ITERATED LAGUERRE AND TURÁN INEQUALITIES. Thomas Craven and George Csordas

Transcription

1 ITERATED LAGUERRE AND TURÁN INEQUALITIES Thomas Craven and George Csordas Abstract. New inequalities are investigated for real entire functions in the Laguerre-Pólya class. These are generalizations of the classical Turán and Laguerre inequalities. They provide necessary conditions for certain real entire functions to have only real zeros.. Introduction and notation. Definition.. A real entire function ϕx := γ =0! x is said to be in the Laguerre- Pólya class, written ϕx L-P, ifϕx can be expressed in the form. ϕx =cx n e αx +βx ω = + xx e x x, 0 ω, where c, β, x R, α 0, n is a nonnegative integer and by convention, the product is defined to be. = /x <. Ifω= 0, then, 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 [, p. 00], [] or [9, Kapitel II]. If ϕx := γ =0! x L-P, then the Turán inequalities γ γ γ + 0and the Laguerre inequalities ϕ x ϕx ϕ + x 0 are nown to hold for all =,,... and for all real x see [] and the references contained therein. In this paper we consider generalizations of both of these inequalities. For some of these generalizations to hold, we must restrict our investigation to the following subclass of L-P. 99 Mathematics Subect Classification. Primary 6D05; Secondary 6C0, 30C5. Key words and phrases. Laguerre-Pólya class, multiplier sequence, Hanel matrix, real zeros. Typeset by AMS-TEX

2 THOMAS CRAVEN AND GEORGE CSORDAS Definition.. A real entire function ϕx := γ =0! x in L-P is said to be in L-P + if γ 0 for all. In particular, this means that all the zeros of ϕ lie in the interval, 0]. Now if ϕx L-P +,thenϕcan be expressed in the form ω. ϕx =cx n e + βx xx, 0 ω, = where c, β 0, x > 0, n is a nonnegative integer and = /x < [9, 9]. If ϕx = γ =0! x L-P, then, following the usual convention, we call the sequence of coefficients, {γ } =0,amultiplier sequence. In Section, the Laguerre inequality ϕ x ϕxϕ x 0 is generalized to a system of inequalities L n ϕx 0 for all n =0,,,... and for all x R, wherel ϕx = ϕ x ϕxϕ x andϕx L-Pcf. [0, Theorem ]. This system of inequalities characterizes functions in L-P Theorem.. We show that the nonlinear operators L n satisfy a simple recursive relation Theorem. and use this fact to give a different proof of a result of Patric [0, Theorem ]. This, together with the converse of Patric s theorem cf. [5, Theorem.9], yields a necessary and sufficient condition for a real entire function with appropriate restrictions on the order and type of the entire function to belong to the Laguerre-Pólya class Theorem.. In Section 3, we consider a different collection of inequalities based on the Laguerre inequality, namely an iterated form of them. Our original proof of the second iterated inequalities for functions in L-P + [, Theorem.3] was based on the study of certain polynomial invariants. In Section 3, we give a shorter and a conceptually simpler proof of these inequalities Proposition 3.4 and Theorem 3.5. Moreover, the proof of Proposition 3.4 leads to new necessary conditions for entire functions to belong to L-P + Corollary 3.6. Evaluating these at x = 0 yields the classical Turán inequalities and iterated forms of them, considered in Section 4. In Section 4, we show that for multiplier sequences which decay sufficiently rapidly all the higher iterated Turán inequalities hold Theorem 4.. Our main result Theorem 5.5 asserts that the third iterated Turán inequalities are valid for all functions of the form ϕx =x ψx, where ψx L-P +. An examination of the proof of Theorem 5.5 see also Lemma 5.4 shows that the restriction that ϕx has a double zero at the origin is merely a ploy to render the, otherwise very lengthy and involved, computations tractable.. Characterizing L-P via extended Laguerre inequalities. Let ϕx denote a real entire function; that is, an entire function with only real Taylor coefficients. Following Patric [0], we define implicitly the action of the nonlinear operators {L n } n=0,taingϕxtol nϕx, by the equation. ϕx + iy = ϕx + iyϕx iy = L n ϕxy n, x, y R. n=0

