Cent. Eur. J. Math. • 11(9) • 2013 • 1643-1650
DOI: 10.2478/s11533-013-0269-x


Central European Journal of Mathematics

The generalized Laguerre inequalities
and functions in the Laguerre–Pólya class
                                                                                                             Research Article

George Csordas1∗ , Anna Vishnyakova2†



1 Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA

2 Department of Mechanics & Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61022, Ukraine




                                                            Received 6 June 2012; accepted 14 October 2012

Abstract: The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to be-
          long to the Laguerre–Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require
          the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving
          the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel–Carathédodory
          Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre
          inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 con-
          cludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent,
          ingenious extension of the Laguerre-type inequalities.

MSC:               30D15, 30D35

Keywords: Laguerre–Pólya class • Generalized Laguerre-type inequalities
                   © Versita Sp. z o.o.




1.      Introduction
The main leitmotif of this note pertains to certain inequalities used in the characterization of functions in the Laguerre–
Pólya class. In the present investigation, it will be convenient to adopt the following, classical definition of this family
of functions.

∗
    E-mail: george@math.hawaii.edu
†
    E-mail: anna.m.vishnyakova@univer.kharkov.ua




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                                                                          The generalized Laguerre inequalities and functions in the Laguerre–Pólya class




       Definition 1.1.                 P∞
       A real entire function φ(x) =     k=0 (γk /k!)x
                                                         k
                                                             is said to be in the Laguerre–Pólya class, written φ(x) ∈ L-P, if φ(x) can
       be expressed in the form
                                                                    ω                
                                                                    Y            x
                                        φ(x) = cx n e−αx +βx                              e−x/xk ,      0 ≤ ω ≤ ∞,
                                                              2
                                                                           1+
                                                                    k=1
                                                                                 xk
                                                                                                 P∞
       where c, β, xk ∈ R, xk 6= 0, α ≥ 0, n is a nonnegative integer and                         k=1 1/xk < ∞.
                                                                                                         2
                                                                                                                     If ω = 0, then, by convention, the
       product is defined to be 1.


       It has been oft stated that the significance of the Laguerre–Pólya class in the theory of entire functions stems from the
       fact that functions in this class, and only these, are the uniform limits, on compact subsets of C, of polynomials with
       only real zeros. For various properties and algebraic and transcendental characterizations of functions in this class we
       refer the reader to Pólya and Schur [12, p. 100], [13] or [10, Kapitel II].
       A real entire function φ(x) is said to satisfy the Laguerre inequalities if L1 (x) = (φ0 (x))2 − φ(x)φ00 (x) ≥ 0 for all x ∈ R.
       These inequalities are only necessary conditions for φ(x) to belong to the Laguerre–Pólya class. Indeed, if, for example,
                                                                            / L-P. On the other hand, as the following theorem
       φ(x) = ex /2 cos x, then L1 (x) = ex sin2 x ≥ 0 for all x ∈ R, but φ ∈
                2                          2


       shows, the so-called generalized real Laguerre inequalities (see also Theorem 3.1) are both necessary and sufficient for
       membership in the Laguerre–Pólya class.


       Theorem 1.2 (Generalized Real Laguerre Inequalities [4, Theorem 2.9]).
       Let φ denote a real entire function, φ 6≡ 0. For n ∈ N0 = N ∪ {0} and x ∈ R, set


                                                                        (−1)j+n 2n
                                                                     2n
                                                                     X          
                                           Ln (x) = Ln (x, φ) =                                   φ(j) (x)φ(2n−j) (x).
                                                                     j=0
                                                                              (2n)!          j


       If φ(x) ∈ L-P, then Ln (x) ≥ 0 for all n ∈ N0 and for all x ∈ R. Conversely, suppose that


                                  φ(x) = e−ax φ1 (x),             a ≥ 0,       where the genus of φ1 (x) is 0 or 1.
                                               2




       If Ln (x) ≥ 0 for all n ∈ N0 and for all x ∈ R, then φ(x) ∈ L-P.


