= Solution
Because the quartic <coupling constant (physics)> has <mass dimension> $\epsilon=4-D$, introduce a <renormalization scale> $\mu$ and define the dimensionless <renormalized coupling> $g$ by <minimal subtraction scheme>:
$$
\boxed{\lambda_0=\mu^\epsilon\left[g+\frac{3g^2}{16\pi^2\epsilon}+O(g^3)\right].}
$$
Here $\lambda_0$ is the fixed bare <coupling constant (physics)>. Equivalently the tree-level quartic term is $-\mu^\epsilon g\phi^4/4!$, and its one-loop <counterterm> is $-\mu^\epsilon3g^2\phi^4/(16\pi^2\epsilon\,4!)$. This states explicitly the scale convention implicit in <4/c/solution>.
Put $a=3/(16\pi^2)$. Differentiating the bare relation at fixed $\lambda_0$ gives
$$
0=\epsilon\left(g+\frac{ag^2}{\epsilon}\right)
+\widehat\beta(g)\left(1+\frac{2ag}{\epsilon}\right)+O(g^3).
$$
Write $\widehat\beta=-\epsilon g+b g^2+O(g^3)$. The coefficient of $g^2$ in the last identity is $a+b-2a$, so $b=a$. Therefore
$$
\boxed{\widehat\beta(g)=-\epsilon g+\frac{3g^2}{16\pi^2}+O(g^3).}
$$
The first term is the <engineering term in a dimensionally continued beta function>. In four dimensions it disappears, leaving the positive <one-loop quartic scalar beta function> $3g^2/(16\pi^2)$. Thus the <renormalized coupling> increases with increasing <renormalization scale> near weak positive coupling. This quantum running is the obstruction to extending the classical <scale invariance> in <1/d/solution> to the interacting quantum theory.
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