Mean matrices and conditional negativity

Transcription

1 Electronic Journal of Linear Algebra Volume 9 Secial volume for Proceedings of the International Conference on Linear Algebra and its Alications dedicated to Professor Ravindra B. Baat Article Mean matrices and conditional negativity Raendra Bhatia Indian Statistical Institute, rbh@isid.ac.in Tanvi Jain Indian Statistical Institute, tanvi@isid.ac.in Follow this and additional works at: Recommended Citation Bhatia, Raendra and Jain, Tanvi. (015), "Mean matrices and conditional negativity", Electronic Journal of Linear Algebra, Volume 9, DOI: htts://doi.org/ / This Article is brought to you for free and oen access by Wyoming Scholars Reository. It has been acceted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Reository. For more information, lease contact scholcom@uwyo.edu.

2 MEAN MATRICES AND CONDITIONAL NEGATIVITY RAJENDRA BHATIA AND TANVI JAIN To R. B. Baat on his sixtieth birthday Abstract. In earlier aers R. Bhatia and H. Kosaki have shown that certain matrices associated with means are infinitely divisible. In this aer it is shown that many of them ossess a stronger roerty: their Hadamard recirocals have exactly one ositive eigenvalue. Key words. Positive definite matrix, conditionally negative definite matrix, infinitely divisible matrix, Hadamard inverse, means. AMS subect classifications. 15A57, 15B57, 15A48 1. Introduction. This note on some structured matrices is a continuation of earlier work Bh, BhK. In those aers it was shown that certain secial matrices are infinitely divisible. Here we show that many of them ossess a stronger roerty. Let A = a i be an n n real symmetric matrix. Then A is said to be ositive semidefinite (sd for short), if for all x R n we have x,ax 0. It is said to be conditionally ositive definite { (cd for short), } if x,ax 0 for all x in the (n 1)- n dimensionalsaceh 1 = x : x = 0. If Aiscd, wesaythataisconditionally =1 negative definite (cnd for short). LetAbeasymmetricmatrix, andsuosea i 0foralli,.WesayAisinfinitely divisible if for all r > 0 the entrywise ower (Hadamard ower) A r = a r i Received by the editors on May, 015. Acceted for ublication on January 0, 016 Handling Editor: Bryan L. Shader. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, New Delhi , India (rbh@isid.ac.in.) Theoretical Statistics and Mathematics Unit, Indian Statistical Institute New Delhi , India (tanvi@isid.ac.in). 06

3 Mean matrices and conditional negativity 07 is sd. If A is a symmetric matrix such that a i > 0 for all i,, and A has exactly one ositive eigenvalue, then we say that A is in the class A. (Here we follow the notation and terminology in BR.) Every cnd matrix with all entries ositive is in the class A. The secial classes defined above are imortant in diverse areas of mathematics. For examle, cd and infinitely divisible kernels are imortant in robability theory. A famous theorem of Schoenberg in distance geometry says that a symmetric matrix A with a ii = 0 and a i 0, is cnd if and only if there exist vectors u 1,...,u n in some Euclidean sace R d satisfying u i u = a i. We call such a matrix a Euclidean distance matrix. See BR. The followingtheorem due to R. BaatB is the startingoint for our discussion. Theorem 1.1. If A = a i is any matrix in class A, then its Hadamard inverse matrix 1/a i is infinitely divisible. In earlier aers Bh, BhK it was shown that several matrices arising naturally in the study ofmatrixmeans areinfinitely divisible. Herewe showthat many ofthese, but not all, are Hadamard recirocals of matrices in class A. We will reeatedly use some elementary facts. Let X stand for the transose of X. If X is a nonsingular matrix, then the transformation A X AX is called a congruence. If A is sd, then so is every matrix congruent to it; and if A A, then every matrix congruent to A lies in A. This is a consequence of Sylvester s law of inertia Bh1. If A is cnd, then for every ermutation matrix P, the matrix P AP is cnd. If A 1,...,A k are cnd matrices, and α 1,...,α k are nonnegative real numbers, then α 1 A α k A k is cnd. On the other hand ositive linear combinations of matrices in A need not be in A. The matrix E with all its entries equal to 1 is a sd matrix of rank 1. The sace H 1 is the null sace of E. Consequently, E is both cd and cnd. Most of the examles we discuss arise as follows. Let m : R + R + R + be a mean; i.e. a ma that satisfies the following roerties: (i) m(a,b) = m(b,a). (ii) min(a,b) m(a,b) max(a,b). (iii) m(αa,αb) = αm(a,b) for all α > 0. (iv) m(a,b) is an increasing function of a and b. (v) m(a,b) is a continuous function of a and b. Let λ 1,...,λ n be given ositive numbers. We consider the n n matrix M =