3 ITERATED LAGUERRE AND TURÁN INEQUALITIES 3 In the sequel, it will become clear that L n ϕx is also a real entire function cf. Remar.4. In [0], Patric shows that if ϕx L-P, thenl n ϕx 0 for all n =0,,,... and for all x R. The novel aspect of our approach to these inequalities is based on the remarable fact that the operators L n satisfy a simple recursive relation Theorem.. By virtue of this recursion relation, we obtain a short proof of Patric s theorem. This, when combined with the nown converse result [5, Theorem.9], yields a complete characterization of functions in L-P Theorem.. We remar that generalizations of the operators defined in. are given by Dilcher and Stolarsy in [6]. These authors study the distribution of zeros of L m n ϕx for their generalized operators L m n and certain functions ϕx. Theorem.. Let ϕx be any real entire function. Then the operators L n satisfy the following: L n x + aϕx = x + a L n ϕx + L n ϕx, fora Rand n =,,...; L 0 ϕ =ϕ ; 3 L n c =0for any constant c and n. Proof. Parts and 3 are clear from the definition. To chec, we compute as follows: x + a + iyϕx + iy =x+a +y L n ϕxy n =x+a =x+a n=0 n=0 n=0 =x+a L 0 ϕx + L n ϕxy n + L n ϕxy n + L n ϕxy n+ n=0 L n ϕxy n n= [x + a L n ϕx + L n ϕx]y n, n= from which follows. Using the recursion of Theorem., we obtain the following characterization of functions in L-P. Theorem.. Let ϕx 0 be a real entire function of the form e αx ϕ x, where α 0and ϕ x has genus 0 or. Then ϕx L-P if and only if L n ϕ 0 for all n =0,,,... Proof. First assume that all L n ϕ 0. If ϕ/ L-P,thenϕhas a nonreal zero z 0 = x 0 +iy 0

4 4 THOMAS CRAVEN AND GEORGE CSORDAS with y 0 0. Hence 0= ϕz 0 = n=0 L n ϕx 0 y n 0. Since all terms in the sum are nonnegative and y 0 0,wemusthaveL n ϕx 0 = 0 for all n. But this gives ϕx 0 + iy = n=0 L nϕx 0 y n = 0 for any choice of y R, whence ϕ itself must be identically zero. Conversely, assume that ϕ L-P. Sinceϕcan be uniformly approximated on compact sets by polynomials with only real zeros, it will suffice to prove that L n ϕ 0forpolynomials ϕ. For this we use induction on the degree of ϕ. From Theorem. and 3, we see that L n ϕ 0 for any n if ϕ has degree 0. If the degree of ϕ is greater than zero, we can write ϕx =x+agx, where a R and gx is a polynomial. By the induction hypothesis, L n gx 0 for all n 0andallx R. Hence, Theorem. gives the desired conclusion for ϕ. Next we show that the explicit form of L n ϕ given in [0] can also be obtained from Theorem.. Theorem.3. For any ϕ L-P, operatorsl n ϕsatisfying the recursion and initial conditions of Theorem. are uniquely determined and are given by. L n ϕx = +n n! n ϕ xϕ n x. Proof. As before, it will suffice to prove the result for polynomials in L-P. It is clear that the recursion formula of Theorem., together with the initial conditions given there, uniquely determine the value of L n on any polynomial with only real zeros. Thus it will suffice to show that the formula given in. satisfies the conditions of Theorem.. We do a double induction, beginning with an induction on n. If n =0,thenL 0 ϕ=ϕ by Theorem. and this agrees with.. Assume that n>0 and that. holds for L n. Now we begin an induction on the degree of ϕ. If ϕ is a constant, then L n ϕ = 0 by Theorem., which agrees with.. Assume the formula holds for polynomials of degree less than deg ϕ. Then we can write ϕx = x+agx, where a R and L n g andl n g are given by.. The conclusion now

