Some Equivalent Forms of Bernoulli’s Inequality: A Survey

Affiliation(s)

Department of Applied Mathematics, National Chung-Hsing University, Taiwan.

Department of Mathematics, National Central University, Taiwan; Department of Information Management, Lunghwa University of Science and Technology, Taiwan.

Department of Applied Mathematics, National Chung-Hsing University, Taiwan.

Department of Mathematics, National Central University, Taiwan; Department of Information Management, Lunghwa University of Science and Technology, Taiwan.

ABSTRACT

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

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

Cite this paper

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.

Y. Li and C. Yeh, "Some Equivalent Forms of Bernoulli’s Inequality: A Survey,"

References

[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.

[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.