4 08 R. Bhatia and T. Jain m(λ i,λ ). With the earlier works as motivation, we ask the following questions: If m(a,b) ab for all a,b, then is the matrix M in class A for all λ 1,...,λ n? And if m(a,b) ab for all a,b, then is the Hadamard recirocal of M in class A for all λ 1,...,λ n? For several classes of means this is answered in the next section. Some related matrices are also considered. After the aers of Bhatia and Kosaki cited earlier, Kosaki has carried out a very extensive and dee analysis of infinite divisibility of several matrices arising from means. See K1, K, K3. General references for matrix analysis used here arebh1 andhj. An exosition of min matrices in a lighter vein is given in Bh3.. Examles..1. The arithmetic and harmonic means. Let λ 1,...,λ n be ositive numbers, and let a i = 1 (λ i +λ ). Then the matrix A = a i can be exressed as A = 1 (DE + ED), where D is the diagonal matrix diag(λ 1,...,λ n ). So, for all x H 1, we have x,ax = 0. This shows that the matrix ( A is both cd and cnd. Likewise, the matrix B with entries b i = 1 λ 1 i +λ 1 ) is both cd and cnd. Averysecialcorollaryofthisisthat thefamoushilbert matrix divisible, a fact that is very well known. 1 i+ is infinitely.. The maximum. Theorem.1. Let λ 1,...,λ n be ositive numbers. Then the matrix M = max(λ i,λ ) is cnd. Proof. First assume λ 1 λ λ n. In this case we have M = λ 1 E +(λ λ 1 )F 1 +(λ 3 λ )F + +(λ n λ n 1 )F n 1, (.1)

5 Mean matrices and conditional negativity 09 where F, 1 n 1, is the matrix with all entries zero in the first rows and columns, and all other entries equal to 1. The matrix λ 1 E is cnd. The matrices F are sd and the coefficients (λ +1 λ ), 1 n 1 are negative. So, the decomosition (.1) shows that M is cnd. If λ 1,...,λ n are any ositive numbers, then we can choose a ermutation matrix P so that P MP = max ( λ i,λ ), where λ 1 λ n is a decreasing rearrangement of λ 1,...,λ n. So M is cnd. Corollary.. Let λ 1,...,λ n be ositive numbers. The matrix min(λ i,λ ) is infinitely divisible. Proof. This follows from Theorems 1.1,.1 and the observation min(λ i,λ ) = ( ( 1 max, 1 )) 1. λ i λ Corollary.3. Let λ 1,...,λ n be ositive numbers. Then the matrix λ i λ is cnd, as is the matrix 1+ λ i λ. Proof. The first statement follows from the relation λ i λ = max(λ i,λ ) (λ i +λ ), and the facts that max(λ i,λ ) is cnd and λ i +λ cd. The second statement follows from the first, since the matrix E is cnd. As a consequence, the matrix 1 1+ λ i λ is infinitely divisible. See Exercise in Bh1 for another roof of this..3. Heinz means. These means are defined as H ν (a,b) = aν b 1 ν +a 1 ν b ν,0 ν 1.

