Abstract
We determine nontrivial intervals \({I \subset(0,+\infty)}\), numbers \({\alpha\in\mathbb R}\) and continuous bijections \({f \colon I \to I}\) such that f(x)f −1(x) = x α for every \({x\in I}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anschuetz R., Scherwood H.: When is a function’s inverse equal to its reciprocal?. College Math. J. 27, 388–393 (1996)
Boros, Z.: Talk Given During the Fifty International Symposium on Functional Equations. Hajdúszoboszló, Hungary, 17–24 June (2012)
Brillouët–Belluot N.: Problem posed during The Forty-ninth International Symposium on Functional Equations. Aequat. Math. 84, 312 (2012)
Chen L., Shi Y.: The real solutions of functional equation f [m] = 1/f. J. Math. Res. Exposition 28, 323–330 (2008)
Cheng R., Dasgupta A., Ebanks B.R., Kinch L.F., Larson L.M., McFadden R.B.: When does f −1 = 1/f?. Amer. Math. Monthly 105, 704–716 (1998)
Daepp U., Gorkin P.: Reading, Writing, and Proving: A Closer Look at Mathematics. Undergraduate Texts in Mathematics. Springer, New York (2003)
Euler R., Foran J.: On functions whose inverse is their reciprocal. Math. Mag. 54, 185–189 (1981)
Jarczyk W., Morawiec J.: Note on an equation occurring in a problem of Nicole Brillouët–Belluot. Aequat. Math. 84, 227–233 (2012)
Kuczma M.: Functional equations in a single variable. Monografie Matematyczne, vol. 46. PWN, Polish Scientific Publishers, Warsaw (1968)
Kuczma M., Choczewski B., Ger R.: Iterative functional equations. Encyclopedia of mathematics and its applications, vol. 32. Cambridge University Press, Cambridge (1990)
Massera J.L., Petracca A.: Sobre la ecuación funcional f(f(x)) = 1/x. Rev. Un. Mat. Argentina. 11, 206–211 (1946)
Morawiec J.: On a problem of Nicole Brillouët–Belluot. Aequat. Math. 84, 219–225 (2012)
Ng C.T., Zhang W.: When does an iterate equal a power?. Publ. Math. Debrecen 67, 79–91 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Morawiec, J. Around a problem of Nicole Brillouët–Belluot. Aequat. Math. 88, 175–181 (2014). https://doi.org/10.1007/s00010-013-0216-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-013-0216-8