Journal of Approximation Theory 132 (2005) 212–223 www.elsevier.com/locate/jat Almost strict total positivity and a class of Hurwitz polynomials Dimitar K. Dimitrova,∗,1, Juan Manuel Peñab,2 aDepartamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, Brazil bDepartamento de Matemática Aplicada, Universidad de Zaragoza, Spain Received 7 February 2004; received in revised form 11 October 2004; accepted in revised form 27 October 2004 Communicated by Allan Pinkus Available online 28 December 2004 Abstract We establish sufficient conditions for a matrix to be almost totally positive, thus extending a result of Craven and Csordas who proved that the corresponding conditions guarantee that a matrix is strictly totally positive. Then we apply our main result in order to obtain a new criteria for a real algebraic polynomial to be a Hurwitz one. The properties of the corresponding “extremal” Hurwitz polynomials are discussed. © 2004 Elsevier Inc. All rights reserved. Keywords:Totally positive matrix; Strictly totally positive matrix; Shadows’ lemma; Hurwitz polynomial; Entire function in the Laguerre–Pólya class 1. Introduction A real matrix is calledtotally positive(TP) if all its minors are nonnegative andstrictly totally positive(STP) if they are positive. Many properties and a variety of applications of these matrices can be found in the book of Karlin[17] and in the comprehensive survey ∗ Corresponding author. E-mail address:dimitrov@dcce.ibilce.unesp.br(D.K. Dimitrov). 1 Partially supported by the Brazilian Science Foundations CNPq and FAPESP. 2 Partially supported by BFM2003-03510 Research Grant, Spain. 0021-9045/$ - see front matter © 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jat.2004.10.010 http://www.elsevier.com/locate/jat mailto:dimitrov@dcce.ibilce.unesp.br D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 213 paper ofAndo[1] (see also[25]).An interesting sufficient condition for strict total positivity was established by Craven and Csordas in[10]: Theorem A (Craven and Csordas[10] , Theorem 2.2). LetA = (aij )1� i,j�n be a matrix with positive entries and aij ai+1,j+1��ai,j+1ai+1,j , 1� i, j�n− 1, (1) where� ≈ 4.0795956235is the unique real root ofx3 − 5x2 + 4x − 1 = 0. Then A is strictly totally positive. Let us observe that (1) is far from being a necessary condition for strict total positivity. However, it is a rather simple and convenient sufficient condition because it allows the total positivity to be affirmed only by verifying (1) and the positivity of the elements of the matrix, and the inequalities (1) themselves are a condition for the 2× 2 minors ofA composed by consecutive rows and columns. We prove an extension of this result without the requirement that the entries ofA are positive. Applications to the theory of entire function and to the Hurwitz stable polynomials are discussed. We formulate the conjecture that the smallest possible value of the constant� to set in (1) is 4 if one considers matrices of any order and it is 4 cos2(�/(n+ 1)) for n× n matrices. Arguments in support of the conjecture are provided. A special subclass of totally positive matrices, calledalmost strictly totally positive (ASTP), which include those that are strictly totally positive was introduced by Gasca et al.[13]. In order to provide the formal definition of ASTP matrices we need to introduce some notions. Fork, n ∈ N, 1�k�n, byQk,n we denote the set of all increasing sequences of k natural numbers, not exceedingn. By Q0 k,n we shall mean the set of sequences ofk consecutivenatural numbers less than or equal ton. For a realn × n matrixA and a pair of multiindeces� = (�1, . . . , �k), � = (�1, . . . ,�k), �,� ∈ Qk,n, we denote byA[�|�] thek × k submatrix ofA composed by rows�1, . . . , �k and columns�1, . . . ,�k of A. In particular, when� = �, we setA[�] := A[�|�]. Thus, anonsingularmatrixA of ordern is called ASTP if it is totally positive and satisfies the following property: a minor ofA formed by consecutive rows and consecutive columns is positive if