6 10 R. Bhatia and T. Jain H 0 = H 1 is the arithmetic mean and H 1/ is the geometric mean. Let M = H ν (λ i,λ ). Note that m i = 1 = λ 1 ν i ( λ ν i λ 1 ν ( λ ν 1 i +λ 1 ν i λ ν ) +λ ν 1 ) λ 1 ν. So M = DAD, where D = diag(λ 1 ν ( 1,...,λ 1 ν n ) and a i = 1 λ ν 1 i +λ ν 1 ). In Section.1 we have noted that the matrix A is cnd. So M, being congruent to A, is in the class A..4. The logarithmic mean. This mean is defined as We have also a b L(a,b) = log a log b, a b, L(a,a) = a. L(a,b) = 1 0 H ν (a,b) dν. Our discussion so far may lead us to seculate that the matrix L(λ i,λ ) may be in A for all λ 1,...,λ n. However, this fails in the most sectacular way. For n 3, no such matrix is in A. Theorem.4. Let n 3 and let λ 1,...,λ n be any ositive numbers. Then the matrix L(λ i,λ ) has at least two ositive eigenvalues. The roof of this theorem is given in the Aendix..5. The binomial means. This is the family of means defined as ( a α +b α ) 1/α B α (a,b) =, α.

7 Mean matrices and conditional negativity 11 The values at α =,0, are understood in the sense of limits. These are, resectively, min(a,b), ab and max(a,b). It has been shown in BhJ1 that for any ositive numbers λ 1,...,λ n the matrix (λ i +λ ) r is cnd for 0 r 1. If n 3, then this matrix has at least two ositive eigenvalues for all r > 1. Using this one can see that the matrix B α (λ i,λ ) is cnd for α 1, but not for 0 < α < 1. The secial cases α = 1 and have been studied in Section.1 and.. It may be aroriate here to discuss a related matrix. Let x i be real numbers with x i < 1. Consider the matrix We have a ower series exansion S = (1 x i x ) r, 0 < r < 1. (1 x i x ) r = m=0 a m (x i x ) m, in which a 0 = 1 and a m < 0 for all m 1. The matrices (x i x ) m are sd for all m 1, and the matrix E is cnd. It follows that S is cnd for 0 < r < 1. The Cayley transformation λ = 1 x 1+x is a biection from the interval ( 1,1) onto R +. With this substitution we see that λ i +λ = 1 x i x (1+x i )(1+x ). So, the matrices (λ i +λ ) r and (1 x i x ) r are congruent..6. Lehmer means. These are defined as L (a,b) = a +b a 1 +b 1,. For = 0, 1,1 these are the harmonic, geometric and arithmetic means, resectively. For 1 < <, L (a,b) is not a monotone function of a and b, and therefore is not

8 1 R. Bhatia and T. Jain a mean in the sense we have defined. However, it is customary to study the entire family together. For = ±, L (a,b) is max(a,b) and min(a,b), resectively. Theorem.5. Let λ 1,...,λ n be ositive numbers. If n 3, then the matrix L (λ i,λ ) is in the class A for all 1. For n > 3, the matrix L (λ i,λ ) is in A if and only if is in the set 1, 3 4 {1} 3,. If n 3, then the matrix L (λ i,λ ) 1 is in the class A for all 1. For n > 3, the matrix L (λ i,λ ) 1 is in A if and only if is in the set, 1 {0} 1 4, 1. Proof. Let x 1,...,x n be distinct ositive realnumbers and let r be a real number. In our ongoing work BhJ we have studied the eigenvalue distribution of the matrix x r i +x r. (.) x i +x We have shown that this matrix has exactly one ositive eigenvalue if and only if r is in the set { 3, 1 1,3 if n > 3 (, 1 1, ) if n 3. (The roof of these statements use ideas similar to those in BhFJ and BhJ1.) Let 1. The matrix under consideration λ i L (λ i,λ ) = +λ (.3) λ 1 i +λ 1 reduces to the form in (.) with the substitution x i = λ 1 i, and r = 1. Thus for n > 3, the matrix in (.3) is in the class A if and only if 1 1,3 3, 1. Now if and only if 3, and if and only if We have already seen that for = 1, the matrix (.3) is in A. By following the same arguments as above, we see that for n 3, the matrix (.3) is in the class A for all 1. This roves the first statement of the theorem. The second can be derived from this using the relation L 1 (a,b) = ab L (a,b)..7. Power difference means.