       In accordance with the editor’s suggestion to make this paper self-contained, we recall here the following standard
       definitions and nomenclature. Let f(z) denote an entire function and let M(r, f) = max|z|=r |f(z)|. Then the order of the
       entire function f is defined as
                                                                     log log M(r, f)
                                                 ρ = ρ(f) = lim sup                  .
                                                              r→∞         log r

       Let 0 < |z1 | ≤ |z2 | ≤ |z3 | ≤ . . . denote the absolute values of the (non-zero) zeros (if any) of f(z). Then the convergence
       exponent of the zeros of f is the infimum of positive numbers α for which ∞                    α
                                                                                         P
                                                                                           n=1 1/|zn | converges. The smallest positive
       integer α for which this series converges is denoted by p + 1 and p is called the genus of the set of zeros of f (see, for
       example, [1, p. 14] or [6, p. 55]). Now, if f(z) is an entire function of finite order ρ with an m-fold zero at the origin, then
       by the Hadamard Factorization Theorem (cf. [1, p. 22]) f(z) = z m eQ(z) Π(z), where Q(z) is a polynomial of degree q ≤ ρ
       and Π(z) is the canonical product (of genus p; that is, p is the genus of the zeros) formed from the (non-zero) zeros
       of f(z). Then the genus of f(z) is defined as max (p, q).


       Remarks 1.3.
       Observe that L0 (x) = φ2 (x) and to justify the appellation “generalized Laguerre expression”, note that L1 (x) = (φ0 )2 (x) −
       φ(x)φ00 (x). In addition, we remark that if the real entire function φ(x) satisfies the generalized real Laguerre inequalities,
       Ln (x) ≥ 0, n ∈ N0 , x ∈ R, then φ(x) has only real zeros (cf. [4, p. 343]). For the sake of completeness, we briefly outline




1644
G. Csordas, A. Vishnyakova




the proof. Here, and in the sequel, we will need to refer to the following representation of |φ(x + iy)|2 which can be
derived by a direct calculation (see, for example, [4, 7, 11] or by using a recursion relation [3]):

                                                                          ∞
                                                                          X
                              |φ(x + iy)|2 = φ(x + iy)φ(x − iy) =               Ln (x)y2n ,       x, y ∈ R.                         (1)
                                                                          n=0



We proceed with a reductio adPabsurdum argument. Thus, suppose that z0 = x0 + iy0 is a non-real zero of φ(x), where
φ 6≡ 0. Then 0 = |φ(z0 )|2 = ∞      n=0 Ln (x0 )y0 (cf. (1)).
                                                 2n
                                                          P Since y0 6=         0 and Ln (x0 ) ≥ 0, it follows that Ln (x0 ) = 0 for all
n = 0, 1, 2, . . . Thus, for any y ∈ R, |φ(x0 + iy)|2 = ∞     n=0 Ln (x 0 )y 2n
                                                                                = 0. But this implies that φ(x) ≡ 0, contrary to our
assumption, and whence φ(x) has only real zeros.


Remarks 1.4.
The action of the non-linear operators {Ln }∞   n=0 taking a real entire function φ(x) to Ln (x, φ) = Ln (x) is given by
equation (1). We mention here, parenthetically, a couple facts about these operators. It is known that the operators Ln
satisfy a simple recursive relation [3, Theorem 2.1] and that Ln (x) is also a real entire function [3, Remark 2.4]. Interesting
generalizations of these operators are given by Dilcher and Stolarsky [5] and Cardon [2] (see also Section 3).



2.     Proof of the main result
The proof of the main result (Theorem 2.3) hinges on the Borel–Carathéodory inequality for the half-plane, C+ = {z ∈ C :
Im z > 0}, and on a theorem of Lindelöf. For ease of reference, we commence here with the statements (and citations)
of these theorems.


Theorem 2.1 (Borel–Carathéodory Inequality [9, p. 18]).
Suppose that f : C+ → C+ is an analytic function. Then for z ∈ C+ and |z| > 1, f satisfies the inequalities

                                         sin θ                      r
                                                                                   z = reiθ ,
                                1
                                  |f(i)|       < |f(z)| < 5|f(i)|       ,                       0 < θ < π.
                                5          r                      sin θ


As usual, n(r) will denote the number of zeros of f(z) in the closed disk {z ∈ C : |z| ≤ r}. Our proof will also require the
determination of the type of an entire function f(z) of positive integral order. This is a subtle matter, since the order can
be larger than the number of zeros, n(r), would indicate. In order to handle this issue, we will appeal to the following
theorem of Lindelöf.


Theorem 2.2 (Lindelöf’s Theorem [1, p. 27]).
An entire function f(z) of integral order ρ ≥ 1 is of normal (finite) type if and only if (i) n(r) = O(r ρ ), r → ∞, and (ii)
the sums
                                                              X          1
                                              |S(r)| =
                                                                         zρ
                                                              {z:f(z)=0, |z|≤r, z6=0}


are bounded as r → ∞.