and only if all its diagonal entries are positive. Equivalently, detA[�|�] > 0 ⇐⇒ a��,�� > 0, � = 1, . . . , k (2) and it must hold for any�,� ∈ Q0 k,n. It was proved in[13] that, ifA is ASTP, then (2) holds not only for the multiindeces inQ0 k,n but for any�,� ∈ Qk,n. Consequently, for this type of matrices we know exactly the minors which are positive and the ones which are zero. Characterization of ASTP matrices by means of the Neville elimination, in terms of their LU-factorizations, as a product of bidiagonal elementary matrices, as well as in terms of positivity of certain minors determined through the so-called zero patterns, were provided in [14]. Important ASTP matrices are the Hurwitz matrices[3,19] and the B-splines collocation matrices[4]. Some examples of applications of these matrices in Approximation Theory can be seen in[6]. Recently Garloff[12] proved that, whenA1 andA2 are ASTP matrices 214 D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 with A1 ≺ A2, where≺ denotes the so-called chequerboard partial ordering, so are allA satisfyingA1 ≺ A ≺ A2. It is known that no nonsingular TP matrix can have zeros as diagonal entries[1, Corollary 3.8]. Then we can deduce from the shadows’ lemma (see[5, Lemma A]) that, ifA = (aij ) is a nonsingularn× n TP matrix, then aii > 0 for i = 1, . . . , n if aij = 0, i > j thenahk = 0 for all h� i andk�j If aij = 0, i < j thenahk = 0 for all h� i andk�j. (3) Before we state our extension of Theorem A to the class of ASTP matrices, recall that a matrix is called nonnegative (positive) if all its entries are nonnegative (positive). Theorem 1. LetA = (aij ) be a nonnegativen× n matrix satisfying(3). Assume that, for any1� i, j�n− 1, the following condition holds: if aij ai+1,j+1 > 0, thenaij ai+1,j+1��ai,j+1ai+1,j , (4) where� is given in Theorem A. Then A is TP. Moreover, if the second inequality in(4) is strict, then A is nonsingular ASTP. One of the consequences of this result is for the theory of entire functions with real zeros. A real entire function�(x) is said to belong to theLaguerre–Pólya class, written� ∈ L−P, if �(x) can be represented in the form �(x) = cxme−�x2+�x �∏ k=1 (1 + x/xk)e −x/xk , (0���∞), (5) wherec,�, xk are real,��0,m is a nonnegative integer, ∑ x−2 k < ∞ and where the canonical product reduces to 1 when� = 0. Pólya and Schur[28] called the real entire function�(x) a function oftype Iin the Laguerre–Pólya class, written� ∈ L−PI , if �(x) or �(−x) can be represented in the form �(x) = cxme x �∏ k=1 (1 + x/xk), (0���∞), (6) wherec is real, �0,m is a nonnegative integer,xk > 0, and ∑ 1/xk < ∞. It is clear that L − PI ⊂ L − P. The importance of the Laguerre–Pólya classL − P (L − PI , respectively) is revealed by the fact that the functions inL − P (L − PI ), and only these, are the uniform limits, on compact subsets ofC, of polynomials with only real (nonpositive) zeros[21, Chapter VIII]. Pólya and Schur[28] observed that, if a function �(x) := ∞∑ k=0 k xk k! (7) is inL−P and its Maclaurin coefficients k are nonnegative, then� ∈ L−PI . In the same fundamental paper[28] Pólya and Schur introduced the notionmultiplier sequencecalling D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 215 by this any sequence{ k}∞0 of Maclaurin coefficients of a function inL − PI . The reader may consult[8,9], [21, Chapter VIII], [24, Kapitel II], [27] and the references therein for more information about the properties of the functions in the Laguerre–Pólya class. We only mention that a necessary condition for an entire function�(x), defined by (7), to belong to L − PI is that the following Turán inequalities 2 k − k−1 k+1�0, k = 1,2, . . . , hold. As an immediate consequence of Theorem1, we obtain the following sufficient con- ditions of a function to be inL − PI . Corollary 2. If the coefficients k in the formal power series ∑∞ k=0 kx k/k! are positive and satisfy 2 k − k k + 1 � k−1 k+1�0, k = 1,2, . . . , (8) then it represents an entire function�(x) of genus0 and� ∈ L − PI . In particular, if the coefficients