A new look at Bernoulli’s inequality

Ferreira, Rui

doi:10.1090/proc/13798

A new look at Bernoulli’s inequality
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by Rui A. C. Ferreira PDF

Proc. Amer. Math. Soc. 146 (2018), 1123-1129 Request permission

Addendum: Proc. Amer. Math. Soc. 151 (2023), 85-86.

Abstract:

In this work, a generalization of the well-known Bernoulli inequality is obtained by using the theory of discrete fractional calculus. As far as we know our approach is novel.

References

Ricardo Alfaro, Lixing Han, and Kenneth Schilling, A very elementary proof of Bernoulli’s inequality, College Math. J. 46 (2015), no. 2, 136–137. MR 3361760, DOI 10.4169/college.math.j.46.2.136
Ferhan M. Atici and Paul W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), no. 3, 981–989. MR 2457438, DOI 10.1090/S0002-9939-08-09626-3
Rui A. C. Ferreira, A discrete fractional Gronwall inequality, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1605–1612. MR 2869144, DOI 10.1090/S0002-9939-2012-11533-3
Christopher Goodrich and Allan C. Peterson, Discrete fractional calculus, Springer, Cham, 2015. MR 3445243, DOI 10.1007/978-3-319-25562-0
Michael Holm, Sum and difference compositions in discrete fractional calculus, Cubo 13 (2011), no. 3, 153–184 (English, with English and Spanish summaries). MR 2895482, DOI 10.4067/s0719-06462011000300009
Baoguo Jia, Lynn Erbe, and Allan Peterson, Comparison theorems and asymptotic behavior of solutions of discrete fractional equations, Electron. J. Qual. Theory Differ. Equ. , posted on (2015), Paper No. 89, 18. MR 3434219, DOI 10.14232/ejqtde.2015.1.89
Riku Klén, Vesna Manojlović, Slavko Simić, and Matti Vuorinen, Bernoulli inequality and hypergeometric functions, Proc. Amer. Math. Soc. 142 (2014), no. 2, 559–573. MR 3133997, DOI 10.1090/S0002-9939-2013-11781-8
Kenneth S. Miller and Bertram Ross, Fractional difference calculus, Univalent functions, fractional calculus, and their applications (K\B{o}riyama, 1988) Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989, pp. 139–152. MR 1199147
Dragoslav S. Mitrinović and Josip E. Pečarić, Bernoulli’s inequality, Rend. Circ. Mat. Palermo (2) 42 (1993), no. 3, 317–337 (1994). MR 1283347, DOI 10.1007/BF02844624
Roger B. Nelsen, Proof without Words: Bernoulli’s Inequality (two proofs), Math. Mag. 69 (1996), no. 3, 197. MR 1573168