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Functional equations involving means and their Gauss composition. (English) Zbl 1082.39018

Let \(I\subseteq\mathbb{R}\) be a nonempty open interval and \(M_j: I^2\to I\) strict means (continuous and \(\min\{x,y\}< M_j(x,y)<\max\{x,y\}\;(j=1,2)\)). The two sequences \(\{x_n\},\,\{y_n\},\) defined by \(x_1=x,\: y_1=y,\; x_{n+1}=M_1(x_n,y_n), y_{n+1}=M_2(x_n,y_n)\; (n=1,2,\dots)\) converge to the same strict mean \(M_3(x,y),\) the Gauss composite of \(M_1\) and \(M_2.\)
The authors offer among others the following results on the relation between the functional equations \[ f[M_1(x,y)]+f[M_2(x,y)]=f(x)+f(y)\tag{1} \] and \[ 2f[M_3(x,y)]=f(x)+f(y)]\tag{2} \] (no regularity assumption about \(f\)). If \(I=]0,\infty[\) and \(M_1,M_2\) are the (symmetric) arithmetic and geometric mean, respectively, then each equation has only trivial solutions \(f=\) constant. Equations (1) and (2) have the same solutions also if \(M_j(x,y)=g_j^{-1}(\frac{g_j(x)+g_j(y)}{2})\; (j=1,2,3)\) (quasi-arithmetic means). If \(M_1(x,y)=px+(1-p)y,\,M_2(x,y)=(1-p)x+py\;(p\in]0,1[)\) then (2) follows from (1) if \(p\) is rational but (2) does not follow from (1) if \(p\) is transcendental or an algebraic number such that \(p/(2p-1)\) is an algebraic conjugate of \(p.\)
The proof of the latter statement relies heavily on a result of the first author in [Acta Sci. Math. (Szeged) 22, 31–41 (1961; Zbl 0097.32102)], surprising at the time, stating that the equation \[ f(ax+y)=Af(x)+f(y)\: (aA\neq 0) \] has nonconstant solutions iff \(a\) and \(A\) are either both transcendental or they are conjugate algebraic numbers.

MSC:

39B22 Functional equations for real functions
39B12 Iteration theory, iterative and composite equations
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
26E60 Means

Citations:

Zbl 0097.32102
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References:

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