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Let $P_1,P_2\in\mathbb{R}^{n\times n}$ be symmetric positive-definite. Let $x,y\in\mathbb{R}^n$. Can we prove or disprove by counter example that the inequality \begin{equation} |x^{\rm T}P_1P_2y|\leq \lambda_\max(P_1)|x^{\rm T}P_2y| \end{equation} holds?

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I don't think the inequality holds in general. Suppose that $x$ is orthogonal to $P_2 y$, and further suppose that $P_1$ is a diagonal matrix with different diagonal entries. In that case right hand side is zero, and left hand side is strictly positive. Thus the inequality is violated.

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