A Concise proof for properties of three functions involving the exponential function?

Applied Mathematics E - Notes

Volumn 9, Issue , 2009, Pages 177-183

  Zhang, Shi Qin ^a   Guo, Bai Ni ^b   Qi, Feng ^b  

^a NANYANG NORMAL UNIVERSITY   (China)

^b HENAN POLYTECHNIC UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 68949156248     PISSN: 16072510     EISSN: 16072510     Source Type: Journal    
DOI: None     Document Type: Article

References (21)

1. P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman Limited, 1998.
2. Ch.-P. Chen and F. Qi, Best constant in an inequality connected with exponential functions, Octogon Math. Mag., 12(2)(2004), 736-737.
3. nd ed., Problem Books in Mathematics, Springer, New York, 2001.
4. B.-N. Guo, A.-Q. Liu and F. Qi, Monotonicity and logarithmic convexity of three functions involving exponential function, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., 15(4)(2008), 387-392.
5. B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms (2009), in press; Available online at http://dx.doi.org/10.1007/s11075-008-9259-7.
6. S. Guo and F. Qi, A class of completely monotonic functions related to the remainder of Binet's formula with applications, Tamsui Oxf. J. Math. Sci., 25(1)(2009), in press.
7. rd ed., Shandong Science and Technology Press, Ji'nan City, Shandong Province, China, 2004. (Chinese)
8. A.-Q. Liu, G.-F. Li, B.-N. Guo and F. Qi, Monotonicity and logarithmic concavity of two functions involving exponential function, Internat. J. Math. Ed. Sci. Tech., 39(5)(2008), 686-691.
9. F. Qi, A monotonicity result of a function involving the exponential function and an application, RGMIA Res. Rep. Coll., 7(3)(2004), Art. 16, 507-509; Available online at http://www.staff.vu.edu.au/rgmia/v7n3.asp.
10. F. Qi, P. Cerone, S. S. Dragomir and H. M. Srivastava, Alternative proofs for monotonic and logarithmically convex properties of one-parameter mean values, Appl. Math. Comput., 208(1)(2009), 129-133.
11. F. Qi and B.-N. Guo, Logarithmic convexities of Gini's mean values, Available online at http://arxiv.org/abs/0903.1208.
12. x: Logarithmic convexity and applications to extended mean values, Available online at http://arxiv.org/abs/0903.1203.
13. F. Qi and B.-N. Guo, Some properties of extended remainder of Binet's first formula for logarithmof gamma function, Math. Slovaca, 60(2010), in press; Available online at http://arxiv.org/abs/0904.1118.
14. x: Logarithmic convexity, RGMIA Res. Rep. Coll., 11(1)(2008), Art. 5; Available online at http://www.staff.vu.edu.au/rgmia/v11n1. asp.
15. F. Qi, Z.-L. Wei and Q. Yang, Generalizations and refinements of Hermite-Hadamard's inequality, Rocky Mountain J. Math., 35(1)(2005), 235-251. RGMIA Res. Rep. Coll., 5(2)(2002), Art. 10, 337-349; Available online at http://www.staff.vu.edu.au/rgmia/v5n2.asp.
16. F. Qi and S.-L. Xu, Refinements and extensions of an inequality, II, J. Math. Anal. Appl., 211(2)(1997), 616-620.
17. x: Inequalities and properties, Proc. Amer. Math. Soc., 126(11)(1998), 3355-3359.
18. F. Qi and M.-L. Yang, Comparisons of two integral inequalities with Hermite-Hadamard-Jensen's integral inequality, Internat. J. Appl. Math. Sci., 3(1)(2006), 83-88.
19. Octogon Math. Mag., 14(1)(2006), 53-58.
20. RGMIA Res. Rep. Coll., 8(3)(2005), Art. 18, 535-540; Available online at http://www.staff.vu.edu.au/rgmia/v8n3.asp.
21. Y.-D. Wu and Zh.-H. Zhang, The best constant for an inequality, Octogon Math. Mag., 12(1)(2004), 139-141. RGMIA Res. Rep. Coll., 7(1)(2004), Art. 19; Available online at http://www.staff.vu.edu.au/rgmia/v7n1.asp.