New proofs of weighted power mean inequalities and monotonicity for generalized weighted mean values

Mathematical Inequalities and Applications

Volumn 3, Issue 3, 2000, Pages 377-383

  Qi, Feng ^a   Mei, Jia Qiang ^b   Xia, Da Feng ^c   Xu, Sen Lin ^d  

^a HENAN POLYTECHNIC UNIVERSITY   (China)

^b NANJING UNIVERSITY   (China)

^c FUYANG NORMAL UNIVERSITY   (China)

^d UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

Author keywords

Cauchy Schwarz Buniakowski's inequality; Generalized weighted mean values; Inequality; Monotonicity; Power mean; Weighted mean

Indexed keywords

EID: 0001002556     PISSN: 13314343     EISSN: None     Source Type: Journal    
DOI: 10.7153/mia-03-38     Document Type: Article

References (23)

1. H. ALZER, A proof of the arithmetic mean-geometric mean inequlity. Amer. Math. Monthly 103 (1996), 585.
2. P. S. BULLEN, D. S. MITRINOVIĆ, AND P. M. VASIĆ, Means and Their Inequalities, D. Reidel Publ. Company, Dordrecht/Boston/Lancaster/Tokyo, 1988.
3. JI-CHANG KUANG, Applied Inequalities, 2nd edition, Hunan Education Press, Changsha, China, 1993. (Chinese)
4. E. B. LEACH AND M. C. SHOLANDER, Extended mean values, Amer. Math. Monthly 85 (1978), 84-90.
5. LÁSZLÓ LOSONCZI, Inequalities for integral mean values, J. Math. Anal. Appl. 61 (1977), 586-606.
6. D. S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, Berlin, 1970.
7. D. S. MITRINOVIĆ, J. E. PEČARIC, AND A. M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.
8. J. PEČARIĆ, Nejednakosti, Element, Zagreb, 1996.
9. J. PEČARIĆ, FENG QI, V ŠIMIĆ, AND SEN-LIN XU, Refinements and extensions of an inequality, III, J. Math. Anal. Appl. 227 (1998), no. 2, 439-448.
10. J. PEČARIĆ AND S. VAROŠANEC, A new proof of the arithmetic mean-the geometric mean inequlity, J. Math. Anal. Appl. 215 (1997), 577-578.
11. FENG QI, Generalized weighted mean values with two parameters, Proc. Roy. Soc. London Ser. A 454 (1998), no. 1978, 2723-2732.
12. FENG QI, On a two-parameter family of nonhomogeneous mean values, Tamkang J. Math. 29 (1998), no. 2, 155-163.
13. FENG QI, Generalized abstracted mean values, Journal of Inequalities in Pure and Applied Mathematics 1 (2000), no. 1, Article 4. http://jipam.vu.edu.au/v1n1/013.99.html. RGMIA Research Report Collection 2 (1999), no. 5, Article 4. http://rgmia.vu.edu.au/v2n5.html.
14. FENG QI, Generalized abstracted mean values, Journal of Inequalities in Pure and Applied Mathematics 1 (2000), no. 1, Article 4. http://jipam.vu.edu.au/v1n1/013.99.html. RGMIA Research Report Collection 2 (1999), no. 5, Article 4. http://rgmia.vu.edu.au/v2n5.html.
15. FENG QI, Logarithmic convexities of the extended mean values, RGMIA Research Report Collection 2 (1999), no. 5, Article 5. http.//rgmia.vu.edu.au/v2n5.html.
16. FENG QI AND QIU-MING LUO, A simple proof of monotonicity for extended mean values, J. Math. Anal. Appl. 224 (1998), 356-359.
17. FENG QI, JIA-QIANG MEI, AND SEN-LIN XU, Other proofs of monotonicity for generalized weighted mean values, RGMIA Research Report Collection 2 (1999), no. 4, Article 6. http://rgmia.vu.edu.au/v2n4.html.
18. FENG QI, SEN-LIN XU, AND LOKENATH DEBNATH, A new proof of monotonicity for extended mean values, Intern. J. Math. Math. Sci. 22 (1999), no. 2, 415-420.
19. FENG QI AND SEN-LIN XU, Refinements and extensions of an inequality, II, J. Math. Anal. Appl. 211 (1997), 616-4520.
20. x)/x: Inequalities and properties, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3355-3359.
21. FENG QI AND SHI-QIN ZHANG, Note on monotonicity of generalized weighted mean values, Proc. Roy. Soc. London Ser. A 455 (1999), no. 1989, 3259-3260.
22. DA-FENG XIA, SEN-LIN XU, AND FENG QI, A proof of the arithmetic mean-geometric mean-harmonic mean inequalities, RGMIA Research Report Collection 2 (1999), no. 1, Article 10, 99-102. http://rgmia.vu.edu.au/v2n1.html.
23. REN-ER YANG AND DONG-JI CAO, Generalizations of the logarithmic mean, J. Ningbo Univ. 2 (1989), no. 2, 105-108.