The proof of Lindelöf’s Theorem in [1, p. 27] is quite involved and so we refer the interested reader to [6, Chapter 2] or
to an alternate proof suggested in [8, p. 22, Problem 3].


Theorem 2.3.
Let φ(x) denote a real entire function, φ 6≡ 0. If Ln (x) ≥ 0 for all n ∈ N0 and for all x ∈ R, then (i) φ(x) has only real
zeros and (ii) φ(x) = e−ax φ1 (x), a ≥ 0, where the genus of φ1 (x) is 0 or 1. In particular, φ(x) ∈ L-P.
                          2




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                                                                                   The generalized Laguerre inequalities and functions in the Laguerre–Pólya class




       Proof.    Since the real entire function φ(x) (φ 6≡ 0) satisfies the generalized real Laguerre inequalities Ln (x) ≥ 0 for
       all n ∈ N0 and for all x ∈ R, φ(x) has only real zeros (cf. Remarks 1.3). Thus, φ(x) does not vanish on the simply
       connected domain C+ and whence φ(x) has an analytic logarithm there; that is, f(z) = log φ(z) is analytic in C+ . Now
       set u(z) = Re f(z) = log |φ(z)| and v(z) = Im f(z). Then f 0 (z) = ∂u(z)/∂x + i∂v(z)/∂x and so by the Cauchy–Riemann
       equations
                                                           ∂u(z)      ∂v(z)     ∂v(z)     ∂u(z)
                                       g(z) = −f 0 (z) = −       −i         =−        +i        .                             (2)
                                                            ∂x         ∂x        ∂y        ∂y

       Now for z ∈ C+ , ef(z) = φ(z) and e2Ref(z) = |φ(z)|2 ≥ 0. Hence, by (1) and the assumption that Ln (x) ≥ 0, x ∈ R,
       n = 0, 1, 2, . . . , we infer the inequality

                                                                     ∞
                                         ∂ 2Ref(z)           ∂u(z) X
                                           e       = 2e2u(z)      =     Ln (x)2ny2n−1 ≥ 0,                                       x ∈ R,     y > 0.                        (3)
                                        ∂y                    ∂y    n=0



       Therefore, if z = x + iy ∈ C+ , then, by (3), ∂u(z)/∂y ≥ 0 and hence, consulting (2), g : C+ → C+ . Thus, by the
       Borel–Carathéodory inequality (Theorem 2.1), there exists a positive constant C1 , such that


                                                  |g(z)| ≤ C1 |z|     for all        z ∈ Ω = {z : Im z ≥ |Re z| + 1}.


       Integrating g(z) = −f 0 (z) along a linear segment [i, z], where z ∈ Ω, we obtain the following upper estimate with a new
       constant C2 > 0:

                                         Z                                     Z                
                             |f(z)| =             f 0 (w) dw + f(i) ≤ C1 |z|                 |dw| + |f(i)| ≤ C2 |z|2                      for all   z ∈ Ω.                (4)
                                          [i,z]                                      [i,z]



       Next, we fix |z| = r ≥ 1 and note that since φ is a real entire function |φ(x + iy)| = |φ(x − iy)| for all x, y ∈ R. Since by
       assumption Ln (x) ≥ 0 for all n ∈ N0 and for all x ∈ R, it follows from (1) that for each fixed x, |φ(x + iy)| is an increasing
       function of y, for y ≥ 0. With the aid of these observations, and choosing r ≥ 1 we deduce the following estimates:


                  max |φ(x + iy)| ≤ max |φ(x + i(r + 1))| = max eRef(x+i(r + 1)) ≤ max e |f(x+i(r+1))| ≤ e5C2 r
                                                                                                                                                         2
                                                                                                                                                             (cf. (4)).
                |x + iy|≤r                   −r≤x≤r                            −r≤x≤r                                  −r≤x≤r




       Thus, we have established that the real entire function φ is of order ρ(φ) ≤ 2 and that if ρ(φ) = 2, then φ is of normal
       type. Accordingly, we consider two cases. In the first place, if ρ(φ) < 2, then by the Hadamard Factorization Theorem,
       the canonical product of the zeros of φ has genus at most 1 (see, for example, [1, p. 22]). Secondly, suppose that ρ(φ) = 2
       and that φ has an infinite number of zeros {xk }∞
                                                       k=1 , xk 6= 0, k ≥ 1. In this case, since φ is of normal type, we can invoke
       Lindelöf’s Theorem (Theorem 2.2) to conclude that the sums