k of the polynomialp(z) = ∑n k=0 kx k/k! are positive and satisfy(8) for k = 1, . . . , n− 1, then all the zeros ofp(z) are real and negative. While we were not able to prove Theorem1 with the best possible value 4 instead of the constant� and we provide a short proof of Corollary 2 only for the sake of completeness and as an illustrative application of Theorem1, results corresponding to Corollary2, already with the constant 4 instead of�, are known. In 1923 Hutchinson[16], extending the work of Petrovitch[26] and Hardy[15], proved the following beautiful result for entire function f (x) = ∞∑ k=0 akx k, whose coefficientsak are given bya0 = 1 and ak = 1 b1b2 · · · bk , k = 1,2, . . . . Theorem B ([16, Theorem A. p. 327]). The relations bk�4bk−1, k = 2,3, . . . , (9) are the necessary and sufficient conditions that the seriesf (x) may have the properties: 1. The zeros off (x) are real, simple and negative; and 2. The zeros of any polynomialamxm + · · · + anx n formed by taking any number of con- secutive terms off (x) are all real, simple, and negative(exceptingx = 0). It is worth mentioning a small gap in Hutchinson’s proof. Theorem B is correct either without the statement for simplicity of the zeros of the polynomials in part 2 or if we substitute (9) by the corresponding strict inequalities. Indeed, if we takef (x) = 1 + x + x2/4+· · ·, then the partial sumf2(x) = 1+x+x2/4 has a double root at−2. Observe that 216 D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 the inequalities (9) are equivalent to the inequalitiesa2 k − 4ak−1ak+1�0 for the Maclaurin coefficients off (x) = ∑∞ k=0 akx k, or to 2 k −4 k k+1 k−1 k+1�0 if f (x) = ∑∞ k=0 kx k/k!. Craven and Csordas[9] proved extensions of Hutchinson’s result. Recently Kurtz[20] considered only the polynomial case, and proved that, ifn�2 and the coefficientsak of the polynomial Pn(x) = a0 + a1x + · · · + anx n are all positive and satisfy the inequalities a2 k − 4ak−1ak+1 > 0, k = 1, . . . , n− 1, (10) then all the zeros ofPn(x) are negative and distinct. Moreover, Kurtz observed the sharpness of (10) showing that, for any givenε > 0 andn ∈ N, n�2, there exists a polynomial of degreen, which has some nonreal zeros and whose coefficients are positive and satisfy a2 k − (4 − ε)ak−1ak+1 > 0 for k = 1, . . . , n− 1. However, if one considers entire functions with positive coefficients, i.e. when property 2 in Hutchinson’s theorem is omitted, then the constant� in the inequalities a2 k − �ak−1ak+1 > 0, k = 1,2, . . . for its Maclaurin coefficients may have somehow smaller value than 4. In a very recent paper Katkova et al.[18], studied in details the extremal value of the constant� as well the properties of the corresponding extremal entire function, the one for which inequalities reduce to equalities. Another application of Theorem1 concerns the so-called Hurwitz (stable) polynomials, namely, polynomialsf (z) = cnz n + cn−1z n−1 + · · · + c0 with real coefficientscj , whose zeros have negative real parts. We refer to[11, Chapter 15], [23, Chapter 9]for compre- hensive information on the stability theory. We only mention that a necessary condition for a polynomialf (z) with positive leading coefficient to be Hurwitz one is that all its coefficients are positive. Theorem 3. Let� be defined as in Theorem A. If the coefficients of f (z) = cnz n + cn−1z n−1 + · · · + c0 are positive and satisfy the inequalities ckck+1�� ck−1ck+2 for k = 1, . . . , n− 2, (11) thenf (z) is a Hurwitz polynomial. In particular, the conclusion is true if c2 k� √ � ck−1ck+1 for k = 1, . . . , n− 1. (12) Observe that inequalities (12) imply that the zeros off (z) have zeros with negative real parts while the similar but stronger requirements (10) guarantee that these zeros are real, negative and distinct.We refer to[29,30]for some necessary conditions for a real polynomial to be stable. D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 217 2. Proof of the main result Proof of Theorem 1. Given the matrixA satisfying (4), let us construct ann× n