5 ITERATED LAGUERRE AND TURÁN INEQUALITIES 5 follows from a computation using Theorem.. Indeed, L n ϕx = L n x + agx = x + a L n gx + L n gx =x+a +n n g xg n x n! + n +n n! n Also, using Leibniz s formula for higher derivatives of a product, +n n! = n ϕ xϕ n x +n n! g xg n x. n [n g g n +x+ag g n +x+an g g n +x+a g g n ] =x+a +n n g xg n x n! + n +n n! n g xg n x, because the coefficient of x + a is shown to be zero by since +n n! = n n! = n n! =0 n [g g n +n g g n ] [ n n [ + n += + n n g g n + n + n +g g n ] n n g g n n n. n g g n ]

6 6 THOMAS CRAVEN AND GEORGE CSORDAS The main emphasis here has been that the result of Theorem. depends only on the recursive condition of Theorem., and this seems to be the easiest way to prove Theorem.. However, the operators L n ϕ can be explicitly computed more easily than from the recursive condition as was done in Theorem.3, as well as in greater generality. Remar.4. For any real entire function ϕ, the operators L n ϕ defined by equation. are given by the formula L n ϕx = +n n! n ϕ xϕ n x. Proof. By Taylor s theorem, for each fixed x R, hy := ϕx+iy = ϕx + iyϕx iy = n=0 h n 0 y n, n! wherewehaveusedthefactthathy is an even function of y. Let D y = d/dy denote differentiation with respect to y. Then by Leibniz s formula, for higher derivatives of a product, we have h n 0 = = =0 =0 n D y ϕx + iy y=0 D n y ϕx iy y=0 n n+ ϕ xϕ n x =n!l n ϕx, by the uniqueness of the Taylor coefficients. 3. Iterated Laguerre inequalities. Definition 3.. For any real entire function ϕx, set and for n, set T n T ϕx := ϕ x ϕ xϕ + x if, ϕx := T n ϕx T n ϕx T n + ϕx if n. Remars. a Note that with the notation above, we have T n n + ϕ =T ϕ for nand =0,,...

7 ITERATED LAGUERRE AND TURÁN INEQUALITIES 7 b The authors investigations of functions in the Laguerre-Pólya class [], [3] have led to the following problem. Open Problem If ϕx L-P +, are the iterated Laguerre inequalities valid for all x 0? That is, is it true that 3. T n ϕx 0 for all x 0 and n? c If we assume only that ϕx L-P, then the inequality T n ϕx 0, x 0, need not hold in general, as the following example shows. Consider, for example, ϕx = x x+ L-P. ThenT ϕx = 6x +3x+x 3 and so we see that T ϕx is negative for all sufficiently small positive values of x. d There are, of course, certain easy situations for which the iterated Laguerre inequalities can be shown to always hold. For example, if ϕx =x+ae x,a 0or ϕx=x+ax + be x, a, b 0, this is true. Since the derivative of such a function again has the same form, the remars above indicate that it suffices to show that T ϕx 0 for =,,... and all x 0. For the quadratic case, we obtain T ϕx { e x a + x +b+x + x + a + b +, for odd = e x x + ax + b+x + a + b +, for even and each expression is clearly nonegative for all real x. e A particularly intriguing open problem is the case of ϕx =x m in 3.. Special cases, such as the iterated Turán inequalities discussed in the next section, can be easily established i.e. T n n x n =n! n, but the general case of T n n x n+, =0,,,..., seems surprisingly difficult. In [, Theorem.3] it is shown that 3. is true when n = ; that is the double Laguerre inequalities are valid. Here we present a somewhat different and shorter proof which still depends on Theorems. and.3 in the hope that it will shed light on the general case. Proposition 3.4. If ϕx is a polynomial with only real, nonpositive zeros and positive leading coefficient so that ϕx L-P + R[x], then 3. T ϕx 0 for all x 0 and. Proof. First we prove 3. by induction, in the special case when =. Ifdegϕ= 0 or, then T ϕ = 0. Now suppose that 3. holds with = for all polynomials g L-P +

