An inequality between ratio of the extended logarithmic means and ratio of the exponential means

Taiwanese Journal of Mathematics

Volumn 7, Issue 2, 2003, Pages 229-237

  Qi, Feng ^a   Guo, Bai Ni ^a  

^a HENAN POLYTECHNIC UNIVERSITY   (China)

Author keywords

Exponential mean; Extended logarithmic mean; Extended mean values; Inequality; Logarithmic mean; One parameter mean; Ratio

Indexed keywords

EID: 0037770285     PISSN: 10275487     EISSN: None     Source Type: Journal    
DOI: 10.11650/twjm/1500575060     Document Type: Article

References (47)

1. H. Alzer, On an inequality of H. Mine and L. Sathre, J. Math. Anal. Appl. 179 (1993), 396-402.
2. P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman Limited, 1998.
3. T. H. Chan, P. Gao and F. Qi, On a generalization of Martins' inequality, RGMIA Res. Rep. Coll. 4 (2001), no.1, Article 12, 93-101. Available online at http://rgmia.vu.edu.au/v4n1.html.
4. S. S. Dragomir and J. van der Hoek, Some new analytic inequalities and their applications in guessing theory, J. Math. Anal. Appl 225 (1998), 542-556.
5. N. Elezović and J. Pecarić, On Alzer's inequality, J. Math. Anal. Appl. 223 (1998), 366-369.
6. L. Galvani, Dei limiti a cui tendono alcune media, Boll. Un. Mat. Ital. (1927), no. 6, 173-179.
7. B. Gavrea and I. Gavrea, An inequality for linear positive functional, J. Inequal. Pure Appl. Math. 1 (2000), no. 1, Article 5. Available online at http://jipam.vu.edu.au/v1n1/004_99.html.
8. B.-N. Guo and F. Qi, An algebraic inequality, II, RGMIA Res. Rep. Coll. 4 (2001), no. 1, Article 8, 55-61. Available online at http://rgmia.vu.edu.au/ v4n1.html.
9. B.-N. Guo, Sh.-Q. Zhang, and F. Qi, Elementary proofs of monotonicity for extended mean values of some functions with two parameters, Shuxue de Shijian yu Renshi (Mathematics in Practice and Theory) 29 (1999), no. 2, 169-174. (Chinese)
10. J.-Ch. Kuang, Some extensions and refinements of Minc-Sathre inequality, Math. Gaz. 83 (1999), 123-127.
11. E. Leach and M. Sholander, Extended mean values, Amer. Math. Monthly 85 (1978), 84-90.
12. J. S. Martins, Arithmetic and geometric means, an applications to Lorentz sequence spaces, Math Nachr. 139 (1988), 281-288.
13. 1/r, Proc. Edinburgh Math. Soc. 14 (1964/65), 41-46.
14. N. Ozeki, On some inequalities, J. College Arts Sci. Chiba Univ. 4 (1965), no. 3, 211-214. (Japanese)
15. F. Qi, An algebraic inequality, J. Inequal. Pure Appl. Math. 2 (2001), no. 1, Article 13. Available online at http://jipam.vu.edu.au/v2n1/006_00.html.
16. RGMIA Res. Rep. Coll. 2 (1999), no. 1, Article 8, 81-83. Available online at http://rgmia.vu.edu.au/v2n1.html.
17. F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl Math. 1 (2000), no. 1, Article 4. Available online at http://jipam.vu.edu.au/v1n1/013_9. html.
18. RGMIA Res. Rep. Coll. 2 (1999), no. 5, Article 4, 633-642. Available online at http://rgmia.vu.edu.au/v2n5.html.
19. F. Qi, Generalizations of Alzer's and Kuang's inequality, Tamkang J. Math. 31 (2000), no. 3, 223-227.
20. RGMIA Res. Rep. Coll. 2 (1999), no. 6, Article 12, 891-895. Available online at http://rgmia.vu.edu.au/v2n6.html.
21. F. Qi, Generalization of H. Alzer's inequality, J. Math. Anal. Appl. 240 (1999), 294-297.
22. F. Qi, Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1978, 2723-2732.
23. n√(n + k)!/k!, RGMIA Res. Rep. Coll. 2 (1999), no. 5, Article 8, 685-692. Available online at http://rgmia.vu.edu.au/v2n5.hrml.
