PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 126, Number 7, July 1998, Pages 2033–2037
S 0002-9939(98)04438-4

HIGHER ORDER TURÁN INEQUALITIES

DIMITAR K. DIMITROV

(Communicated by J. Marshall Ash)

Abstract. The celebrated Turán inequalities P 2
n(x) − Pn−1(x)Pn+1(x) ≥

0, x ∈ [−1, 1], n ≥ 1, where Pn(x) denotes the Legendre polynomial of
degree n, are extended to inequalities for sums of products of four classical
orthogonal polynomials. The proof is based on an extension of the inequalities
γ2
n − γn−1γn+1 ≥ 0, n ≥ 1, which hold for the Maclaurin coefficients of the

real entire function ψ in the Laguerre-Pólya class, ψ(x) =
∑∞
n=0 γnx

n/n!.

1. Introduction and statement of results

For any sequence of polynomials {pn}∞n=0 the quantities ∆n(p;x) := p2
n(x) −

pn−1(x)pn+1(x) are called Turán determinants, associated with {pn}∞n=0. Szegő [18]
was the first to call attention to the following beautiful inequalities of P. Turán:

∆n(P ;x) = P 2
n(x)− Pn−1(x)Pn+1(x) ≥ 0, x ∈ [−1, 1], n ≥ 1.(1)

In the same paper Szegő obtained extensions of (1) to Gegenbauer (ultraspheri-
cal), Laguerre and Hermite polynomials. Karlin and Szegő [10] proved that certain
higher order Turán determinants for the same classes of classical orthogonal poly-
nomials do not change their sign in the interval of orthogonality. Gasper [9] proved
the analog of (1) for a class of Jacobi polynomials. Askey’s comments on [10]
and [18] in Volume 3 of Szegő’s collected papers survey further contributions and
developments.

The reason for the recent interest in Turán determinants is that for the orthogo-
nal polynomials {pn}∞n=0 in a subclass of the class M(0, 1) the quantities ∆n(p;x)
converge uniformly on the compact subsets of (−1, 1) to 2(1 − x2)1/2/(πα′(x)),
where α′(x) is the absolutely continuous part of the measure, with respect to which
the pn are orthogonal [5, 6, 7, 12, 19].

Szegő [18] gives Turan’s proof and three additional proofs of (1). The third
proof is particularly ingenious and allows the extension of (1) to the ultraspherical,
Laguerre and Hermite polynomials. Szegő atributes the idea of this proof to Pólya.

Received by the editors December 12, 1996.
1991 Mathematics Subject Classification. Primary 30D10, 33C45.
Key words and phrases. Turán inequalities, Turán determinants, entire functions in the

Laguerre-Pólya class, Riemann hypothesis.
Research supported by the Brazilian foundation CNPq under Grant 300645/95-3 and the Bul-

garian Science Foundation under Grant MM-414.

c©1998 American Mathematical Society

2033

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2034 DIMITAR K. DIMITROV

A real entire function

ψ(x) =

∞∑
n=0

γn
xn

n!
(2)

is said to belong to the Laguerre-Pólya class (ψ ∈ L-P) if

ψ(x) = cxme−αx
2+βx

∞∏
k=1

(1 + x/xk)e
−x/xk ,

where c, β, xk are real, α ≥ 0,m is a nonnegative integer and
∑
x−2
k < ∞. We

have adopted the notations in [2, 4, 14], which one may consult for the important
properties of the functions in Laguerre-Pólya class. Generally, L-P consists of entire
functions which are uniform limits on the compact sets of the complex plane of real
polynomials with only real zeros. A necessary condition that ψ ∈ L-P is that its
Maclaurin coefficients satisfy (cf. [2, 4, 16])

γ2
n − γn−1γn+1 ≥ 0, n ≥ 1.(3)

Then, in order to prove the inequalities

∆n(p;x) ≥ 0, n ≥ 1,

where
a) pn(x) = P

(λ)
n (x)/P

(λ)
n (1) for x ∈ [−1, 1], λ > −1/2,

b) pn(x) = L
(α)
n (x)/L

(α)
n (0) for x ∈ [0,∞), α > −1,

or
c) pn(x) = Hn(x) for x ∈ (−∞,∞),

where P
(λ)
n , L

(α)
n and Hn denote the ultraspherical, Laguerre and Hermite poly-

nomials, one uses (3) together with the fact that the generating functions which
appear on the right-hand sides of

∞∑
n=0

P
(λ)
n (x)

P
(λ)
n (1)

zn

n!
= 2λ−1/2Γ(λ+ 1/2)exz

Jλ−1/2((1 − x2)1/2z)

((1− x2)1/2z)λ−1/2
, λ > −1/2,

∞∑
n=0

L
(α)
n (x)

L
(α)
n (0)

zn

n!
= Γ(α+ 1)ez

Jα(2(xz)1/2)

(xz)α/2
, α > −1,

and
∞∑
n=0

Hn(x)
zn

n!
= e2xz−z

2

are in the Laguerre-Pólya class.
Another reason that inequalities (3) are interesting is their connection to the

celebrated Riemann hypothesis [17] about the zeros of the Riemann ζ-function. It
is well known and easy to see that the Riemann hypothesis holds true if and only
if the Riemann ξ-function, defined by

ξ(iz) =
1

2
(z2 − 1/4)π−z/2−1/4Γ(z/2 + 1/4)ζ(z + 1/2),

has only real zeros. It is known that ξ is a real entire function of order one. It can
be represented in the form

ξ(x/2) = 8

∫ ∞

0

Φ(t) cosxt dt,

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HIGHER ORDER TURÁN INEQUALITIES 2035

where

Φ(t) =

∞∑
n=1

(2n4π2e9t − 3n2πe5t) exp(−n2πe4t).(4)

Then

1

8
ξ(x/2) =

∞∑
k=0

(−1)kb̂k
x2k

(2k)!

with

b̂k =

∫ ∞

0

t2kΦ(t)dt, for k ≥ 0.(5)

On setting z = −x2 we obtain the entire function

ξ1(z) =

∞∑
k=0

γ̂k
zk

k!
, γ̂k =

k!

