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Symmetric polynomials and lp 1 inequalities for certain intervals of p. 1 0 2 n Ivo Klemeˇs a J 9 Department of Mathematics and Statistics, 805 Sherbrooke Street West, ] A McGill University, Montréal, Québec, H3A 2K6, Canada. C Email: [email protected] . h t a m Abstract. We prove some sufficient conditions implying lp inequalities of [ the form ||x|| ≤ ||y|| for vectors x,y ∈ [0,∞)n and for p in certain pos- p p 2 itive real intervals. Our sufficient conditions are strictly weaker than the v usual majorization relation. The conditions are expressed in terms of certain 8 3 homogeneous symmetric polynomials in the entries of the vectors. These 9 polynomials include the elementary symmetric polynomials as a special case. 3 . We also give a characterization of the majorization relation by means of 2 0 symmetric polynomials. 0 1 : v i X r a A.M.S. Mathematics Subject Classifications: 47A30 (26B25, 52A40). Key words: inequality; p-norm; symmetric polynomial; majorization. Date: February 2010. Revised 18 April 2010. 1 §1. Introduction Let x and y be given vectors in Rn having nonnegative entries. We will investigate sufficient conditions on x and y for lp inequalities of the form ||x|| ≤ ||y|| simultaneously for all 0 ≤ p ≤ 1. Under the additional as- p p sumption that ||x|| = ||y|| , our conditions also imply ||x|| ≥ ||y|| for 1 1 p p 1 ≤ p ≤ 1+r, where r ≥ 1 is a freely adjustable integer parameter appearing in the conditions. As will be seen in Theorem 1, the conditions are expressed using a finite number of symmetric polynomials in x or y with positive coefficients, whose degrees are controlled by r in some way. In particular, the special case r = 1 of these conditions involves just the elementary symmetric polynomials. This case is a kind of “folk theorem”. It has typically been used in order to obtain lp estimates for the eigenvalues of some operator A, via the determinant of (I + tA) [6, Ch. 4, p. 211-212, Lemma 11.1], [13, Theorem 4], [7, Theorem 1.2]. Such polynomial conditions may be viewed as expressing certain averaged properties of the kth tensor powers x⊗k and y⊗k for various k. As a comple- ment to Theorem 1, we will present in §3 an almost trivial characterization of the usual majorization relation x ≻ y from the same point of view, that is by means of certain symmetric polynomials in x or y (Theorem 2). More 2 precisely, we supply a converse to a previous result by Proschan and Sethura- man [12, Theorem 3.J.2, Example 3.J.2.b] regarding a class of Schur-concave symmetric polynomials. A considerable amount of literature exists concerning the larger set of simultaneous lp inequalities given by ||x|| ≤ ||y|| for −∞ ≤ p ≤ 1 and p p ||x|| ≥ ||y|| for 1 ≤ p ≤ ∞. This relation is implied by, but strictly weaker p p than x ≻ y, and has been called “power majorization” [3]. It has been studied in the context of some concrete numerical sequences [4], [5], and also inquantum informationtheory, where certaincharacterizations have recently been obtained [10], [14], [1], [2]. It is interesting that the latter quantum information literature is concerned with relations of the form x⊗k ≻ y⊗k, and also x⊗z ≻ y⊗z for some z (the “catalyst”). However, the characterizations themselves are more in the spirit of existence proofs, rather than explicit conditions that can be checked in concrete situations. Theorems 1and2andtheirproofswere originallypresented bytheauthor inthe2002preliminaryreport[8]alongwithanumber ofrelatedresults. This and some further results were submitted to a journal in February 2007 in the formofpreprint [9]. Two yearslater(January2009)thejournalreportedthat it had been unable to recruit any referees. Also, during the latter waiting 3 process the author decided to post [9] on arXiv (June 2008). §2. The main result. Let us fix the following notation for the lp means of a vector x ∈ Rn: ||x|| := 1 n |x |p 1/p ,0 6= p ∈ R(withtheconventionthatforanegative p n i=1 i (cid:0) P (cid:1) p we set ||x|| = 