positive matrixB = (bij ) such that, for 1� i, j�n− 1, bij bi+1,j+1��bi,j+1bi+1,j . (13) For any(i, j) such thataij �= 0, we definebij := aij . If {(i, j)| aij = 0, |i − j | = 1} = {(i11, j1 1 ), . . . , (i 1 r1 , j1 r1 )} with i11 � i12 � · · · � i1r1 and, if i1k = i1k+1 for somek, thenj1 k < j1 k+1, clearly we can choose positive numbers bi11,j 1 1 , . . . , bi1r1,j 1 r1 such that (13) holds for all 1� i = j�n− 1. Let us now continue to fill in the lower triangular part ofA. If {(i, j)| aij = 0, i − j = 2} = {(i21, j2 1 ), . . . , (i 2 r2 , j2 r2 )} with i21 < i 2 2 < · · · < i2r2, then we can choose positive numbersbi21,j 2 1 , . . . , bi2r2,j 2 r2 such that (13) holds for all 1� i, j�n− 1 with i − j = 1. Analogously, we can iterate the previous procedure until we obtain all elementsbij > 0 (with i�j ) satisfying (13) for 1� i, j�n−1 andi�j . In a similar way, we can fill in the upper triangular part ofA in order to obtain a positive matrixB satisfying (13) for 1� i, j�n− 1. Let 0< ε < 1 and letBε be the matrix obtained fromB by replacing the elementsbisk ,j sk by the elementsbisk ,j sk ε 2s−1 . Then it can be checked that the entries ofB satisfy a condition analogous to (13). SinceBε is positive and satisfies (13), we deduce from Theorem A that Bε is an STP matrix for eachε. Taking limits asε → 0, we deduce that the matricesBε converge toA. Since the set of TP matrices is closed, we conclude thatA is TP. Now, suppose that the second inequality in (4) is strict and let us prove thatA is nonsingular ASTP. For this purpose, it is sufficient to get a contradiction after assuming that there exists anh × h submatrixC = (cij ) formed by consecutive rows and columns ofA and whose positive diagonal entries are positive and detC = 0. Leth > 1 be the least integer satisfying the previous property. SinceA is nonnegative and satisfies (4) with the second inequality strict, we can find� > 0 such that (c11 − �)c22 > �c12c21. Let C� be the matrix with the entries ofC but with c11 − � instead ofc11. SinceC is a submatrix ofA formed by consecutive rows and columns and its diagonal entries are positive, we deduce thatC, and soC� too, satisfy the hypotheses ofA. Thus, by the first part of the proof,C� is TP, and so detC��0. Taking into account that detC�[2, . . . , h] = detC[2, . . . , h] > 0 by our choice ofh, we can deduce by the expansion of detC� on its first row that detC� < detC = 0: a contradiction which proves the result.� 3. The smallest value of the constant� Before we prove the applications of Theorem1 to entire functions and to stable poly- nomials, we shall discuss in this section the smallest possible value of the constant� in Theorems A and1. First, we consider the case when the dimension of the matrix is fixed. 218 D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 Theorem 4. Let n ∈ N, n�2. Then, for any ε > 0 there exist ann × n positive matrix An,ε = (aij ) for which aij ai+1,j+1�4(1 − ε) cos2(�/(n+ 1)) ai+1,j ai,j+1, 1� i, j�n− 1, (14) butAn,ε is not STP. Proof. Consider then× n Jacobi matrix An(ε,�) =   √ 1 − ε � 1/2 O 1/2 . . . . . . . . . . . . . . . . . . . . . 1/2 O 1/2 √ 1 − ε �   , whereε is any real number with 0< ε < 1 and letQm(x) be the characteristic polynomial of Am(ε,�), m�1. Then the sequence of polynomials{Qm(x)}∞m=0 is generated by the three term recurrence relation Q0(x) := 1; Q1(x) = √ 1 − ε � − x; Qm+1(x) = ( √ 1 − ε � − x)Qm(x)− (1/4)Qm−1(x), m = 1,2, . . . . On the other hand, the Chebyshev polynomials of the second kindUm(x), defined by Um(cos ) = sin((m+ 1) )/ sin , satisfy the recurrence relationUm+1(x) = 2xUm(x)− Um−1(x), m = 1,2, . . ., with initial conditionsU0(x) = 1 andU1(x) = 2x. Thus, the characteristic polynomial ofAn(ε,�) is the Chebyshev polynomialUn(x) with shifted argument, Qn(x) = (−1/2)nUn(x − √ 1 − ε �). Then, since the zeros ofUn(x) are cos(k�/(n + 1)), k = 1, . . . , n, those ofQn(x) are �k = √ 1 − ε � + cos(k�/(n+ 1)). Therefore, for� = �n := cos(�/(n+ 1)), if ε > 0, at least the smallest zero�n of Qn(x) is