art. ID 149286, 12 pages, doi:10.1155/2008/149286.

Cited by

  1. Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean vol.2016, pp.1, 2016, https://doi.org/10.1186/s13660-016-1113-1
  2. SCHUR POWER CONVEXITY OF GINI MEANS vol.50, pp.2, 2013, https://doi.org/10.4134/BKMS.2013.50.2.485
  3. Sharp bounds for the arithmetic-geometric mean vol.2014, pp.1, 2014, https://doi.org/10.1186/1029-242X-2014-192
  4. Three families of two-parameter means constructed by trigonometric functions vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-541