9 These means are defined as Mean matrices and conditional negativity 13 K (a,b) = 1 a b a 1 b 1. When = 1, 1,1,, they give, resectively, the harmonic, geometric, logarithmic, and arithmetic means. When = ±, they are the max and min, resectively. Theorem.6. Let λ 1,...,λ n be ositive numbers. Then the matrix K (λ i,λ ) is in the class A if and only if is in 1, 3,. The matrix K (λ i,λ ) 1 is in A if and only if 1. Proof. The matrix under consideration is K (λ i,λ ) = 1 (λ 1 i ) / 1 (λ 1 λ 1 i λ 1 ) / 1. (.4) The matrix x r i x r L r = x i x, r R (.5) is the much studied Loewner matrix. It is a classical result that for 0 < r < 1, this matrix is sd. It was shown by Bhatia and Holbrook BhH that this matrix is in the class A when 1 r. Bhatia and Sano BhS1 carried this further and showed that this matrix is cnd when 1 r, and cd when r 3. More generally, the eigenvalue distribution of this matrix was studied by Bhatia et al BhFJ for all real r. It follows from this work that the matrix L r has exactly one ositive eigenvalue if and only if r 1, 3,, and it has exactly one negative eigenvalue if and only if r,3, 1. By the arguments given in Section.6 it follows that for > 1 the matrix (.5) is of the form 1 r L r. Hence it is in A if and only if if and only if. 1 1,; i.e., Let 0 < < 1. Then 1 is a negative number. So, in this case the matrix (.5) is of the form 1 r L r. Hence it has exactly one ositive eigenvalue if and only if L r has 1; i.e., if and only if 1 3. Hence the matrix (.5) is in A if and only if, 1, 3. This comletes the roof of the first assertion of the Theorem. exactly one negative eigenvalue. This haens if and only if 1 To rove the second assertion, note that 1 K (λ i,λ ) = 1 (λ ) 1 i (λ ) 1 λ. (.6) i λ

10 14 R. Bhatia and T. Jain Suose < 0. Then 1 is ositive. So, the matrix (.6) is in A if and only if 1 1 ; i.e., if and only if 1. Finally if 0 < < 1, then this matrix is in A if and only if 1 1; i.e., Next we consider a matrix obtained from a ratio of two means..8. The generalised Lehmer matrix. Let 0 < λ 1 < λ < < λ n, and consider the matrix A = max(λi,λ ). (.7) min(λ i,λ ) The Hadamard recirocal of A, for the secial choice λ i = i, is called the Lehmer matrix. The Lehmer matrix is infinitely divisible Bh. The following theorem dislays a stronger roerty. Theorem.7. The matrix A defined by (.7) is in the class A. Proof. Let X be any symmetric matrix with ositive entries, and let t n +a 1 t n 1 + +a n (.8) be the characteristic olynomial of X. By the Descartes rule of signs, X has only one ositive eigenvalue if and only if the coefficients of the olynomial (.8) change signs only once. Since a 1 = trx < 0, this condition is equivalent to saying that X is in A if and only if a k < 0 for all 1 k n. Since a k equals ( 1) k times the sum of k k rincial minors of X, this condition is satisfied if for each k all the k k rincial minors have the same sign, and all the (k + 1) (k + 1) rincial minors have the sign oosite to this. Aly these considerations to A. All rincial submatrices of A have the same form as A. Let A k denote a k k matrix of the form A; i.e. A k = for some max(λi,λ ) min(λ i,λ ) 0 < λ 1 < < λ k. If we show that for each such matrix sgn(det A k ) = ( 1) k 1, it would follow that A is in the class A. This is a consequence of the LU decomosition of A that is obtained in Lemma.8. We shall resume the roof of the Theorem after roving the Lemma.