                                                                                              X 1
                                                                          |S(r)| =
                                                                                                        xk2
                                                                                              |xk |≤r



       are bounded as r → ∞. Therefore, ∞
                                        P
                                           k=1 1/xk < ∞ and the genus of the canonical product of the zeros of φ is again at
                                                  2

       most 1. Consequently, another appeal to the Hadamard Factorization Theorem shows that φ(z) has the representation

                                                                                             ∞                    
                                                                                             Y                z
                                                              φ(z) = ceaz +bz z m                       1−             ez/xk ,
                                                                               2
                                                                                                                                                                          (5)
                                                                                             k=1
                                                                                                              xk

                                                                                                                                                    P∞
       where a, b ∈ R, xk ∈ R \ {0}, m is a non-negative integer, c is a non-zero real number, and                                                   k=1 1/xk < ∞.
                                                                                                                                                            2




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G. Csordas, A. Vishnyakova




In order to complete the proof of the theorem, we need to show that in the above representation (see (5)) a ≤ 0. The
proof hinges on the fact, noted above, that |φ(iy)| is an increasing function of y, y > 0. Now, a calculation shows that
                                                                                  ∞               1/2
                                                                                  Y          y2
                                             h(y) = |φ(iy)| = |c||y|m e−ay                             ,
                                                                              2
                                                                                        1+
                                                                                  k=1
                                                                                             xk2

where we adhere to the usual convention, whereby the empty product has value 1. Then for y > 0, logarithmic
differentiation and some algebraic manipulation yield the following expression
                                                                           ∞
                                                      1 h0 (y)   m        X       1
                                             H(y) =            = 2 − 2a +              .                                    (6)
                                                      y h(y)    y         k=1
                                                                              x 2
                                                                                k + y2

Since h(y) is a positive increasing function for y > 0, it follows that H(y) ≥ 0 for all y > 0 (cf. the left-hand side of (6)).
We next demonstrate that the assumption that a > 0 is untenable for it leads to a contradiction. Let 0 < ε < a/4. Since
the series ∞
          P
             k=1 1/xk converges, there exists a positive integer N such that for all y > 0,
                    2


                                                     ∞                    ∞
                                                     X        1          X    1   ε
                                                                    ≤            < .                                        (7)
                                                         x 2 + y2
                                                    k=N+1 k
                                                                             x
                                                                        k=N+1 k
                                                                               2  3

Fixing N, there exists a positive number y0 , y0 sufficiently large, such that for all y ≥ y0 ,

                                                                          N
                                                m    ε                    X      1     ε
                                                   <         and                      < .                                   (8)
                                                y2   3                       x
                                                                          k=1 k
                                                                               2
                                                                                 + y2  3

Inequalities (7) and (8) show that, if a > 0, then for all y ≥ y0 ,
                                                           ∞
                                      1 h0 (y)   m        X     1             a
                             H(y) =            = 2 − 2a +           < ε − 2a < − 2a < 0.
                                      y h(y)    y            x +y
                                                              2
                                                          k=1 k
                                                                  2           4

This is the desired contradiction and hence a > 0. Thus, φ ∈ L-P.


Remark 2.4.
We remark that an often overlooked fact is that a real entire function of order 2 having only real zeros need not belong
to the Laguerre–Pólya class. It is curious that our detailed, albeit elementary, proof that the coefficient a of z 2 (in eaz )
                                                                                                                            2


is non-positive required the non-negativity of Ln (x) only at x = 0. Perhaps this part of the proof of Theorem 2.3 can be
made more elegant and perspicacious by noting that if the canonical product of the zeros of φ is of order 2, then (by
virtue of the aforementioned argument) φ is of minimal type. However, such an approach may be less transparent if it
suffers from the omission of certain relevant details.



3.     Scholia
In this section, we include the proofs of some cognate results, state an open problem and formulate two conjectures, one
of which is based on Cardon’s recent extension of the Laguerre-type inequalities [2]. To begin with, we consider the
following classical characterization of functions in the Laguerre–Pólya class via the complex Laguerre inequalities.


