Intrinsic metrics defined with arithmetic and logarithmic mean values

We introduce several new functions that measure the distance between two points $x$ and $y$ in a domain $G\subsetneq\mathbb{R}^n$ by using the arithmetic or the logarithmic mean of the Euclidean distances from the points $x$ and $y$ to the boundary of $G$. We study in which domains these functions are metrics and find sharp inequalities between them and the hyperbolic metric. We also present one result about their distortion under quasiregular mappings.