8 8 THOMAS CRAVEN AND GEORGE CSORDAS of degree at most n. Let ϕx :=x+agx, where a 0. For notational convenience, set hx :=T gx = g x gxg x and note that hx is ust L gx in Theorem.3. Then some elementary, albeit involved, calculations which can be readily verified with the aid of a symbolic program yield { } 3.3 ϕxt ϕx = ϕ x x + a 4 T hx + ϕx[ϕxl gx + Ax], where L gx is given by. and Ax =8g x 3 gxg xg x+4gx g x. Since ϕx,ϕ x L-P +, ϕx 0andϕ x 0 for all x 0. Also, by Theorem., L gx 0 for all x R. Now, another calculation shows that g xt hx = gxt gx and so T hx 0forx 0, since by the induction assumption T gx 0for x 0. Therefore, it remains to show that Ax 0forx 0. Let gx =c n x + x, where c>0andx 0for n. Then using logarithmic differentiation and the product rule we obtain 3.4 Ax =4gx 3 d dx g x. =4gx 3 gx n 0 for all x>0. x + x 3 Thus, the right-hand side of 3.3 is nonnegative for all x 0 and whence T ϕx 0if x>0. But then continuity considerations show that T ϕx 0 for all x 0. Finally, since L-P + is closed under differentiation and since T n + =0,,... see Remar a, we conclude that 3. holds. n ϕ =T ϕ for nand Recall from the introduction, that if ϕx L-P +,thenϕx can be expressed in the form ω 3.5 ϕx =ce σx + x/x, 0 ω, where c 0, σ 0, x > 0and /x <. Now set ϕ N x =c + σx N N minn, ω + xx. Then ϕ N x ϕx asn, uniformly on compact subsets of C. Moreover, the class L-P + is closed under differentiation, and so the derivatives of ϕx can also be expressed in the form 3.5. Therefore, the following theorem is an immediate consequence of Proposition 3.4.

9 ITERATED LAGUERRE AND TURÁN INEQUALITIES 9 Theorem 3.5. If ϕx L-P +, then for =0,,..., T ϕ x 0 for all x 0 and. In the course of the proof of Proposition 3.4, we have shown see 3.4 that for polynomials gx L-P +, the following inequality holds 3.6 g x 3 3gxg xg x+gx g x 0 for all x 0. Next, we employ the foregoing limiting argument see the paragraph preceding Theorem 3.5 and the fact that L-P + is closed under differentiation, to deduce from 3.6 the following corollary. Corollary 3.6. If ϕx = =0 γ! x L-P +, then for p =0,,... and for all x 0, 3 ϕ 3.7 ϕ x p+ 3ϕ p xϕ p+ xϕ p+ x+ ϕ x p p+3 x 0. The interest in inequality 3.7 stems, in part, from the fact that for x = 0 it provides a new necessary condition for a real entire function to belong to L-P +. Indeed, for x =0, inequality 3.7 may be expressed in the form 3.8 γ p+ γ p+ γ p γ p+ γp γ p+ γ p+ γ p γ p+3 p =0,,... The Turán inequalities γ p+ γ pγ p+ 0 imply that γ p+ γ p+ γ p γ p+3 0. Thus, if γ p > 0 for all p 0, then γ p γ p+ γ p+ γ p γ p+3 /γ p+ is a nontrivial positive lower bound for the Turán expression γ p+ γ pγ p+. 4. Iterated Turán inequalities. Let Γ = {γ } =0 be a sequence of real numbers. We define the r-th iterated Turán sequence of Γ via γ 0 = γ, =0,...,andγ r =γ r γ r γr +, = r, r +,... Thus, if we write ϕx = γ x /!, then γ r is ust T r ϕx evaluated at x = 0. Under certain circumstances, we can show that all of the higher iterated Turán expressions are positive for a multiplier sequence. In Section 3 we mentioned some simple cases in which we could, in fact, show that all of the iterated Laguerre inequalities hold. In this section we establish the iterated Turán inequalities for a large class of interesting multiplier sequences.

