An asymptotic estimate of Aoki’s function

Vito Lampret

doi:10.56754/0719-0646.2801.099

Authors

Vito Lampret Faculty of Civil and Geodetic Engineering, University of Ljubljana, Ljubljana 1000, Slovenia. https://orcid.org/0000-0002-2921-3451

DOI:

https://doi.org/10.56754/0719-0646.2801.099

Keywords:

Approximation, estimate, expansion, exponential function, inequality

Abstract

The Aoki's function \(A(x):=\left(1+\frac{1}{x}\right)^x+\left(1-\frac{1}{x}\right)^{-x}\) is sharply estimated for \(x\gg1\). For example, we have the zero approximation given as
\[
2e\left(1+\frac{1}{4x^2-1}\right)<A(x)
<2e\left(1+\frac{3}{4x^2-1}\right),\quad x\ge \frac{29}{14}.
\]

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References

M. Aoki, Y. Nishizawa, R.Suzuki, and G. Takamori, “Approximations by the fractional function of the sum of two functions converging to e,” Appl. Math. E-Notes, vol. 24, pp. 19–26, 2024.

C.-P. Chen and R. B. Paris, “An inequality involving the constant e and a generalized Carleman-type inequality,” Math. Inequal. Appl., vol. 23, no. 4, pp. 1197–1203, 2020, doi: 10.7153/mia-2020-23-92.

V. Lampret, “Sharp, double inequalities bounding the function (1 + x)(1/x) and a refinement of Carleman’s inequality,” Math. Inequal. Appl., vol. 26, no.1, pp. 93–107, 2023, doi: 10.7153/mia-2023-26-08.

Wolfram Research, Inc., “Mathematica,” Version 7.0, Champaign, IL, USA, Nov. 2008, computer software.