(2k)!
b̂k,(6)

of order 1/2. Therefore, by the Hadamard theorem, the Riemann hypothesis is
equivalent to the statement that ξ1 ∈ L-P (cf. Pólya and Schur [16] and Boas [1, p.
24]). Hence the inequalities γ̂2

n− γ̂n−1γ̂n+1 ≥ 0, n ≥ 1, which are equivalent to the

inequalities (2n+1)b̂2n− (2n−1)b̂n−1b̂n+1 ≥ 0, n ≥ 1, are necessary conditions for
the Riemann hypothesis to be true. Craven, Norfolk and Varga [3] proved the latter
inequalities, thus verifying a conjecture of Pólya [15] (see also Varga [20, Chapter
3]).

In this paper we obtain, in a very simple way, new necessary conditions for a
real entire function to belong to L-P . These condition are extensions of (3). Then
the idea of Pólya, sketched above, immediately yields extensions of (1).

Theorem 1. Let the real entire function ψ, defined by (2), be in the Laguerre-
Pólya class. Then

4(γ2
n − γn−1γn+1)(γ

2
n+1 − γnγn+2)− (γnγn+1 − γn−1γn+2)

2 ≥ 0 for n ≥ 1.

Corollary 1. Let γ̂n be defined by (4), (5) and (6). A necessary condition that
the Riemann hypothesis holds true is that the inequalities

4(γ̂2
n − γ̂n−1γ̂n+1)(γ̂

2
n+1 − γ̂nγ̂n+2)− (γ̂nγ̂n+1 − γ̂n−1γ̂n+2)

2 ≥ 0, n ≥ 1,(7)

hold.

Corollary 2. The inequalities

δn(p;x) := 4
(
p2
n(x)− pn−1(x)pn+1(x)

) (
p2
n+1(x) − pn(x)pn+2(x)

)
− (pn(x)pn+1(x) − pn−1(x)pn+2(x))

2 ≥ 0, n ≥ 1,

hold for the classes of orthogonal polynomials a), b) and c), described above.

2. Proof of the theorem and remarks

Proof of the theorem. Let the real entire function ψ, defined by (2), be in the
Laguerre-Pólya class. Then, for any positive integer n, the n-th associated Jensen
polynomial

gn(x) :=

n∑
k=0

(
n
k

)
γkx

k

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2036 DIMITAR K. DIMITROV

has only real zeros (cf. [2, 4, 15]). Observe that for any q ≤ n

g(q)
n (x) =

n!

(n− q)!
gn−q,q(x),

where

gn,q(x) :=

n∑
k=0

(
n
k

)
γk+qx

k, n = 0, 1, . . . ,

are the Jensen polynomials associated with ψ(q). Then Rolle’s theorem implies that
for any positive integer n and any nonnegative integer q the polynomial gn,q(x) has
only real zeros. The latter follows also from the fact that the class L-P is closed
under differentiation (cf. Pólya and Schur [16]). Now the assertion of the theorem
follows from a result of Mař́ık [11] (see also [13, Theorem 1.3.3 on p. 99]). It states
that if the real polynomial

p(x) =

n∑
k=0

akx
k/(k!(n− k)!)

of degree n ≥ 3 has only real zeros, then the inequalities

4(a2
k − ak−1ak+1)(a

2
k+1 − akak+2)− (akak+1 − ak−1ak+2)

2 ≥ 0, 1 ≤ k ≤ n− 2,

hold.

Corollary 1 is immediate. In order to prove Corollary 2 one uses the statement
of Theorem 1 and the idea of Pólya, described in the first section.

A natural conjecture is that inequalities (7) hold true. Numerical calculations,

based on the values of the first twenty coefficients b̂n, given in [3], support the
conjecture.

It is interesting to see what is the limit of the quantities δn(p;x) for the class of
orthogonal polynomials whose associated Jacobi matrix is a compact perturbation
of the Jacobi matrix corresponding to the Chebyshev polynomials of the second
kind. The above mentioned results on convergence of Turán determinants for the
polynomials in the class M(0, 1) and their extension to the so-called shifted Turán
(or Geronimo and Van Assche) determinants [8, Theorem 6 ] yield:

Proposition 1. Let the sequence of orthogonal polynomials {pn} be defined by the
three-term recurrence relation

xpn(x) = an+1pn+1(x) + bnpn(x) + anpn−1(x), n ≥ 0,

p−1(x) = 0, p0(x) = 1,

with real bn and positive an. Suppose that the recurrence coefficients satisfy an →
1/2 and bn → 0 as n diverges, and

∞∑
k=0

(|bk+1 − bk|+ |ak+2 − ak+1|) <∞.

Then the measure α, with respect to which the pn are orthogonal, is absolutely
continuous in (−1, 1), α′(x) > 0 for all x ∈ (−1, 1), and α′ is continuous in (−1, 1).

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HIGHER ORDER TURÁN INEQUALITIES 2037

Moreover,

lim
n→∞ δn(p;x) =

8

π2

1− x2

[α′(x)]2

uniformly on the compact subsets of (−1, 1).

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Departamento de Ciências de Computação e Estat́ıstica, IBILCE, Universidade Es-
tadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

E-mail address: dimitrov@nimitz.dcce.ibilce.unesp.br

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