Let \(I\subseteq \mathbb {R}\) be a nonempty open subinterval. We say that a two-variable mean \(M:I\times I\rightarrow \mathbb {R}\) enjoys the balancing property if, for all \(x,y\in I\), the equality

holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that M is analytic, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes regular quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are continuously differentiable. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of Matkowski means.

令\（I \ subseteq \ mathbb {R} \）为非空打开子间隔。我们说，如果对于所有\（x，y \ I \）等于，则二元均值\（M：I \ times I \ rightarrow \ mathbb {R} \）享有均衡属性

持有。上面的方程式已经由几位作者研究过。第一步是Georg Aumann于1935年做出的。除其他外，假设M是解析性的，他求解（1）并获得了拟算术平均值作为解。然后，两年后，他证明（1）刻画了柯西方法中的常规拟算术方法的特征，其中，可微性假设自然而然地出现了。2015年，Lucio R. Berrone研究了具有对称性和严格单调性的更一般的方程，证明了一般解是准算术方法，只要所讨论的方法是连续可微的。本文的目的是解决（1）一类二元均值的情况，其中没有可微性假设，其中包含Matkowski均值类。