标题： 关于均值横坐标为中点的函数，以及与调和函数的联系 显示英文标题

标题： On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions

摘要： 我们研究具有这样的性质的函数：对于每个区间，区间的中点的斜率与区间的平均斜率相同。 更一般地，我们找到其在区间上的平均斜率由这些区间的端点的加权平均处的斜率给出的函数。 这等价于寻找满足加权中值性质的函数。 在我们的探索过程中，我们发现了与调和函数的联系，这促使我们探讨多变量的类似情况以及“加权调和函数”的存在性。 显示英文摘要

摘要： We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogues and the existence of "weighted harmonic functions."