10 0 THOMAS CRAVEN AND GEORGE CSORDAS Theorem 4.. Fix c and d 0. Consider the set M c of all sequences of positive numbers {γ } =0 satisfying 4. γ cγ γ + 0, for all. Then 4. γ γ γ + c + dγ γ γ γ + γ γ + 0 for all and all sequences in M c if and only if c d. Proof. To see necessity, consider the specific sequence γ 0 =,γ =,γ = /b, γ 3 = /cb,γ =0for 4. This satisfies 4. for any b c. But 4. yields b cb = c b 4 c + d b c b 4 c 3c++ c b d+d. b Since b may be made as large as desired, this is only guaranteed to be nonnegative if c d, the larger root of c 3c + d. The other alternative, of c 3 5+4d does not occur for d> and, in particular, for d 0. Conversely, assume 4. holds with c d. An upper bound for γ is γ. From 4., we obtain the lower bound γ = γ γ γ + c γ γ +. Estimating the expression in 4., we obtain γ c + dγ γ + [c γ γ + ] c + dγ γ + =[c 3c+ d]γ γ + 0 by the condition on c. The set M 4 is of particular interest. Condition 4. forces the numbers γ to decrease rather quicly, leading us to term such sequences rapidly decreasing sequences. They are nown to be multiplier sequences and were first investigated in some detail in [8]. These interesting sequences are discussed at some length in [3, Section 4] and [4, Section 4].

11 ITERATED LAGUERRE AND TURÁN INEQUALITIES Corollary 4.. For a sequence as in 4. with c> , the corresponding constant for the sequence of Turán expressions γ = γ γ γ + is strictly greater than c by the amount d = c 3c +. If we then iterate this, forming the sequence {γ },the corresponding constant again increases by more than d. After a finite number of steps, it will reach 4 in the normalized case γ 0 = γ = and the sequence of higher Turán expressions {γ r } =r,forrfixed and sufficiently large, will be a rapidly decreasing sequence. In particular, if the original sequence is a rapidly decreasing sequence, the sequence of Turán inequalities is again a rapidly decreasing sequence and we obtain an infinite sequence of multiplier sequences by iterating this process. Although the iterated Turán expressions seem to be positive for all multiplier sequences an open question in general, it follows from the Theorem 4. that inequality 4. with c = is not sufficient to achieve this since M contains sequences that fail to satisfy 4. for d = 0. But then, the specific sequence used in the proof is not a multiplier sequence if c =, as it violates condition 3.8 for p =. 5. The third iterated Turán inequality. In this section we establish the third iterated Turán inequality γ 3 0=3,4,5... for multiplier sequences, {γ } =0, of the form γ = α, =,,3...,where {α } =0 is an arbitrary multiplier sequence. With the notation adopted in Section 4, we have 5. γ 3 =γ γ γ +, =3,4,5,..., or equivalently 5. where T 3 ϕx = T ϕx T ϕxt + ϕx, =3,4,5,..., x=0 x=0 5.3 ϕx := =0 γ! x L-P +. Before embaring on the proof of the third iterated Turán inequality, we briefly discuss a representation of the third iterated Turán expression γ 3 = T 3 ϕx in terms x=0 of Wronsians and determinants of Hanel matrices Proposition 5.. We recall that the n th order Wronsian determinant W ϕx,ϕ x,...,ϕ n x, where ϕx isan entire function, is defined as ϕx ϕ x ϕ n x 5.4 W ϕx,ϕ x,...,ϕ n ϕ x ϕ x ϕ n x x :=,... ϕ n x ϕ n x ϕ n x