negative. Hence, for� = �n, the matrixAn(ε,�n) is not positive definite, and then it is not a TP matrix. On the other hand, the inequalities (14) for i = j , reduce to equalities for this matrix. Let � be any positive number with � < (1 − ε)−1/2�−1 n . (15) Setk := |i − j | and let us define then × n matrix An(ε,�n,�) whose elementsaij coincide with those ofAn(ε,�n) whenk�1 and are given byaij := �k−1/2k 2 whenk�2. The matrixAn(ε,�n,�) is positive. As it was pointed out, (14) holds fork = 0. The above requirements on� guarantee that it holds fork = 1. Fork�2 (14) is obviously satisfied even for any real�. Observe that lim�→0 An(ε,�n,�) = An(ε,�n). Since the set of TP matrices is closed, if the matricesAn(ε,�n,�) were STP for all values of� which satisfy (15), thenAn(ε,�n) D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 219 would be aTP matrix.This contradiction implies that there exist positive matricesAn(ε,�n,�) satisfying (14) which are not STP matrices and the result follows.� Lettingn to tend to infinity, we see that the bound� of Theorem A cannot be reduced to less than 4 when we consider matrices of any ordern. Corollary 5. For anyε > 0 there existn ∈ N,n�2,andann×npositivematrixA = (aij ) such that aij ai+1,j+1�4(1 − ε)ai+1,j ai,j+1, and which is not STP. We strongly believe that the matrices constructed in the proof of Theorem4 are in some sense the extremal ones and we venture to suggest the following conjecture. Conjecture 6. LetA = (aij ) be a nonnegativen × n matrix satisfying(3). Assume that, for any1� i, j�n− 1, the following condition holds: if aij ai+1,j+1 > 0, thenaij ai+1,j+1 > 4 cos2(�/(n+ 1))ai,j+1ai+1,j . (16) Then A is nonsingular ASTP. In particular, if A = (aij ) is a positiven× n matrix whose entries satisfy aij ai+1,j+1 > 4 cos2(�/(n+ 1))ai,j+1ai+1,j , 1� i, j�n− 1, then A is strictly totally positive. Needless to say, when we consider matrices of any order, the above conditions reduce to aij ai+1,j+1�4ai,j+1ai+1,j and, as seen from Corollary5, the constant 4 cannot be reduced. 4. Entire functions in the Laguerre–Pólya class and Hurwitz polynomials We begin this section with some additional information about entire functions in the Laguerre–Pólya class. Recall that an infinite sequence{ak}∞k=0 is said to betotally positive (or Pólya frequency sequence) if ∑∞ k=0 akx k is an entire function and the infinite upper triangular matrix  a0 a1 a2 . . . . . . . . . a0 a1 a2 . . . . . . a0 a1 . . . . . . a0 . . . . . . . . . . . .   (17) 220 D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 is totally positive. Corollary2 is an immediate consequence of Theorem1 and the follow- ing characterization of functions in the Laguerre–Pólya class with nonnegative Maclaurin coefficients in terms of totally positive sequences, due to Aisen et al.[2]: Theorem C. The real entire function�(x) = ∑∞ k=0 akx k with nonnegative coefficientsak is in the Laguerre–Pólya class if and only if the sequence{ak}∞k=0 is totally positive. Indeed, the Maclaurin coefficients of �(x) = ∞∑ k=0 akx k = ∞∑ k=0 k xk k! , (18) satisfy inequalities a2 k ��ak−1ak+1, k = 1,2, . . . , (19) which are equivalent to (8) and so, by Theorem1, the sequence{ak}∞k=0 is totally positive provided�(x) is an entire function. Thus, in order to prove Corollary2 we only need to prove that�(x) is an entire function of order zero. We shall prove that, if a positive sequence {ak}∞k=0 satisfies inequalities (19), then ak� ak1 ak−1 0 �−k(k−1)/2 for k�2. (20) If we set bk = ak+1/ak, then the inequalities (19) are equivalent to the inequalities bk��−1bk−1. These immediately yield bk�(a1/a0)� −k. (21) Now, we are in a position to prove (20) by induction with respect tok. Inequality (20) for k = 2 is exactly (19) for k = 1. Suppose that (20) holds for some natural numberk. Then, the induction passage follows from the following simple chain of inequalities where we use (19), (21) and the induction hypothesis (20): ak+1��−1 a2 k ak−1 = �−1bk−1ak��−1a1 a0 �−k+1 ak1 ak−1 0 �−k(k−1)/2 = ak+1 1 ak0 �−k(k+1)/2. It is well known