11 Mean matrices and conditional negativity 15 Writing out the entries of A shows 1 λ /λ 1 λ 3 /λ 1 λ n /λ 1 λ /λ 1 1 λ 3 /λ λ n /λ A = λ 3 /λ 1 λ 3 /λ 1 λ n /λ 3, (.9).... λ n /λ 1 λ n /λ. 1 i.e., a i = λ /λ i if 1 i n. We rove the following: Lemma.8. The matrix A in (.11) can be factored as A = LU, where L is the lower triangular matrix with entries l i = { ai for all 1 i n 0 for > i, and U is the uer triangular matrix with entries a i for i = 1, 1 n u i = (λ i λ i 1 )λ λ i for i λ i 1 0 for i >. (.10) (.11) Proof. Let i, be any two indices with i. If L and U are as in (.10) and (.11), then the (i,) entry of LU is equal to n l ik u k = k=1 l ik u k k=1 = l i1 u 1 + = λ i λ 1 λ λ 1 = λ iλ λ 1 = λ iλ λ 1 = λ i λ = a i. l ik u k k= k= +λ i λ λ i (λ k λ k 1 )λ λ k λ k λ k 1 ( 1 1 ) k= λ λ k ( ) 1 +λ i λ 1 λ 1 λ k 1 We have shown that for i the (i,) entry of LU agrees with that of A. A similar calculation shows that this haens also when i <.

12 16 R. Bhatia and T. Jain As a consequence of Lemma.8 we have det A = ( 1) n 1 n i= λ i λ i 1 λ. (.1) i 1 Hence sgn(det A) = ( 1) n 1. This comletes the roof of Theorem.7. Corollary.9. Let x 1,...,x n be any real numbers. Then the matrix e xi x is in the class A. The LU factoring of the Lehmer matrix is obtained in the aer KS..9. Stolarsky means. This is the family defined as ( a γ b γ ) ( 1/(γ 1) 1 S γ (a,b) = = γ(a b) b a b a t γ 1 dt) 1/(γ 1), < γ <. For γ = 1,0,1,, this yields the geometric, the logarithmic, the identric, and the arithmetic means, resectively. It was shown in BhK that for every choice of ositive numbers λ 1,...,λ n, the matrix is infinitely divisible if 1 S γ(λ i,λ ) γ 1, and the matrix S γ (λ i,λ ) is infinitely divisible if γ 1. This family does not yield readily to our analysis. We have some evidence for the following: Conecture 1. Let λ 1,λ,...,λ n be ositive numbers. The matrix S γ (λ i,λ ) is in the class A if γ but this is not always so when 0 γ <. Conecture 1 is known to be true for some secial values of γ. (i) S 0 is the logarithmic mean, and we have studied this in Section.4. (ii) S 1/ is the same as the binomial mean B 1/ which we have studied in Section.5. (iii) S is the arithmetic mean discussed in Section.1. We show that the statement of Conecture 1 is true when γ = 3 or 4. Note that S 3 (λ i,λ ) = ( 1 ( λ 3 i +λ i λ +λ ) ) 1/.

13 Mean matrices and conditional negativity 17 The following theorem includes the assertion that S 3 (λ i,λ ) is a cnd matrix. Theorem.10. Let λ 1,...,λ n be ositive numbers. Then for t, and 0 r 1/, the matrix (λ A = i +t λ i λ +λ r ) (.13) is cnd. Proof. Let x < 1 and 0 < r < 1. Then we have the binomial exansion (1 x) r = m=0 a m x m, where a 0 = 1 and a m < 0 for all m 1. We have the identity If < t <, then λ i + t λ iλ +λ = (λ i +λ ) ( 1 ( t)λ iλ (λ i +λ ) So for 0 < r < 1, we have (λ i +t λ i λ +λ ) r = = The Cauchy matrix the matrix is cnd. (λ i +λ ) r { (λ i +λ ) r + 1 λ i+λ ( t)λ i λ (λ i +λ ) < 4 λ iλ (λ i +λ ) a m ( t) m λ m i λm (λ m=1 i +λ ) m a m ( t) m m=1 } λ m i λm, ). (λ i +λ ) (m r). (.14) is infinitely divisible. So, for each m 1, the last matrix on the right-hand side of (.14) is sd. The coefficients a m are all negative. So, the last infinite sum reresents a negative definite matrix. Further, if 0 r 1/, then (λ i +λ ) r is cnd by the results in BhJ1. Hence the matrix in (.15) We remark that in the secial case t =, the matrix A in (.13) reduces to A = λ i λ r. (.15)

