= Solution
Use the <Wirtinger derivatives> $\partial_z=(\partial_x-i\partial_y)/2$ and $\partial_{\bar z}=(\partial_x+i\partial_y)/2$, for which $\Delta=4\partial_z\partial_{\bar z}$. Since $H$ and $G$ are <holomorphic>,
$$
\partial_{\bar z}F=H(z),\qquad \Delta F=4H'(z),\qquad \Delta^2F=16\partial_z\partial_{\bar z}H'(z)=0.
$$
The <Laplacian> has real coefficients, so taking real and imaginary parts proves
$$
\boxed{\Delta^2U=0,\qquad\Delta^2V=0.}
$$
This is the <complex analytic representation of biharmonic functions>; $F$ itself is generally not holomorphic because of its $\bar z$ factor.
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