12 THOMAS CRAVEN AND GEORGE CSORDAS and that the n th order Hanel matrices, associated with the sequence {γ } =0, are matrices of the form H n =γ +i+ n i,,thatis H n = γ γ +... γ +n γ + γ +... γ +n+... γ +n γ +n... γ +n n =,,3,..., =0,,,... We note that if we set A n =deth n,thenwϕ 0,ϕ + 0,...,ϕ +n 0 = A n and for n = 3 the following relation holds 5.5 γ + A 3 = γ + =0,,... Furthermore, if ϕx L-P + is given by 5.3 and if γ > 0, then by Theorem 3.5, T ϕx 0forx 0=,3,4,... and whence, in light of 5.5, A 3 x=0 0for=0,,... A straightforward, albeit lengthy, calculation yields the following representation of the third iterated Turán expression γ 3. Proposition 5.. Let ϕx:= γ =0! x be an entire function. Then for x R, 5.6 T 3 ϕx = T ϕx W ϕ 3 x,ϕ x,ϕ x,ϕ x ϕ x + T ϕxt + ϕx ϕ xϕ + x for =3,4,5... In particular, if x =0and =0,,...,then 5.7 γ 3 +3 = T 3 +3 x=0 ϕx =γ +3 γ + γ +4 A 4 γ +3 + A 3 A3 +, where A n =deth n denotes the determinant of the Hanel matrix H n. Remars 5.. a Since the equalities 5.6 and 5.7 are formal identities, the assumption that ϕx is an entire function is not needed. b With the aid of some nown identities see, for example, [3, VII, Problem 9], equation 5.7 can be recast in the following suggestive form 5.8 γ 3 +3 = A 3 + γ +3 A 3 A3 + γ +γ +4. Now, suppose that ϕx := whenever A A A3 + =0 γ! x L-P +. Then, by virtue of 5.8, γ , =0,,... However, this inequality is not valid,

13 ITERATED LAGUERRE AND TURÁN INEQUALITIES 3 in general, as the following example shows. Let ϕx := γ =0! x = x x +. Here, γ 0 = γ =0,γ =,γ 3 =66,γ 4 = 30,γ 5 = 9800 and γ 6 = Then A 3 0 A3 = A 3 c Let ϕx L-P + be given by 5.3. Since A 3 A3 + 0 cf. 5.5 and Theorem 3.5, 5.7 shows that γ whenever A4 0. However, A 4 may be negative, as may be readily verified using the function ϕx defined in part b. Examples of this sort are subtle as they depict a heretofore inexplicable phenomenon. The technique used below sheds light on this and at the end of this paper we provide a sufficient condition which guarantees that A 4 < 0. In connection with the investigations of a conecture of S. Karlin, additional examples are considered in [7] and []. Furthermore, to highlight the intricate nature of Karlin s conecture, it was pointed out in these papers, in particular, that A 4 0ifγ =α /!, =0,,...,where{α } =0 is any multiplier sequence. d In the sequel we will prove that if ϕx := =0 α x L-P +,! where {α } =0 is any multiplier sequence, then γ It is not hard to see that this is equivalent to proving the result only for multiplier sequences that begin with two zeros. However, the assumption that {α } =0 is a multiplier sequence is not necessarily required to mae the inequalities hold. To see this, we consider once again the example in part b. Let α 0 = α = 0 and for, set α = γ. We claim that {α } =0 is not a multiplier sequence. Indeed, consider the fourth Jensen polynomial defined, for example, in [] associated with the sequence {α } =0,thatis, g 4 x= 4 =0 4 α x =x 3 + x +55x. Since g 4 x has two nonreal zeros, {α } =0 is not a multiplier sequence, though our main theorem will establish the third iteration of the Turán inequalities for {γ } =0. The proof of the main theorem requires that we express T 3 3 ϕx in terms of x=0 sums of powers of the logarithmic derivatives of ϕx. Accordingly, we proceed to establish the following preparatory result. Lemma 5.3. Let ϕx= n x + x, x > 0,,,...,n, be a polynomial in L-P +. For fixed x 0 and =,,...,n,seta := x+x and let 5.9 A := a, B := a, n C := n a 3, and D := a 4.

