Let \({\mathbb{X}}\) be a normed space and \(M:{\mathbb{R}}^+\times{\mathbb{R}}^+\to{\mathbb{R}}^+\) be a symmetric function satisfying \(M(|x|,|y|)>0\) if \(|x||y|>0\) for \(x,y\in{\mathbb{X}}\). A function \(\rho_M\) is called an \(M\)-relative distance if it has the form \(\rho_M(x,y)=\frac{|x-y|}{M(|x|,|y|)}\). In this paper, the author obtains necessary and sufficient conditions under which \(\rho_M\) is a metric in two special cases.
The first case is when \(M\) equals a power of the power mean, \(M=A_p^q\). In this case \(\rho_M\) is denoted by \(\rho_{p,q}\). He proves that if \(q\not=0\), then the \((p,q)\)-relative distance \(\rho_{p,q}(x,y)=\frac{|x-y|}{A_p(|x|,|y|)^q}\) is a metric in \({\mathbb{R}}^n\) if and only if \(0<q\leq 1\) and \(p\geq\max\{1-q,(2-q)/3\}\).
The second case is when \(M(x,y)=f(x)f(y)\), where \(f:{\mathbb{R}}^n\to(0,\infty)\). He shows that \(\rho_M\) is a metric in \({\mathbb{R}}^n\) if and only if (i) \(f\) is increasing, (ii) \(f(x)/x\) is decreasing for \(x>0\) and (iii) \(f\) is convex.