# On the invariance of the arithmetic mean with respect to generalized Bajraktarević means

• Published:

## Abstract

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation

$$f^{-1}\Bigl(\frac{p_1(x)f(x) +p_2(y)f(y)}{p_1(x)+p_2(y)}\Bigr)+g^{-1}\Bigl(\frac{q_1(x)g(x) +q_2(y)g(y)}{q_1(x)+q_2(y)}\Bigr)=x + y \ \ (x,y\in I),$$

where I is a nonempty open real interval and $$f,g \colon I \to\mathbb{R}$$ are continuous, strictly monotone and $$p_1,p_2,q_1,q_2 \colon I \to\mathbb{R}_+$$ are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of f, g, twice continuous differentiability of p1 and p2 and assuming that p1 differs from p2 on a dense subset of I, a necessary and sufficient condition for the equality above is that the unknown functions are of the form

$$f=\frac{u}{v}, \quad g=\frac{w}{z}, \quad \mbox{and} \quad p_1q_1=p_2q_2=vz,$$

where $$u,v,w,z \colon I \to\mathbb{R}$$ are arbitrary solutions of the second-order linear differential equation $$F''=\gamma F (\gamma\in\mathbb{R}$$ is arbitrarily fixed) such that v > 0 and z > 0 holds on I and $$\{u,v\}$$ and $$\{w,z\}$$ are linearly independent.

This is a preview of subscription content, log in via an institution to check access.

## Subscribe and save

Springer+ Basic
\$34.99 /Month
• Get 10 units per month
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

## References

1. M. Bajraktarević, Sur une équation fonctionnelle aux valeurs moyennes, Glasnik Mat.-Fiz. Astronom. Drutvo Mat. Fiz. Hrvatske Ser. II, 13 (1958), 243–248

2. M. Bajraktarević, Sur une généralisation des moyennes quasilinéaires, Publ. Inst. Math. (Beograd) (N.S.), 3 (1963), 69–76

3. Sz. Baják and Zs. Páles, Computer aided solution of the invariance equation for two-variable Gini means, Comput. Math. Appl., 58 (2009), 334–340

4. Sz. Baják and Zs. Páles, Invariance equation for generalized quasi-arithmetic means, Aequationes Math., 77 (2009), 133–145

5. Sz. Baják and Zs. Páles, Computer aided solution of the invariance equation for two-variable Stolarsky means, Appl. Math. Comput., 216 (2010), 3219–3227

6. Burai, P.: A Matkowski-Sutô type equation. Publ. Math. Debrecen 70, 233–247 (2007)

7. Z. Daróczy and Zs. Páles, Gauss-composition of means and the solution of the Matkowski–Sutô problem, Publ. Math. Debrecen, 61 (2002), 157–218

8. R. Grünwald and Zs. Páles, On the equality problem of generalized Bajraktarević means, Aequationes Math., 94 (2020), 651–677

9. Jarczyk, J.: Invariance of weighted quasi-arithmetic means with continuous generators. Publ. Math. Debrecen 71, 279294 (2007)

10. Jarczyk, J., Matkowski, J.: Invariance in the class of weighted quasi-arithmetic means. Ann. Polon. Math. 88, 39–51 (2006)

11. Matkowski, J.: Invariant and complementary quasi-arithmetic means. Aequationes Math. 57, 87–107 (1999)

12. Matkowski, J.: Solution of a regularity problem in equality of Cauchy means. Publ. Math. Debrecen 64, 391–400 (2004)

13. Matkowski, J.: Generalized weighted and quasi-arithmetic means. Aequationes Math. 79, 203–212 (2010)

14. Zs. Páles and A. Zakaria, On the invariance equation for two-variable weighted nonsymmetric Bajraktarević means, Aequationes Math., 93 (2019), 37–57

15. Sutô, O.: Studies on some functional equations I. Tôhoku Math. J. 6, 1–15 (1914)

16. Sutô, O.: Studies on some functional equations II. Tôhoku Math. J. 6, 82–101 (1914)

## Author information

Authors

### Corresponding author

Correspondence to Zs. Páles.

The research of the first author was supported by the ÚNKP-20-3 New National Excellence Program of the Ministry of Human Capacities.

The research of the second author was supported by the K-134191 NKFIH Grant and the 2019-2.1.11-TÉT-2019-00049 and the EFOP-3.6.1-16-2016-00022 projects. The last project is cofinanced by the European Union and the European Social Fund.

## Rights and permissions

Reprints and permissions

Grünwald, R., Páles, Z. On the invariance of the arithmetic mean with respect to generalized Bajraktarević means. Acta Math. Hungar. 166, 594–613 (2022). https://doi.org/10.1007/s10474-022-01230-5

• Revised:

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1007/s10474-022-01230-5