Functional equations involving means. (English) Zbl 1174.39006
Summary: The functional equation
\[
f(px+(1-p)y)+f(1-p)x+py)=f(x)+f(y), \quad (x,y\in I)
\]
is considered, where \(0 < p < 1\) is a fixed parameter and \(f: I \to \mathbb R\) is an unknown function. The equivalence of this and Jensen’s functional equation is completely characterized in terms of the algebraic properties of the parameter \(p\). As an application, solutions of certain functional equations involving four weighted arithmetic means are also determined.
MSC:
| 39B22 | Functional equations for real functions |
| 26E10 | \(C^\infty\)-functions, quasi-analytic functions |
Keywords:
\(p\)-Wright affine function; Jensen-affine function; linear functional equation; biadditive functionReferences:
| [1] | Z. Daróczy, Notwendige und hinreichende Bedingungen für die Existenz von nichtkonstanten Lösungen linearer Funktionalgleichungen, Acta Sci. Math. (Szeged), 22 (1961), 31–41. · Zbl 0097.32102 |
| [2] | Z. Daróczy, Gy. Maksa and Zs. Páles, Functional equations involving means and their Gauss composition, Proc. Amer. Math. Soc., 134 (2006), 521–530. · Zbl 1082.39018 · doi:10.1090/S0002-9939-05-08009-3 |
| [3] | A. Gilányi and Zs. Páles, On Dinghas-type derivatives and convex functions of higher order, Real Anal. Exchange, 27 (2001/2002), 485–493. |
| [4] | M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Prace Naukowe Universitetu Śląskiego w Katowicach Vol. CDLXXXIX, Państwowe Wydawnictwo Naukowe – Universitet Śląski(Warszawa-Kraków-Katowice, 1985). |
| [5] | K. Lajkó, On a functional equation of Alsina and García-Roig, Publ. Math. Debrecen, 52 (1998), 507–515. · Zbl 0912.39010 |
| [6] | Gy. Maksa, K. Nikodem and Zs. Páles, Results on t-Wright convexity, C. R. Math. Rep. Acad. Sci. Canada, 13 (1991), 274–278. · Zbl 0749.26007 |
| [7] | L. Székelyhidi, Convolution Type Functional Equations on Topological Abelian Groups, World Scientific Publishing Co. Inc. (Teaneck, NJ, 1991). · Zbl 0748.39003 |
| [8] | E. M. Wright, An inequality for convex functions, Amer. Math. Monthly, 61 (1954), 620–622. · Zbl 0057.04801 · doi:10.2307/2307675 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