14 4 THOMAS CRAVEN AND GEORGE CSORDAS Then 5.0 ϕ x ϕx = A, ϕ x ϕx = A B, ϕ x ϕx = A3 3AB +C and ϕ 4 x ϕx = A 4 6A B +3B +8AC 6D. Proof. Logarithmic differentiation yields ϕ x n ϕx = and ϕ x =ϕ x ϕx x + x x + x x + x. Hence, x + x = x + x ϕ x ϕ ϕx = x ϕx x + x x + x = A B. Continuing in this manner, similar calculations yield 3 ϕ x ϕx = 3 + x + x x + x x + x x + x 3 =A 3 3AB +C and ϕ 4 x ϕx 4 = 6 x + x x + x x + x +3 x + x +8 6 x + x x + x 3 x + x x + x 4 =A 4 6A B +3B +8AC 6D. The next lemma gives an explicit expression for γ 3 3 = T 3 3 ϕx,whereϕxis x=0 of the form ϕx =x ψx. While the verification involves only simple algebraic manipulations, the expression obtained is sufficiently involved to warrant the use of a computer.

15 ITERATED LAGUERRE AND TURÁN INEQUALITIES 5 Lemma 5.4. Let ψx:= α =0! x be an entire function. Let ϕx =x ψx= =0 γ! x, so that γ 0 = γ =0and γ = α,for=,3,...then 5. γ 3 3 = T 3 3 ϕx = ψ 0 ψ0 ψ 0 E0, where x=0 Ex :=79 ψ x ψx ψ x 4 ψ x + 34 ψx ψ x ψ x + 6 ψx 3 ψ x 3 +54xψx ψ x 3 ψ 3 x 360 ψx 3 ψ x ψ x ψ 3 x +00 ψx 4 ψ 3 x 90 ψx 4 ψ x ψ 4 x. Preliminaries aside, we are now in a position to prove the principal result of this section. Theorem 5.5. Let ψx:= α =0! x L-P +.Let ϕx=x ψx= =0 γ! x, so that γ 0 = γ =0and γ = α,for=,3,...then 5. γ 3 3 = T 3 3 ϕx 0. x=0 Proof. In view of 5. of Lemma 5.4, since ψx L-P +, T ψx x=0 0=,,3..., we only need to establish that E0 0. Also, since ψx L-P +, ψx can be uniformly approximated, on compact subsets of C, by polynomials having only real, nonpositive zeros. Therefore, it suffices to prove inequality 5. when ψx = n x + x, x 0, is a polynomial in L-P +.Nowifψ0 = 0, then E0 = 79 ψ 0 6 0and so in this case inequality 5. is clear. Thus, henceforth we will assume that ψ0 0 and, for fixed x 0, consider Ex as given in Lemma 5.4. We will prove a stronger result, namely, that for all x 0, Ex ψx 6 =79 ψ x 6 ψx ψ x ψ x ψx ψ x 4 ψ x ψx ψ x ψ x ψ 3 x ψx ψ 3 x ψx ψ3 x ψx ψ x 3 ψ 3 x ψx 4 90 ψ x ψ 4 x ψx 0.