14 18 R. Bhatia and T. Jain It is known BR that this matrix is cnd for 0 r 1. Likewise in the case t = 0, we have (λ A = i +λ r ), (.16) and this matrix is also cnd for 0 r 1; BhJ1. This raises the question whether for t 0, the matrix A in (.13) is cnd for 0 r 1. Next, consider the matrix S 4 (λ i,λ ). Note that S 4 (λ i,λ ) = ( ) 1/3 λ 4 i λ 4 = 4(λ i λ ) ( (λi +λ ) 3 (λ i λ +λ i λ ) ) 1/3. So, the following theorem contains the assertion that S 4 (λ i,λ ) is a cnd matrix. Theorem.11. Let λ 1,...,λ n be ositive numbers. Then for 1 t 3 and 0 r 1/3, the matrix is cnd. A = Proof. If 1 < t 3, then { (λ i +λ ) 3 (3 t) ( λ i λ +λ i λ ) } r, (.17) 4 (3 t) λ i λ +λ i λ (λ i +λ ) 3 < 4 λ iλ (λ i +λ ) 1. So the entries of the matrix A can be written as ( a i = {(λ i +λ ) 3 1 (3 t)(λ i λ +λ i λ ) )} r (λ i +λ ) 3 ) = (λ i +λ ) (1+ 3r a m (3 t) m λ m i λm (λ i +λ ) m, m=1 where the coefficients a m are negative. Hence, A = (λ i +λ ) 3r + a m (3 t) m m=1 λ m i λm (λ i +λ ) m 3r. (.18) BythesameargumentsasintheroofofTheorem.10, theinfinite seriesontherighthand side of (.18) reresents a negative definite matrix, and the matrix (λ i +λ ) 3r is cnd. Hence, the matrix A is cnd.

15 Mean matrices and conditional negativity Remarks. We have seen here many examles of cnd matrices A with ositive diagonal. In each case, relacing the diagonal by zero we get cnd matrices with zero diagonal. By the theorem of Schoenberg cited earlier, all these are Euclidean distance matrices. It is also known BR that if A is in the class A, then log a i is a cnd matrix. So, more examles of distance matrices can be obtained from the matrices studied here. Aendix A. Proof of Theorem.4. It suffices to rove the statement of the Theorem for n = 3. The general case follows from this by Cauchy s interlacing rincile. Let λ 1,λ,λ 3 be ositive numbers and let L be the 3 3 matrix L = L(λ i,λ ). Since L has at least one ositive eigenvalue, if we show that det L < 0, then it will follow that it has exactly one more ositive eigenvalue. The inequality det L < 0 translates to λ 1 λ λ 3 +L(λ 1,λ )L(λ,λ 3 )L(λ 1,λ 3 ) < λ 1 L(λ 1,λ ) +λ L(λ 1,λ 3 ) +λ 3 L(λ 1,λ ) (A.1) Make the substitution λ 1 = e x, λ = e y, λ 3 = e z and then divide both sides of the inequality soobtained bye x+y+z. This showsthat the assertionofthe inequality(a.1) is equivalent to saying that for all real numbers x, y, z, we have sinh(x y)sinh(y z)sinh(z x) 1+ (x y)(y z)(z x) < sinh (x y) (x y) + sinh (y z) (y z) + sinh (z x) (z x) Assume x > y > z, and ut x y = α, y z = β. Then x z = α+β, and the above inequality says that for all α,β > 0 sinh α sinh β sinh(α+β) 1+ αβ(α+β) < sinh α α + sinh β β + sinh (α+β) (α+β). (A.) Let us first consider the secial case α = β. The argument is easier and illuminates the later discussion. In this case the inequality (A.) reduces to ( ) sinh α sinh(α) α 1 < sinh (α) α (α) 1; i.e., to sinh α α < sinh(α) α +1. (A.3)