Theorem 3.1 (Complex Laguerre Inequalities [4, Theorem 3.8]).
Let φ(x), φ(x) 6≡ 0, be a real entire function. Suppose that

                             φ(x) = e−αx φ1 (x),        where     α ≥ 0 and the genus of φ1 (x) is 0 or 1.
                                         2
                                                                                                                            (9)

Then φ(x) ∈ L-P if and only if
                                             |φ0 (z)|2 ≥ Re φ(z)φ00 (z)
                                                                          
                                                                                  for all    z ∈ C.                       (10)


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                                                                   The generalized Laguerre inequalities and functions in the Laguerre–Pólya class




       Once again, our goal is to demonstrate that Theorem 3.1 remains valid if we omit the hypothesis (9). If φ ∈ L-P, then
       φ can be expressed in the form (9) and it is known [4] that φ satisfies the complex Laguerre inequalities (10). Moreover,
       without the hypothesis (9), we can infer from the complex Laguerre inequalities (10) that φ has only real zeros (see, for
       example, [7] or [4, Theorem 3.8]). In light of these remarks, we state the next theorem as follows.


       Theorem 3.2.
       Let φ(x), φ(x) 6≡ 0, be a real entire function. If the complex Laguerre inequalities (10) hold, then φ ∈ L-P.


       Proof.    The proof of this theorem is, mutatis mutandis, the same as the proof of Theorem 2.3. However, the different
       assumptions require appropriate modifications and therefore, even at the risk of pleonasm, we will succinctly sketch
       here the relevant justifications needed. As in the proof of Theorem 2.3, we set f(z) = log φ(z) = u(z) + iv(z), and
       g(z) = −f 0 (z) = −vy (z) + iuy (z), where z ∈ C+ . In order to follow the arguments used in the proof of Theorem 2.3, we
       need to justify two assertions: (i) uy (z) ≥ 0 when z = x + iy ∈ C+ and (ii) |φ(iy)| is an increasing function of y, y > 0.
       In the proof of Theorem 2.3, both claims (i) and (ii) were clear by virtue of the generalized real Laguerre inequalities.
       (i) Here, we consider again the expansion |φ(x + iy)|2 = ∞
                                                                     P
                                                                        n=0 Ln (x)y , x, y ∈ R. Then a calculation shows that, for
                                                                                   2n

       each fixed x ∈ R,

                                           ∞
                      ∂2                  X
                                              (2n + 1)(2n + 2)Ln+1 (x)y2n 2 |φ0 (z)|2 − Re φ(z)φ00 (z) ≥ 0,
                                                                                                     
                          |φ(x + iy)| 2
                                        =                                                                                 z ∈ C.             (11)
                     ∂y 2
                                          n=0



       Therefore, from the convexity condition (11), we deduce that, for each fixed x ∈ R, ∂|φ(z)|2 /∂y = ∂e2u(z) /∂y = 2e2u(z) uy (z)
       is an increasing function of y, y > 0. Now for any x which is not a zero of φ, we have

                                                                                       ∞
                                          ∂e2u(z)                                     X
                                   lim+           = 2 lim+ |φ(z)|2 uy (x + iy) = lim+     2nLn (x)y2n−1 = 0,
                                  y→0      ∂y        y→0                        y→0
                                                                                      n=0



       and hence uy (x + i0) = 0.          Since 2e2u(z) uy (z) is an increasing function of y, 0 = 2e2u(x+i0) uy (x + i0) ≤
       2|φ(x + iy)| uy (x + iy), and uy (x + iy) ≥ 0, y > 0. On the other hand, if φ(x0 ) = 0, then a continuity argument shows
                    2

       again that uy (x + iy) ≥ 0, y > 0. Thus, g : C+ → C+ and the method of proof leading to the product representation
       of φ(z) (cf. (5)) is verbatim the same as in the proof of Theorem 2.3.
       (ii) In order to prove that a ≤ 0 in (5), it suffices to show that |φ(iy)| is an increasing function of y, y > 0. Now, we
       infer from the convexity condition (cf. (11)) that

                                                           Z y
                                                              ∂2                  ∂
                                                    0<             |φ(it)|2 dt =    |φ(iy)|2 ,                                               (12)
                                                            0 ∂t 2               ∂y

       where we have used the fact that ∂/∂y|φ(iy)|2 is an odd function. Thus, it follows from inequality (12) that |φ(iy)| is an
       increasing function of y, y > 0.

       As an immediate consequence of Theorems 2.3 and 3.2, we obtain the following corollary.


       Corollary 3.3.
       Let φ(x), φ(x) 6≡ 0, be a real entire function. Then


                                                (−1)j+n 2n
                                             2n
                                             X          
                                  Ln (x) =                       φ(j) (x) φ(2n−j) (x) ≥ 0,     n ∈ N,     x ∈ R,
                                             j=0
                                                   (2n)!     j

                                                
       if and only if |φ0 (z)|2 ≥ Re φ(z)φ00 (z) for all z ∈ C.