0 whenever some entry x = 0, as would be demanded by p i continuity in x), ||x|| := min |x | , ||x|| := ( n |x |)1/n , ||x|| := −∞ i i 0 i=1 i ∞ Q max |x |, as demanded by continuity in p. i i Definition 1. Let r ≥ 1 be an integer. Let P be the rth degree Taylor r polynomial of exp, that is P (s) = 1+ s1 +···+ sr. If x ∈ Rn and t ∈ R let r 1! r! n n (x t)r i f (x,t) := P (x t) = 1+x t+···+ . (1) r r i i r! i=1 i=1(cid:18) (cid:19) Y Y For each integer k ≥ 1 define F (x) to be the coefficient of tk in f (x,t). k,r r Note that we have not explicitly indicated n in the notation f and F , but r k,r this should not cause any confusion. Clearly, the F can be written out k,r explicitly as n xki F (x ,...,x ) = i , (2) k,r 1 n k ! i Pki=kX, maxki≤r Yi=1 where it is understood that the (k )n range over n-tuples of nonnegative i i=1 integers. Equivalently, F (x) is the sum of those terms in the expansion of k,r 4 1(x +···+x )k in which each variable x has exponent at most r. Clearly k! 1 n i F =: E is the elementary symmetric polynomial of degree k and F = k,1 k k,r (E )k/k! whenever k ≤ r. Also, F (x) = (E (x)r+1 − xr+1)/(r +1)!, 1 r+1,r 1 i i P F (x) = (E (x))r/(r!)n, and F (x) = 0 when k > nr. Our main result is nr,r n k,r the following. Theorem 1. Let x,y ∈ [0,∞)n and fix an integer r ≥ 1. If F (x) ≤ F (y) (3) k,r k,r for all integers k in the interval r ≤ k ≤ nr, then ||x|| ≤ ||y|| whenever 0 ≤ p ≤ 1. (4) p p n n If also x = y , then i i i=1 i=1 X X ||x|| ≥ ||y|| whenever 1 ≤ p ≤ r +1. (5) p p Proof. Fix the integer r ≥ 1. Observe that log(1+s+···+sr) is O(s) when r! s → 0+ and O(logs) when s → +∞. Thus, the integrals (Mellin transforms) ∞ sr ds I (p) := log(1+s+···+ ) s−p r r! s Z0 are finite (and positive) for all p in the interval 0 < p < 1. Replacing s by at for any positive a gives the identity 1 ∞ (at)r dt log(1+at+···+ ) t−p = ap (a ≥ 0, 0 < p < 1). (6) I (p) r! t r Z0 5 Now let x,y ∈ [0,∞)n and F (x) ≤ F (y) for all integers k in the interval k,r k,r r ≤ k ≤ nr. Note that in the case r = k we have F (x) = (E (x))r/r! = r,r 1 (n||x|| )r/r!. Hence ||x|| ≤ ||y|| . Also, F = (E )k/k! for 1 ≤ k ≤ r. 1 1 1 k,r 1 Thus in fact F (x) ≤ F (y) for all integers k in the interval 1 ≤ k ≤ nr, k,r k,r i.e. for all coefficients of tk in the generating functions f (x,t) and f (y,t) r r (see Definition 1). Hence 1 ≤ f (x,t) ≤ f (y,t) , ∀ t ≥ 0. r r Taking logarithms of the f and integrating with respect to t−p dt 1 gives, r t Ir(p) by identity (6), n n xp ≤ yp (0 < p < 1). i i i=1 i=1 X X Normalizing both sides we obtain the first case of the theorem, since the inequalities ||x|| ≤ ||y|| extend to the endpoint case p = 0 by continuity in p p p. Next, if in addition x = y , then x t = y t for all t ≥ 0. i i i i i i i i P P P P Subtracting from this the inequality logf (x,t) ≤ logf (y,t), one obtains r r (x t)r i x t−log(1+x t+···+ ) i i r! i (cid:18) (cid:19) X (y t)r i ≥ y t−log(1+y t+···+ ) . (7) i i r! i (cid:18) (cid:19) X Consider the function δ (s) := s−log(1+s +···+ sr) for s ≥ 0. We have r r! δ (s) ≥ s−log(es) = 0 for s ≥ 0. When s → +∞, we have δ (s) = O(s)+ r r 6 O(log(sr)) = O(s).Whens → 0+ wehaveδ (s) = s−log(es −O(sr+1)) = s− r log(es(1−e−sO(sr+1)) = s−log(es)−log(1−e−sO(sr+1)) = O(e−sO(sr+1)) = O(sr+1). It follows that the integrals ∞ sr ds J (p) := s−log(1+s+···+ ) s−p r r! s Z0 (cid:18) (cid:19) are finite (and positive) for all p in the interval 1 < p < r + 1. Replacing s by at gives the new identity 1 ∞ (at)r dt at−log(1+at+···+ ) t−p = ap , (8) J (p) r! t r Z0 (cid:18) (cid:19) for a ≥ 0, 1 < p < r +1. Thus, when 1 < p < r + 1 we may integrate (7) with respect to t−p dt 1 and use (8) to obtain t Jr(p) xp ≥ yp , (1 < p < r +1). i i i i X X By continuity in p, we obtain ||x|| ≥ ||y|| for 1 ≤ p ≤ r +1. p p Remarks on Theorem 1: (a). Thecaser = 1ofTheorem1employs onlytheelementarysymmetric polynomials E = F and is relatively well known, as mentioned in the k k,1 introduction. We illustrate the cases r = 1,2 in an example following these remarks. 