# If $P_1$ and $P_2$ are positive definite matrices, can we show $|x^{\rm T}P_1P_2y|\leq \lambda_\max (P_1) |x^{\rm T}P_2y|$?

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 $$|x^{\rm T}P_1P_2y|\leq \lambda_\max(P_1)|x^{\rm T}P_2y|$$ holds?

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.