Abstract Pythagorean theorem and corresponding functional equations (Q2843900)

Motivated by a result of \textit{L. R. Berrone} [Am. Math. Mon. 116, 936--939 (2009; Zbl 1229.51017)] the author investigates the functional equation NEWLINE\[NEWLINE f(x+y)=f(x)+f(y)+2f\bigl (\Phi (x, y)\bigr). NEWLINE\]NEWLINE In Berrone's paper it is assumed that \(f\) is continuous and strictly monotonic, \(\Phi \) is reflexive and the associative operation \(F:]0, +\infty [^{2}\to \mathbb {R}\) defined by NEWLINE\[NEWLINE F(x, y)=f^{-1}\left (\frac {f(x)+f(y)}{2}\right)\qquad \left (x, y\in \,]0, +\infty [\right) NEWLINE\]NEWLINE is homogeneous of first order. Due to the motivation of the problem, this latter assumption is rather unnatural. One of the main results of the paper is the following: The only nonzero solutions \(f:]0, +\infty [\to \mathbb {R}\) admitting finite \(\lim _{x\to 0+}\frac {f(x)}{x^{2}}\) of the above functional equation are NEWLINE\[NEWLINE f(x)=cx^{2} \quad \text{and} \quad \Phi (x, y)=\sqrt {xy} \qquad \left (x, y\in \,]0, +\infty [\right). NEWLINE\]NEWLINENEWLINENEWLINEIn the second section the twice continuously differentiable solutions of the equation NEWLINE\[NEWLINE f(x+y)=f(x)+f(y)+2f\Biggl (\varphi ^{-1}\biggl (\frac {\varphi (x)+\varphi (y)}{2}\biggr)\Biggr) NEWLINE\]NEWLINE are described.NEWLINENEWLINEFinally, it is also proved that with the use of a result of \textit{A. Járai} et al. [Publ. Math. 65, 381--398 (2004; Zbl 1071.39026)] this regularity assumption can be weakened to that \(f\) and \(\varphi \) are strictly monotonic and \(\varphi\) is continuous.