7 (b). One can ask some natural questions regarding the sharpness of various aspects of Theorem 1, but we will not go into the details within the space of the present paper. Let us mention only the following without proof (some of these remarks are discussed further in [9]): (i) In the conclusions (4) and (5), the intervals of p cannot be enlarged at either end, at least when n ≥ 3. In particular, one cannot make any general conclusion in the range p < 0. (ii) The converse of Theorem 1 does not hold in general, in the sense that (4) and (5) do not imply the hypotheses (3), when n ≥ 4. There is a strong converse when n = 3 and x = y : Then the two i i P P end point lp inequalities ||x|| ≤ ||y|| and ||x|| ≥ ||y|| imply all of the 0 0 r+1 r+1 hypotheses (3) concerning the F for a fixed r. (And hence they also imply k,r all the interior cases of p in (4) and (5)). (iii) For general n, although there is no converse, there may be some redundancy in the hypotheses (3). That is, perhaps some of the k’s can be omitted from the current list r ≤ k ≤ nr. (iv) When r is increased, do the hypotheses (3) get stronger ? The conclusions suggest that they do. But on the other hand, for r < r the family of 1 2 functions {F }∞ is not simply a subset of the family {F }∞ ; one may k,r1 k=1 k,r2 k=1 need to examine the convex cones spanned by their gradients to answer the question. 8 (c.1). In Theorem 1 the F can be replaced by different choices of k,r special polynomials as follows. Fix the index r ≥ 1. In the proof, only some key properties of the Taylor polynomial P (s) = 1+s1+···+sr were needed: r 1! r! We could have replaced P (s) by any expression of the form r ∞ s1 sr sj Q (s) := (1+ +···+ )+ a r r,j 1! r! j! j=r+1 X foranyfixedsetofconstants0 ≤ a < 1havingthepropertythatlogQ (s) ≤ r,j r K sǫ as s → ∞ for any ǫ > 0, i.e. logQ (s) = O(sǫ) for any ǫ > 0. Thus, ǫ r Q (s) should have “order zero” in the sense of entire functions; see for exam- r ple [11, Ch. 1]. [Moreover, even with the weaker property that as s → ∞, logQ (s) = O(sǫ) for a fixed 1 > ǫ > 0, the proof of Theorem 1 still succeeds r for the lp inequalities in the range ǫ ≤ p ≤ 1 for (4), and the full range 1 ≤ p ≤ r + 1 for (5).] We can then use Q (s) to define a new generating r function f (x,t) = n Q (x t) and re-define F (x) to be the coefficient r i=1 r i k,r Q of tk in f (x,t). Theorem 1 then holds as before (of course, the hypothesis r r ≤ k ≤ nr should be loosened to include all r ≤ k < ∞). (c.2). The following are some natural examples of Remark (c.1). For simplicity we first consider the case r = 1. (i) Notice that the inequality 9 n (1+x t) ≤ n (1+y t) would hold if it was known that i=1 i i=1 i Q Q n n M M (1+x t) ≤ (1+y t) i i i=1 i=1 (cid:0)Y (cid:1) (cid:0)Y (cid:1) for some fixed integer M ≥ 2. So, we could consider the coefficients E (x) k of tk in the expansion of n (1+x t) M, instead of the usual elemenetary i=1 i (cid:0)Q (cid:1) symmetric polynomials E (x). The weaker hypothesis E (x) ≤ E (y) ∀k k k k would clearly suffice in the r = 1 case of Theorem 1. (eii) More geenerally ∞ consider any finite or infinite product Q(s) := (1 + c s) with c = j=0 j 0 Q 1,c ≥ 0 and c → 0 sufficiently fast to guarantee that Q(s) converges and j j logQ(s) = O(sǫ)foranyǫ > 0ass → ∞. Thisiscanbeseentobeequivalent to the simple requirement that the sequence c = {c } belong to lǫ for every j ǫ > 0 [11, Ch. 1, §5]. (Forexample, c = qj with 0 < q < 1.) Then let E (x) j k,c be the coefficient of tk in n Q(x t). The hypotheses E (x) ≤ E (y) for i=1 i k,c k,c Q all k would again suffice in the r = 1 case of Theorem 1. For the general r ≥ 1 in Theorem 1, similar modifications of the F (x) can be constructed k,r ∞ by considering products of the form Q (s) := P (c s) in place of the r j=0 r j Q latter Q(s). (c.3). We note that the discussion in remark (c.2) is equivalent to considering the finite or infinite “catalyst” c = {c } and comparing various j properties of the two vectors x ⊗ c = {x c } and y ⊗ c = {y c }. (See the i j i j 10