Some Equivalent Forms of Bernoulli’s Inequality: A Survey ()

Yuan-Chuan Li^{1}, Cheh-Chih Yeh^{2,3}

^{1}Department of Applied Mathematics, National Chung-Hsing University, Taichung.

^{2}Department of Mathematics, National Central University, Taoyuan.

^{3}Department of Information Management, Lunghwa University of Science and Technology, Taoyuan.

**DOI: **10.4236/am.2013.47146
PDF
HTML
8,052
Downloads
11,950
Views
Citations

The main purpose of this paper is to link some known inequalities which are equivalent to Bernoulli’s inequality.

Share and Cite:

Y. Li and C. Yeh, "Some Equivalent Forms of Bernoulli’s Inequality: A Survey," *Applied Mathematics*, Vol. 4 No. 7, 2013, pp. 1070-1093. doi: 10.4236/am.2013.47146.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | E. F. Beckenbach and R. Bellman, “Inequality,” 4th Edi tion, Springer-Verlag, Berlin, 1984. |

[2] | E. F. Beckenbach and W. Waler, “General Inequalities III,” Birkhauser Verlag, Basel, 1983. |

[3] | P. S. Bullen, “Handbook of Means and Their Inequali ties,” Kluwer Academic Publishers, Dordrecht, 2003. doi:10.1007/978-94-017-0399-4 |

[4] | P. S. Bullen, D. S. Mitrinovic and P. M. Vasic, “Means and Their Inequalities,” D. Reidel Publishing Company, Dordrecht, 1952. |

[5] | P. S. Bullen, “A Chapter on Inequalities,” Southeast Asian Bulletin of Mathematics, Vol. 3, 1979, pp. 8-26. |

[6] | M. J. Cloud and B. C. Drachman, “Inequalities with Ap plications to Engineering,” Springer Verlag, New York, 1998. |

[7] | C. Georgakis, “On the Inequality for the Arithmetic and Geometric Means,” Mathematical Inequalities and Ap plications, Vol. 5, 2002, pp. 215-218. |

[8] | G. Hardy, J. E. Littlewood and G. Pólya, “Inequalities,” 2nd Edition, Cambridge University Press, Cambridge, 1952. |

[9] | Z. Hao, “Note on the Inequality of the Arithmetric and Geometric Means,” Pacific Journal of Mathematics, Vol. 143, No. 1, 1990, pp. 43-46. |

[10] | J. Howard and J. Howard, “Equivalent Inequalities,” The College Mathematics Journal, Vol. 19, No. 4, 1988, pp. 350-354. |

[11] | K. Hu, “Some Problems of Analytic Inequalities (in Chi nese),” Wuhan University Press, Wuhan, 2003. |

[12] | C. A. Infantozzi, “An Introduction to Relations among Inequalities,” Notices of the American Mathematical Society, Vol. 141, 1972, pp. A918-A820. |

[13] | S. Isumino and M. Tominaga, “Estimation in Holder’s Type Inequality,” Mathematical Inequalities and Appli cations, Vol. 4, 2001, pp. 163-187. |

[14] | J. Kuang, “Applied Inequalities (in Chinese),” 3rd Edition, Shandong Science and Technology Press, Shandong, 2004. |

[15] | Y.-C. Li and S. Y. Shaw, “A Proof of Holder’s Inequality Using the Cauchy-Schwarz Inequality,” Journal of Ine qualities in Pure and Applied Mathematics, Vol. 7, No. 2, 2006. |

[16] | C. K. Lin, “Convex Functions, Jensen’s Inequality and Legendre Transformation (in Chinese),” Mathmedia, Academic Sinica, Vol. 19, 1995, pp. 51-57. |

[17] | C. K. Lin, “The Essence and Significance of Cauchy Schwarz’s Inequality (in Chinese),” Mathmedia, Aca demic Sinica, Vol. 24, 2000, pp. 26-42. |

[18] | L. Maligranda, “Why Holder’s Inequality Should Be Called Rogers’ Inequality,” Mathematical Inequalities and Applications, Vol. 1, 1998, pp. 69-83. |

[19] | A. W. Marshall and I. Olkin, “Inequalities: Theory of Majorization and Its Applications,” Academic Press, New York, 1979. |

[20] | D. S. Mitrinovic, “Analytic Inequalities,” Springer-Verlag, Berlin, 1970. |

[21] | D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, “Classical and New Inequalities in Analysis,” Klumer Academic Publisher, Dordrecht, 1993. doi:10.1007/978-94-017-1043-5 |

[22] | D. S. Mitrinovic and J. E. Pevaric, “Bernoulli’s Inequa lity,” Rendiconti del Circolo Matematico di Palermo, Vol. 42, No. 3, 1993, pp. 317-337. |

[23] | D. J. Newman, “Arithmetric, Geometric Inequality,” The American Mathematical Monthly, Vol. 67, No. 9, 1960, p. 886. doi:10.2307/2309460 |

[24] | N. O. Ozeki and M. K. Aoyaki, “Inequalities (in Japa nese),” 3rd Edition, Maki Shoten, Tokyo, 1967. |

[25] | J. E. Pecaric, “On Bernoulli’s Inequality,” Akad. Nauk. Umjet. Bosn. Hercegov. Rad. Odelj. Prirod. Mat. Nauk, Vol. 22, 1983, pp. 61-65. |

[26] | J. Pecaric and K. B. Stolarsky, “Carleman’s Inequality: History and New Generalizations,” Aequationes Mathe maticae, Vol. 61, No. 1-2, 2001, pp. 49-62. doi:10.1007/s000100050160 |

[27] | J. Pecaric and S. Varacance, “A New Proof of the Arith metic Mean—The Geometric Mean Inequality,” Journal of Mathematical Analysis and Applications, Vol. 215, No. 2, 1997, pp. 577-578.doi:10.1006/jmaa.1997.5616 |

[28] | J. Rooin, “Some New Proofs for the AGM Inequality,” Mathematical Inequalities and Applications, Vol. 7, No. 4, 2004, pp. 517-521. |

[29] | N. Schaumberger, “A Coordinate Approach to the AM GM Inequality,” Mathematics Magazine, Vol. 64, No. 4, 1991, p. 273. doi:10.2307/2690837 |

[30] | S.-C. Shyy, “Convexity,” Dalian University of Technolgy Press, Dalian, 2011. |

[31] | X. H. Sun, “On the Generalized Holder Inequalities,” Soochow Journal of Mathematics, Vol. 23, 1997, pp. 241 252. |

[32] | C. L. Wang, “Inequalities of the Rado-Popoviciu Type for Functions and Their Applications,” Journal of Mathe matical Analysis and Applications, Vol. 100, No. 2, 1984, pp. 436-446. doi:10.1016/0022-247X(84)90092-1 |

[33] | X. T. Wang, H. M. Su and F. H. Wang, “Inequalities, Theory, Methods (in Chinese),” Henan Education Publi cation, Zhengzhou, 1967. |

[34] | J. Wen, W. Wang, H. Zhou and Z. Yang, “A Class of Cylic Inequalities of Janous Type (in Chinese),” Journal of Chengdu University, Vol. 22, 2003, pp. 25-29. |

[35] | C. X. Xue, “Isolation and Extension of Bernoulli Ine qualities (in Chinese),” Journal of Gansu Education Col lege, Vol. 13, No. 3, 1999, pp. 5-7. |

Journals Menu

Contact us

+1 323-425-8868 | |

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2023 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.