24. F. Qi, Logarithmic convexity of extended mean values, Proc. Amer. Math. Soc. 130 (2002), no.6, 1787-1796.
25. accepted RGMIA Res. Rep. Coll. 2 (1999), no. 5, Article 5, 643-652. Available online at http://rgmia.vu.edu.au/v2n5.html.
26. F. Qi, On a two-parameter family of nonhomogeneous mean values, Tamkang J. Math. 29 (1998), no. 2, 155-163.
27. F. Qi, Studies on Problems in Topology and Geometry and on Generalized Weighted Abstracted Mean Values, Thesis submitted for the degree of Doctor of Science at University of Science and Technology of China, Hefei City, Anhui Province, China, Winter 1998. (Chinese)
28. F. Qi and L. Debnath, On a new generalization of Alzer's inequality, Internat. J. Math. Math. Sci. 23 (2000), no. 12, 815-818.
29. F. Qi and B.-N. Guo, Monotonicity of sequences involving convex function and sequence, RGMIA Res. Rep. Coll. 3 (2000), no. 2, Article 14, 321-329. Available online at http://rgmia.vu.edu.au/v3n2.html.
30. F. Qi and B.-N. Guo, Some inequalities involving the geometric mean of natural numbers and the ratio of gamma functions, RGMIA Res. Rep. Coll. 4 (2001), no. 1, Article 3, 41-48. Available online at http://rgmia.vu.edu.au/ v4n1.html.
31. B.-N. Guo and F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2003), 239-247.
32. F. Qi and Q.-M. Luo, A simple proof of monotonicity for extended mean values, J. Math. Anal. Appl. 224 (1998), no. 2, 356-359.
33. F. Qi and Q.-M.- Luo, Generalization of H. Mine and J. Sathre's inequality, Tamkang J. Math. 31 (2000), no. 2, 145-148.
34. RGMIA Res. Rep. Coll. 2 (1999), no. 6, Article 14, 909-912. Available online at http://rgmia.vu.edu.au/v2n6.html.
35. F. Qi and Q.-M. Luo, Refinements and extensions of an inequality, Mathematics and Informatics Quarterly 9 (1999), no. 1, 23-25.
36. F. Qi, J.-Q. Mei, Da-F. Xia, and S.-L. Xu, New proofs of weighted power mean inequalities and monotonicity for generalized weighted mean values, Math. Inequal. Appl. 3 (2000), no. 3, 377-383.
37. F. Qi, J.-Q. Mei, and S.-L. Xu, Other proofs of monotonicity for generalized weighted mean values, RGMIA Res. Rep. Coll. 2 (1999), no. 4, Article 6, 469-472. Available online at http://rgmia.vu.edu.au/v2n4.html.
38. F. Qi and N. Towghi, Inequalities for the ratios of the mean values of functions, submitted.
39. F. Qi and S.-L. Xu, Refinements and extensions of an inequality, II, J. Math. Anal. Appl. 211 (1997), 616-620.
40. x)/x: Inequalities and properties, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3355-3359.
41. F. Qi, S.-L. Xu, and L. Debnath, A new proof of monotonicity for extended mean values, Internat. J. Math. Math. Sci. 22 (1999), no. 2, 415-420.
42. F. Qi and Sh.-Q. Zhang, Note on monotonicity of generalized weighted mean values, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1989, 3259-3260.
43. J. Sándor, Comments on an inequality for the sum of powers of positive numbers, RGMIA Res. Rep. Coll. 2 (1999), no. 2, 259-261. Available online at http://rgmia.vu.edu.au/v2n2.html.
44. J. Sándor, On an inequality of Alzer, J. Math. Anal. Appl. 192 (1995), 1034-1035.
45. K. B. Stolarsky, Generalizations of the logarithmic mean, Math. Mag. 48 (1975), 87-92.
46. N. Towghi and F. Qi, An inequality for the ratios of the arithmetic means of functions with a positive parameter, RGMIA Res. Rep. Coll. 4 (2001), no. 2, Article 15, 305-309. Available online at http://rgmia.vu.edu.au/v4n2.html.
47. J. S. Ume, An elementary proof of H. Alzer's inequality, Math. Japon. 44 (1996), no. 3, 521-522.