The celebrated Turán inequalities
P
n
2
(
x
)
−
P
n
−
1
(
x
)
P
n
+
1
(
x
)
≥
0
P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) \geq 0
,
x
∈
[
−
1
,
1
]
x \in [-1,1]
,
n
≥
1
n \geq 1
, where
P
n
(
x
)
P_{n}(x)
denotes the Legendre polynomial of degree
n
n
, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities
γ
n
2
−
γ
n
−
1
γ
n
+
1
≥
0
\gamma _{n}^{2} - \gamma _{n-1} \gamma _{n+1} \geq 0
,
n
≥
1
n \geq 1
, which hold for the Maclaurin coefficients of the real entire function
ψ
\psi
in the Laguerre-Pólya class,
ψ
(
x
)
=
∑
n
=
0
∞
γ
n
x
n
/
n
!
\psi (x) = \sum _{n=0}^{\infty } \gamma _{n} x^{n}/n!
.