Extended power difference means

The extended power difference mean \(f_{a,b}(t):={b\over a}{{t^a-1}\over {t^b-1}}\) $f_{a,b}(t):=\frac{b}{a}\frac{t^a-1}{t^b-1}$ \((a,b\in {\mathbb {R}})\) $(a,b\in \mathbb{R})$ is investigated in this paper. We show some Thompson metric inequalities involving \(f_{a,b}\) $f_{a,b}$ and Tsallis relative operator entropy. We also discuss the behavior of the bivariate function defined as the perspective map for \(f_{a,b}\) $f_{a,b}$ . Finally, the relationship beween \(f_{a,b}\) $f_{a,b}$ and the weighted logarithmic mean is studied.