= Solution
The denominator is positive throughout the domain. Holding $y$ fixed, differentiate with respect to $x$:
$$
k_x(x,y)=\frac{1-y^2}{(1+xy)^2},\qquad k_{xx}(x,y)=\frac{2y(y^2-1)}{(1+xy)^3}.
$$
At the interior point $(1,1/2)$ this second <partial derivative> is $-2/9<0$, so the restriction to a horizontal line is locally strictly <concave>, ruling out <convexity>. At the interior point $(1,2)$ it is $4/9>0$, ruling out <concavity>. Restrictions of a <convex function> or a <concave function> to every line segment must retain that property. Consequently \b[$k$ is neither <convex> nor <concave>] on its stated domain.
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