1.  The points $(x, y)$ that satisfy the inequality $\log _{x} y \geqslant \log _{\frac{x}{y}}(x y)$
$$
\text { II. 1. }\{(x, y) \mid x>1
$$

and $y>x\} \cup\{(x, y) \mid 0<x<1$ and $y<x\} \\
\begin{aligned}
& \Leftrightarrow \{(x, y) \mid x>1 \text { and } y>x\} \cup\{(x, y) \mid 0<x<1 \text { and } y<x\} \\
& \Leftrightarrow >1 .
\end{aligned}
$$

Therefore, the required set is $\{(x, y) \mid x>1$ and $y>x\} \cup$ $\{(x, y) \mid 0<x<1$ and $y<x\}$. The region of the set is the shaded part in the figure (excluding the boundary).