---
tag: 4K3D632K60
kind: statement
role: theorem
name: Turán's inequalities
msc: [33C45, 26D15]
uses: [QFYGJVA996]
about: [QFYGJVA996]
sources:
  - url: https://en.wikipedia.org/wiki/Turán's_inequalities
same_as:
  wikidata: Q7856594
  wikipedia: Turán's inequalities
---

## Statement

Let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$, and let $n \ge 1$.
Then for every $x$ with $-1 < x < 1$,
$$P_n(x)^2 - P_{n-1}(x) P_{n+1}(x) > 0.$$

## Remarks

At $x = \pm 1$ the left side is $0$, since $P_k(1) = 1$ and $P_k(-1) = (-1)^k$; so the strict inequality needs $\abs{x} < 1$.
Check for $n = 1$: $x^2 - \frac{3x^2 - 1}{2} = \frac{1 - x^2}{2} > 0$ for $\abs{x} < 1$.
The known proofs combine the three-term recurrence of the $P_n$ with their differential equation, and show that the left side cannot vanish inside $(-1, 1)$.
The left side is called the Turán determinant, since it is $\det \begin{pmatrix} P_n & P_{n+1} \\ P_{n-1} & P_n \end{pmatrix}$.
Analogues hold for other families: for the [[Chebyshev polynomials|M6EAR1JHPP]], $T_n(x)^2 - T_{n-1}(x) T_{n+1}(x) = 1 - x^2$.
This follows from $\cos^2(n\theta) - \cos((n - 1)\theta) \cos((n + 1)\theta) = \sin^2 \theta$ with $x = \cos \theta$.
For the [[Hermite polynomials|MGACY98E9H]] one has $H_n(x)^2 - H_{n-1}(x) H_{n+1}(x) > 0$ for all real $x$.

## Explanations

### L1 human

Fix a point $x$ strictly between $-1$ and $1$, and look at the sequence of numbers $P_0(x), P_1(x), P_2(x), \dots$.
Turán's inequality says that each term squared is larger than the product of its two neighbours.
For a sequence of positive numbers this means the sequence is log-concave, a kind of smoothness without sudden jumps.

### L2 human

Turán found the inequality for Legendre polynomials, and Szegő published several proofs of it in 1948.
Similar Turán-type inequalities hold for ultraspherical, Laguerre and Hermite polynomials, and for Bessel functions in the order variable.
Such inequalities for the Taylor coefficients of entire functions are related to the Laguerre–Pólya class.
This link is the reason they appear in work on the Riemann hypothesis through the Jensen polynomials of the Riemann $\xi$ function.