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G. Csordas, A. Vishnyakova




It may be of interest to note here that the complex Laguerre expression can be also formulated in terms of two real
Laguerre-type expressions. Indeed, if φ(x + iy) = U(x, y) + iV (x, y) is a real entire function, then a calculation shows
that
                                |φ0 (z)|2 − Re φ(z)φ00 (z) = Ux2 − UUxx + Vx2 − V Vxx .
                                                          


The preceding analysis (cf. the convexity condition (11)) reveals that the complex Laguerre inequalities are, a fortiori, a
direct consequence of the generalized real Laguerre inequalities. It is the converse implication that does not appear to
be obvious and it motivates us to formulate, in the simplest terms, the following (ostensibly) intriguing, related problem.


Open Problem 3.4.
              ∞
P∞{uk (x)}k=0
Let                  be a sequence of real entire functions. For each fixed x ∈ R, consider the even power series F (y; x) =
   k=0 u k (x)y 2k
                   , x, y ∈ R, with radius of convergence R = ∞, where F (y; x) 6≡ 0. Suppose that u0 (x) ≥ 0 for all x ∈ R
and that ∂2 F (y; x)/∂y2 ≥ 0 for all x, y ∈ R. If F (y; x) is not a polynomial, then under what additional assumptions is it
true that uk (x) ≥ 0 for all k ≥ 1 and all x ∈ R?


We next turn to Cardon’s recent extension of the Laguerre-type inequalities [2]. In order to expedite our presentation of
Cardon’s main result [2, Theorem 2.1], we need to introduce the following functions. Let

                                                         m
                                                         X                 m
                                                                           Y
                                                p(z) =          ck z k =         (z + αj ),                             (13)
                                                          k=0              j=1



be an even polynomial with non-negative real coefficients ck ≥ 0, and suppose that p(z) has at least one non-real zero.
Let φ(x), φ(x) 6≡ 0, be a real entire function and define

                                                m
                                                Y                     ∞
                                                                      X
                                    Φ(z, t) =         φ(z + αj t) =         Ak (z)t k ,        where
                                                j=1                   k=0

                                                          1 dk
                                                                         
                                             Ak (z) =                  t)      ,              k ∈ N0 .
                                                          k! dt k
                                                                  Φ(z,                                                  (14)
                                                                           t=0



Then it is easy to see that Ak (z) is also a real entire function and, as Cardon has remarked, the choice p(z) = 1 + z 2
produces A2k+1 (z) ≡ 0. Moreover, in this case, A2k (z) = Lk (z) for k = 0, 1, 2, . . . Preliminaries aside, we are now in
position to state the following theorem.


Theorem 3.5 (Cardon–Laguerre Inequalities [2, Theorem 2.1]).
Let φ(x) = e−ax φ1 (x), where the genus of the real entire function φ1 (x) is 0 or 1, φ1 (x) 6≡ 0 and a ≥ 0. Let p(z) and
                  2


Ak (z) denote the functions defined above (see (13) and (14)). Then φ(x) ∈ L-P if and only if Ak (x) ≥ 0 for all x ∈ R
and k ∈ N0 .


We expect that appropriate modifications of the techniques adopted in this paper, may lead to a proof of the following
conjecture.


Conjecture 3.6.
Let φ(x) be a real entire function, φ(x) 6≡ 0. Let p(z) and Ak (z) denote the functions defined above (see (13) and (14)).
If Ak (x) ≥ 0 for all x ∈ R and k ∈ N0 , then φ(x) ∈ L-P.


Finally, by way of conclusion, we propose here a more challenging question. While this query has a geometric flavor,
it does belong to the circle of ideas advanced in this investigation. We suspect that even a partial answer to Open
Problem 3.7 could adumbrate new insights and results in the theory of distribution of zeros of entire functions.



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                                                                 The generalized Laguerre inequalities and functions in the Laguerre–Pólya class




       Open Problem 3.7.
       Let Sτ = {z : |Im z| ≤ τ, τ ≥ 0} denote the strip of width 2τ located symmetrically about the real axis. Let f(z) be a
       real entire function. Under what additional assumptions is it possible to characterize non-trivial        regions Ω $ C such
       that if the complex Laguerre inequalities are restricted to Ω; that is, |f 0 (z)|2 ≥ Re f(z)f 00 (z) , z ∈ Ω, then all the zeros
       of f(z) lie in the strip Sτ .



       Acknowledgements
       The second author is deeply grateful to her daughter, Liudmyla Kadets, who always wins her wagers.




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