On symmetry of Makó–Páles means

Abstract. Given a nonempty real interval \(I\), a continuous strictly monotonic function \(\varphi: I\rightarrow \mathbb R\), and a Borel probability measure \(\mu\) on \([0,1]\), we characterize all symmetric Makó–Páles means \(M_{\varphi, \mu}\) defined on \(I\times I\) by \[ M_{\varphi,\mu}(x,y)=\varphi^{-1} \left(\int\limits^{1}_{0}\varphi\left(tx+(1-t)y\right)d\mu(t)\right). \]