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  "body": "\n## Statement\n\nLet $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$, and let $n \\ge 1$.\nThen for every $x$ with $-1 \u003c x \u003c 1$,\n$$P_n(x)^2 - P_{n-1}(x) P_{n+1}(x) \u003e 0.$$\n\n## Remarks\n\nAt $x = \\pm 1$ the left side is $0$, since $P_k(1) = 1$ and $P_k(-1) = (-1)^k$; so the strict inequality needs $\\abs{x} \u003c 1$.\nCheck for $n = 1$: $x^2 - \\frac{3x^2 - 1}{2} = \\frac{1 - x^2}{2} \u003e 0$ for $\\abs{x} \u003c 1$.\nThe known proofs combine the three-term recurrence of the $P_n$ with their differential equation, and show that the left side cannot vanish inside $(-1, 1)$.\nThe left side is called the Turán determinant, since it is $\\det \\begin{pmatrix} P_n \u0026 P_{n+1} \\\\ P_{n-1} \u0026 P_n \\end{pmatrix}$.\nAnalogues hold for other families: for the [[Chebyshev polynomials|M6EAR1JHPP]], $T_n(x)^2 - T_{n-1}(x) T_{n+1}(x) = 1 - x^2$.\nThis follows from $\\cos^2(n\\theta) - \\cos((n - 1)\\theta) \\cos((n + 1)\\theta) = \\sin^2 \\theta$ with $x = \\cos \\theta$.\nFor the [[Hermite polynomials|MGACY98E9H]] one has $H_n(x)^2 - H_{n-1}(x) H_{n+1}(x) \u003e 0$ for all real $x$.\n\n## Explanations\n\n### L1 human\n\nFix a point $x$ strictly between $-1$ and $1$, and look at the sequence of numbers $P_0(x), P_1(x), P_2(x), \\dots$.\nTurán's inequality says that each term squared is larger than the product of its two neighbours.\nFor a sequence of positive numbers this means the sequence is log-concave, a kind of smoothness without sudden jumps.\n\n### L2 human\n\nTurán found the inequality for Legendre polynomials, and Szegő published several proofs of it in 1948.\nSimilar Turán-type inequalities hold for ultraspherical, Laguerre and Hermite polynomials, and for Bessel functions in the order variable.\nSuch inequalities for the Taylor coefficients of entire functions are related to the Laguerre–Pólya class.\nThis link is the reason they appear in work on the Riemann hypothesis through the Jensen polynomials of the Riemann $\\xi$ function.\n",
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