16 6 THOMAS CRAVEN AND GEORGE CSORDAS For fixed x 0, by Lemma 5.3 with ψ in place of ϕ, weobtain or Ex ψx 6 =79A6 458A 4 A B + 34A A B +6A B A 3 A 3 3AB +C 360AA BA 3 3AB +C +00A 3 3AB +C 90A BA 4 6A B +3B +8AC 6D Ex ψx 6 =A6 +A 4 B 8 A B +54B 3 +40A 3 C +40 ABC C A D 540 BD =A 6 +B A 3B 63 B A 3 C + 40 ABC+ 400 C A B D. Since x > 0for,...,n,wehaveA, B, C, D > 0 and all the derivatives of ψx are positive for x 0. Therefore we also have A B = ψ x/ψx > 0, and thus Ex > 0forx 0. Remars 5.6. a We wish to point out that in Theorem 5.5 we introduced the factor x in order to simplify the ensuing algebra. In the absence of this factor we would have to calculate ϕ5 x ϕx as well as ϕ6 x ϕx see the proof of Lemma 5.3. Then, as in the proof of Theorem 5.5, we would obtain an expression, analogous to Ex, which has terms rather than nine. Nevertheless, it seems that the technique developed above, should yield the desired result 5. for an arbitrary multiplier sequence rather than one with the first two terms equal to zero. b We briefly indicate here how the foregoing technique can be used to derive a sufficient condition which guarantees that A 4 0 =deth 4 0 < 0. Let ϕx :=x ψx, where ψx = n x + x, x > 0, is a polynomial in L-P +. Then, the determinant of the 4 th order Hanel matrix ϕ i+ 0 4 i, reduces to 5.3 A 4 0 =W ϕ0,ϕ 0,ϕ 0,ϕ 0 =48 7 ψ ψ0 ψ 0 ψ 0 + ψ0 ψ 0 +0 ψ0 ψ 0 ψ ψ0 3 ψ 4 0. Guided by 5.3 and the argument used in the proof of Theorem 5.5, we form the expression Kx = 7 ψ x 4 ψx 4 54 ψ x ψ x ψx 3 + ψ x ψx + 0 ψ x ψ 3 x ψx 5 ψ4 x ψx

17 ITERATED LAGUERRE AND TURÁN INEQUALITIES 7 and with the aid of Lemma 5.4, for fixed x 0, we obtain that Kx = 30 D B, where the quantities B = n a and D = n a4 have the same meaning as in 5.9. Thus, we readily infer that if the the zeros of the polynomial ψx L-P + are distributed such that 0 D<B holds at x =0,thenA 4 0 < 0. By way of illustration, consider ϕx =x ψx=x x+a,wherea>0. Then for x = 0, we find that 0D = 0/a 4 < 44/a 4 = B, and whence by our criterion, A 4 0 < 0. Indeed, a direct computation yields that A 4 0 = 3456a 44. The authors wish to than the referee for extensive comments and suggestions. References. T. Craven, G. Csordas, Karlin s conecture and a question of Pólya to appear.. T. Craven and G. Csordas, Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math , T. Craven and G. Csordas, Problems and theorems in the theory of multiplier sequences, Serdica Math. J. 996, T. Craven and G. Csordas, Complex zero decreasing sequences, Methods Appl. Anal. 995, G. Csordas and R. S. Varga, Necessary and sufficient conditions and the Riemann Hypothesis, Adv. in Appl. Math. 990, K. Dilcher and K. B. Stolarsy, On a class of nonlinear differential operators acting on polynomials, J. Math. Anal. Appl , D. K. Dimitrov, Counterexamples to a problem of Pólya and to a problem of Karlin, East J. Approx , J. I. Hutchinson, On a remarable class of entire functions, Trans. Amer. Math. Soc. 5 93, N. Obreschoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, M. Patric, Extensions of Inequalities of the Laguerre and Turán type, Pacific J. Math , G. Pólya, Collected Papers, Vol. II Location of Zeros, R. P. Boas, ed., MIT Press, Cambridge, MA, G. Pólya and J. Schur, Über zwei Arten von Fatorenfolgen in der Theorie der algebraischen Gleichungen, J. ReineAngew. Math , 89 3.

18 8 THOMAS CRAVEN AND GEORGE CSORDAS 3. G. Pólya and G. Szegö, Problems and Theorems in Analysis, Vols. I and II, Springer-Verlag, New Yor, 976. Department of Mathematics, University of Hawaii, Honolulu, HI 968 Department of Mathematics, University of Hawaii, Honolulu, HI 968 address: tom@math.hawaii.edu, george@math.hawaii.edu