that the function�(x) of the form (18) is entire if its coefficients satisfy limn→∞ |an|1/n = 0 and in this case the order� of �(x) is given by (see[22, Lecture1]) � = lim supn→∞ n log n log(1/|an|) . Observe that the inequalities (20) are equivalent to �k := ak/a0�Ck�−k(k−1)/2, whereC = a1/a0. Then n log n log(1/|�n|)� log n (n− 1) log �1/2 − log C . D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 221 Since the order of an entire function does not depend on multiplication by a constant, then �(x) is an entire function of order zero. The extremal entire function for which the inequalities in Hutchinson’s theorem reduce to equalities turns out to be an interesting one. If we fixa0 = 1 anda1 = 1 2, then obviously we have equalities in (9) (or, equivalently,a2 k = 4ak−1ak+1) providedan = 2−n2 . Then the requirements of Theorem B will be satisfied ifan = qn 2 , n = 0,1, . . ., andq� 1 2. Thus we conclude that ∞∑ n=0 qn 2 xn, (22) is an entire function of order zero which belongs toL − PI whenever 0< q� 1 2. Katkova et al. [18, Theorem 4]proved the existence of a constantq∞ ≈ 0.556415, such that the function (22) has only real zeros if and only ifq�q∞. It is worth mentioning that it was proved recently in[7] that ∞∑ n=0 qn 2 n! x n, is in L − P if |q| < 1. In fact, the equivalent fact that the sequence{qn2} is a multiplier (or zero-increasing) sequence for|q| < 1 was pointed out in[7], while the result in[18] shows {n!qn2} is a multiplier sequence if and only if 0< q�q∞. The proof of Theorem3 is an immediate consequence of Theorem1 and a result of Hurwitz. Here we only provide the necessary definitions and formulate the Hurwitz theorem. With the polynomial f (z) = cnz n + cn−1z n−1 + cn−2z n−2 + cn−3z n−3 + · · · + c0, we associate the Hurwitz matrix which is formed as follows. Setc−1 = c−2 = · · · = 0 and construct the two line block( cn−1 cn−3 . . . cn cn−2 . . . ) , where the first line containscn−2k−1, k = 0,1, . . . , and the second line is composed by the coefficientscn−2k, k = 0,1, . . . , of f (z). Then, the Hurwitz matrixH(f ) of f (z) is composed by the above block in its first two lines, the next two lines ofH(f ) contain the same block shifted one position to the right, the fifth and the sixth lines contain this block shifted two positions to the right, and so forth. Thus H(f ) =   cn−1 cn−3 cn−5 . . . 0 cn cn−2 cn−4 . . . 0 0 cn−1 cn−3 . . . 0 0 cn cn−2 . . . 0 · · · . . . ·   . The following is the Hurwitz theorem which is sometimes called the Routh–Hurwitz criterion. 222 D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 Theorem D. The polynomialf (z) with cn > 0 is stable if and only if the first n principal minors of the corresponding Hurwitz matrixH(f ) are positive. Since the matrixH(f ) satisfies the requirements of the shadows’ lemma, then the fact thatf (z) is a Hurwitz polynomial in Theorem3 does follow immediately from Theorem1. To complete the proof of Theorem3, it remains to observe that the conditions (12) imply (11). Interesting examples of Hurwitz polynomials are those for which the inequalities (12) reduce to equalities. Let� be defined as in Theorem A and̃q = �−1/2 ≈ 0.495098. It follows from Theorem3 that the polynomials fn(z) = n∑ k=0 qk 2 xk are stable whenq� q̃1/2 ≈ 0.703632 and, whenq = q̃1/2, (12) reduce to equalities for the coefficients offn(z). On the other hand, motivated by the results in Section 3, we believe thatfn(z) are still stable forq�1/ √ 2 ≈ 0.70710678 and even for larger values ofq. On the other hand, Theorem 4 in[18] implies that the same polynomials have only real and negative zeros whenq�q∞ ≈ 0.556415, at least for large values ofn ∈ N. These consequences of our results suggest a challenging question about the behaviour of the zeros offn(z). Given a positive integern, which are the largest values of the constantsmn andMn, such that the zeros offn(z) are: • real and negative whenq ∈ (0,mn]? • with negative real parts whenq ∈ (0,Mn]? Obviouslymn < Mn, Theorem 4 in[18] and Theorem3 in the present paper show that