多项式$P_n$的不等式

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对$n = 0, 1, 2, 3...,$ 设$P_n(x)=\dfrac{1}{2^{n} n !} \dfrac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n}$
求证Turán's inequalities
$$P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)\quad\forall x\in(-1,1).$$