16 0 R. Bhatia and T. Jain Use the ower series exansions sinh(α) α = n=1 (α) n (n 1)! (A.4) and sinh α α = (α) n n=1 (n)! (A.5) to see that 1+ sinh(α) sinh α α α = 1+ (α) n ( 1 (n 1)! 4 ) (n)! n=1 (α) n ( 4 ) = 1. (n 1)! n n=3 This is ositive, and that establishes (A.3). Now we rove the inequality (A.1) for the general case. Using the ower series (A.5) and exanding (α+β) n by the binomial theorem, we can exress the right hand side of (A.1) as 3+ where k= k (k +1)! a(k,)(α 1 β k 1 +α k 1 β 1 )+a(k,k)α k 1 β k 1) ( k 1 =1 a(k,) = { k +1 k, = 1 ( k+1 k ) 1 k, k. (A.6) Further since sinh α sinh β sinh(α+β) = sinh((α+β)) sinh(α) sinh(β), using the ower series exansion (A.4), the left hand side of (A.1) can be exressed as 3+ k= k (k +1)! b(k,)(α 1 β k 1 +α k 1 β 1 )+b(k,k)α k 1 β k 1), ( k 1 =1

17 Mean matrices and conditional negativity 1 where Using the identity we have { k +1 k, = 1 b(k,) = ( k+1 ) ( k+1 ) ( 1 + ± k+1 ) 1 k, k. m ( ) r ( 1) i i i=0 b(k,) = {( k ( k ( ) r 1 = ( 1) m, m ) 1 if is even ) +1 if is odd. Since α and β are ositive, (A.1) will be established if we show that a(k,) b(k,) for all k and k, and a(k,) > b(k,) for at least one such k,. where Let k and k. Then straightforward comutations yield a(k,) b(k,) = {( k ) (k) 1 (k ) ( k ) (k) 1 (k ) +1 if is even 1 if is odd, (k) = 4( )k ( )( +1)k. When =, (k) is the constant 4 and a(k,) = b(k,). It is easy to see that a(k,) b(k,) for 3, and we will have a(k,) > b(k,) for all even > 3 if we can show that (k). Let 3. In this case (k) is quadratic olynomial in k with exactly one ositive zero. When k =, () = ( 3)( 1) which is nonnegative. Also (k) is ositive for large k. Hence the ositive zero of (k) is no bigger than, and therefore (k). The first author is suorted by a J. C. Bose National Fellowshi, and the second author is suorted by a SERB Women Excellence Award. REFERENCES

18 R. Bhatia and T. Jain B R. B. Baat, Multinomial robabilities, ermanents and a conecture of Karlin and Rinott, Proc. Amer. Math. Soc., 10 (1988) BR R. B. Baat and T. E. S. Raghavan, Nonnegative Matrices and Alications, Cambridge University Press, Bh1 R. Bhatia, Positive Definite Matrices, Princeton University Press, 007. Bh R. Bhatia, Infinitely divisible matrices, Amer. Math. Monthly, 113 (006) Bh3 R. Bhatia, Min matrices and mean matrices, Math. Intelligencer, 33 (011) -8. BhFJ R. Bhatia, S. Friedland, and T. Jain, Inertia of Loewner matrices, Indiana Univ. Math. J., To aear. BhH R. Bhatia and J. Holbrook, Fréchet derivatives of the ower function, Indiana U. Math. J., 49 (000) BhJ1 R. Bhatia and T. Jain, Inertia of the matrix ( i + ) r, J. Sectral Theory, 5 (015) BhJ R. Bhatia and T. Jain, in rearation. BhK R. Bhatia and H. Kosaki, Mean matrices and infinite divisibility, Lin. Alg. Al., 44 (007) BhP R. Bhatia and K. R. Parthasarathy, Positive definite functions and oerator inequalities, Bull. London Math. Soc., 3 (000) BhS1 R. Bhatia and T. Sano, Loewner matrices and oerator convexity, Math. Ann., 344 (009) BhS R. Bhatia and T. Sano, Positivity and conditional ositivity of Loewner matrices, Positivity, 14 (010) HJ R. A. Horn and C. R. Johnson, Toics in Matrix Analysis, Cambridge University Press, KS E. Kilic and P. Stanica, The Lehmer matrix and its recursive analogue, rerint. K1 H. Kosaki, On infinite divisibility of ositive definite functions arising from oerator means, J. Funct. Anal., 54 (008) K H. Kosaki, Positive definiteness of functions with alications to oerator norm inequalities, Mem. Amer. Math. Soc., 1 (011) No K3 H. Kosaki, Strong monotonicity for various means, J. Funct. Anal. 67 (014)