these constants satisfy the inequalitiesq∞ < mn andq̃1/2 < Mn, and obviouslyMn < 1 for n�4. The polynomialf2(z) is stable for any positiveq and it has real zeros if and only if q� 1 2 which means thatm2 = 1 2. For n = 3 we havem3 = 1/ √ 3 andM3 = 1. Do mn andMn maintain a monotonic behavior and do they converge asn goes to infinity? In particular, is it true thatmn → q∞ asn goes to infinity? Acknowledgments The authors thank the anonymous referees for their valuable comments and suggestions. References [1] T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987) 165–219. [2] M. Aissen, A. Edrei, I.J. Schoenberg, A. Whitney, On the generating functions of totally positive sequences, Proc. Natl. Acad. Sci. 37 (1951) 303–307. [3] B.A. Asner Jr., On the total nonegativity of the Hurwitz matrix, SIAM J. Appl. Math. 18 (1970) 407–414. [4] C. de Boor, Total positivity of the spline collocation matrix, Indiana Univ. J. Math. 25 (1976) 541–551. [5] C. de Boor, A. Pinkus, The approximation of a totally positive band matrix by a strictly positive one, Linear Algebra Appl. 42 (1982) 81–98. [6] J. Carnicer, J.M. Peña, Spaces with almost strictly totally positive bases, Math. Nachr. 169 (1994) 69–79. D.K. Dimitrov, J.M. Peña / Journal of Approximation Theory 132 (2005) 212–223 223 [7] J. Carnicer, J.M. Peña, A. Pinkus, On some zero-increasing operators, Acta Math. Hungar. 94 (2002) 173–190. [8] T. Craven, G. Csordas, Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math. 136 (1989) 241–260. [9] T. Craven, G. Csordas, Complex zero decreasing sequences, Methods Appl. Anal. 2 (1995) 420–441. [10] T. Craven, G. Csordas, A sufficient condition for strict total positivity of a matrix, Linear Multilinear Algebra 45 (1998) 19–34. [11] F.R. Gantmacher, The Theory of Matrices, Matrices, vol. 2, Chelsea, New York, 1959. [12] J. Garloff, Intervals of almost totally positive matrices, Linear Algebra Appl. 363 (2003) 103–108. [13] M. Gasca, C.A. Micchelli, J.M. Peña, Almost strictly totally positive matrices, Numer. Algorithms 2 (1992) 225–236. [14] M. Gasca, J.M. Peña, On the characterization of almost strictly totally positive matrices, Adv. Comput. Math. 3 (1995) 239–250. [15] G.H. Hardy, On the zeros of a class of integral functions, Messenger Math. 34 (1904) 97–101. [16] J.I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc. 25 (1923) 325–332. [17] S. Karlin, Total Positivity, vol. I, Stanford University Press, Stanford, CA, 1968. [18] O.M. Katkova, T. Lobova, A. Vishnyakova, On power series having sections with only real zeros, Comput. Methods Funct. Theory 3 (2003) 425–441. [19] J.H.B. Kempermann, A Hurwitz matrix is totally positive, SIAM J. Math. Anal. 13 (1982) 331–341. [20] D.C. Kurtz, A sufficient condition for all the roots of a polynomial to be real, Amer. Math. Monthly 99 (1992) 259–263. [21] B.Ja. Levin, Distribution of zeros of entire functions Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, RI, 1964; revised ed. 1980. [22] B.Ja. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. [23] M. Marden, Geometry of polynomials, American Mathematical Society Surveys, vol. 3, Providence, RI, 1966. [24] N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. [25] J.M. Peña, Shape Preserving Representations in ComputerAided-Geometric Design, Nova Science Publishers Inc., 1999. [26] M. Petrovitch, Une classe remarquable de séries entières,Atti IV Congr. Internat. Mat. Rome Ser. 1 (2) (1908) 36–43. [27] G. Pólya, Über einen Satz von Laguerre, Jber. Deutsch. Math-Verein. 38 (1929) 161–168. [28] G. Pólya, J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914) 89–113. [29] X. Yang, Necessary conditions of Hurwitz polynomials, Linear Algebra Appl. 359 (2003) 21–27. [30] X. Yang, Some necessary conditions for Hurwitz stability, Automatica 40 (2004) 527–529. Almost strict total positivity and a class of Hurwitz polynomials Introduction Proof of the main result The smallest value of the constant delta Entire functions in the Laguerre--Pólya class and Hurwitz polynomials References