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Recently i'm reading a paper,there is a inequality that confuse me. L is a symmetric,irreducible and semi-positive definite matrix with eigenvalues of $0=\lambda_{1}(L)<\lambda_{2}(L)\leq...\leq\lambda_{m}(L)$(counting the multiplicities),P is a positive definite matrix with maximum eigenvalues $\lambda_{m}(P)$,both of them are square matrix,satisfied

$$x^{T}(L^{2}\otimes P^{2})x\leq \lambda_{m}(L)\lambda_{m}(P)x^{T}(L\otimes P)x$$

I have searched many references ,but hardly find any result.Maybe there needs other properties,welcome to point out!

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