= Solution
Use the normalized <eigenfunctions> with the stated <weighted inner product>. <Orthogonality> gives $G[\hat y]=\sum_n|c_n|^2$. The corrected <Sturm-Liouville problem> and the vanishing boundary term give
$$
F[\hat y]=\int_a^b\overline{\hat y}\,\mathcal L\hat y\,dx=\sum_n\lambda_n|c_n|^2.
$$
For a complex trial the numerator means $\int(p|\hat y'|^2+q|\hat y|^2)dx$, agreeing with the given formula for real trials. Therefore
$$
\boxed{\Lambda[\hat y]=\frac{\sum_n\lambda_n|c_n|^2}{\sum_n|c_n|^2}\geq\lambda_0.}
$$
It is a <convex combination> of the <eigenvalues>, with equality only for a trial proportional to the lowest <eigenfunction>. The <Rayleigh-Ritz variational principle> consequently estimates $\lambda_0$ by choosing a nonzero admissible trial family and minimizing this quotient over its parameters:
$$
\boxed{\lambda_0\leq\min_{\text{trial parameters}}\Lambda[\hat y].}
$$
The result is an upper bound. A more flexible trial space can lower it, while equality requires that the exact lowest <eigenfunction> belong to the trial space. One need not know the expansion coefficients to calculate the quotient directly from the trial function.
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